Homogeneous function

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In mathematics, a homogeneous function is a function which satisfies the condition , for some integer .

It can also be described as a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if ƒ : VW is a function between two vector spaces over a field F, and k is an integer, then ƒ is said to be homogeneous of degree k if

 

 

 

 

(1)

for all nonzero α ∈ F and vV. This implies it has scale invariance. When the vector spaces involved are over the real numbers, a slightly less general form of homogeneity is often used, requiring only that (1) hold for all α > 0.

Homogeneous functions can also be defined for vector spaces with the origin deleted, a fact that is used in the definition of sheaves on projective space in algebraic geometry. More generally, if S ⊂ V is any subset that is invariant under scalar multiplication by elements of the field (a "cone"), then a homogeneous function from S to W can still be defined by (1).

Examples[edit]

A homogeneous function is not necessarily continuous, as shown by this example. This is the function f defined by if or if . This function is homogeneous of degree 1, i.e. for any real numbers . It is discontinuous at .

Example 1[edit]

The function is homogeneous of degree 2
[1]
suppose x = 2, y = 4 and t = 5

  • f(x,y) = 22 + 42 = 4 + 16 = 20
  • f(5x, 5y) = 52 ( 22 + 42) = 25 x 20 = 500

Linear functions[edit]

Any linear function ƒ : VW is homogeneous of degree 1 since by the definition of linearity

for all α ∈ F and vV. Similarly, any multilinear function ƒ : V1 × V2 × ... VnW is homogeneous of degree n since by the definition of multilinearity

for all α ∈ F and v1V1, v2V2, ..., vnVn. It follows that the n-th differential of a function ƒ : XY between two Banach spaces X and Y is homogeneous of degree n.

Homogeneous polynomials[edit]

Monomials in n variables define homogeneous functions ƒ : FnF. For example,

is homogeneous of degree 10 since

The degree is the sum of the exponents on the variables; in this example, 10=5+2+3.

A homogeneous polynomial is a polynomial made up of a sum of monomials of the same degree. For example,

is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions.

Given a homogeneous polynomial of degree k, it is possible to get a homogeneous function of degree 1 by raising to the power 1/k. So for example, for every k the following function is homogeneous of degree 1:

Min/Max[edit]

For every set of weights , the following functions are homogeneous of degree 1:

  • (Leontief utilities)

Polarization[edit]

A multilinear function g : V × V × ... VF from the n-th Cartesian product of V with itself to the underlying field F gives rise to a homogeneous function ƒ : VF by evaluating on the diagonal:

The resulting function ƒ is a polynomial on the vector space V.

Conversely, if F has characteristic zero, then given a homogeneous polynomial ƒ of degree n on V, the polarization of ƒ is a multilinear function g : V × V × ... VF on the n-th Cartesian product of V. The polarization is defined by

These two constructions, one of a homogeneous polynomial from a multilinear form and the other of a multilinear form from a homogeneous polynomial, are mutually inverse to one another. In finite dimensions, they establish an isomorphism of graded vector spaces from the symmetric algebra of V to the algebra of homogeneous polynomials on V.

Rational functions[edit]

Rational functions formed as the ratio of two homogeneous polynomials are homogeneous functions off of the affine cone cut out by the zero locus of the denominator. Thus, if f is homogeneous of degree m and g is homogeneous of degree n, then f/g is homogeneous of degree m − n away from the zeros of g.

Non-examples[edit]

Logarithms[edit]

The natural logarithm scales additively and so is not homogeneous.

This can be proved by noting that , , and . Therefore, there is no such that .

Affine functions[edit]

Affine functions (the function is an example) do not scale multiplicatively.

Positive homogeneity[edit]

In the special case of vector spaces over the real numbers, the notation of positive homogeneity often plays a more important role than homogeneity in the above sense. A function ƒ : V \ {0} → R is positive homogeneous of degree k if

for all α > 0. Here k can be any real number. A (nonzero) continuous function homogeneous of degree k on Rn \ {0} extends continuously to Rn if and only if Re{k} > 0.

Positive homogeneous functions are characterized by Euler's homogeneous function theorem. Suppose that the function ƒ : Rn \ {0} → R is continuously differentiable. Then ƒ is positive homogeneous of degree k if and only if

This result follows at once by differentiating both sides of the equation ƒy) = αkƒ(y) with respect to α, applying the chain rule, and choosing α to be 1. The converse holds by integrating. Specifically, let . Since ,

Thus, . This implies . Therefore, : ƒ is positive homogeneous of degree k.

As a consequence, suppose that ƒ : RnR is differentiable and homogeneous of degree k. Then its first-order partial derivatives are homogeneous of degree k − 1. The result follows from Euler's theorem by commuting the operator with the partial derivative.

One can specialise the theorem to the case of a function of a single real variable (n = 1), in which case the function satisfies the ordinary differential equation

.

This equation may be solved using an integrating factor approach, with solution for some constant real number c.

Homogeneous distributions[edit]

A continuous function ƒ on Rn is homogeneous of degree k if and only if

for all compactly supported test functions ; and nonzero real t. Equivalently, making a change of variable y = tx, ƒ is homogeneous of degree k if and only if

for all t and all test functions ;. The last display makes it possible to define homogeneity of distributions. A distribution S is homogeneous of degree k if

for all nonzero real t and all test functions ;. Here the angle brackets denote the pairing between distributions and test functions, and μt : RnRn is the mapping of scalar multiplication by the real number t.

Application to differential equations[edit]

The substitution v = y/x converts the ordinary differential equation

where I and J are homogeneous functions of the same degree, into the separable differential equation

See also[edit]

References[edit]

  • Blatter, Christian (1979). "20. Mehrdimensionale Differentialrechnung, Aufgaben, 1.". Analysis II (2nd ed.) (in German). Springer Verlag. p. 188. ISBN 3-540-09484-9. 

External links[edit]