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  2. Homogeneous function - Wikipedia

    en.wikipedia.org/wiki/Homogeneous_function

    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 ...

  3. Homogeneous polynomial - Wikipedia

    en.wikipedia.org/wiki/Homogeneous_polynomial

    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.

  4. Homogeneous differential equation - Wikipedia

    en.wikipedia.org/wiki/Homogeneous_differential...

    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

  5. Contact geometry - Wikipedia

    en.wikipedia.org/wiki/Contact_geometry

    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

  6. Hilbert series and Hilbert polynomial - Wikipedia

    en.wikipedia.org/wiki/Hilbert_series_and_Hilbert...

    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.

  7. Homogeneous distribution - Wikipedia

    en.wikipedia.org/wiki/Homogeneous_distribution

    In mathematics, a homogeneous distribution is a distribution S on Euclidean space R n or R n \ {0} that is homogeneous in the sense that, roughly speaking, = ()for all t > 0.. More precisely, let : / be the scalar division operator on R n.

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  9. Homothetic preferences - Wikipedia

    en.wikipedia.org/wiki/Homothetic_preferences

    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 >: