Homogeneous function

In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if its argument is multiplied by a factor, then its value is multiplied by some power of this factor.

For example, a homogeneous function of two variables x and y is a real-valued function that satisfies the condition $f(\alpha x,\alpha y)=\alpha ^{k}f(x,y)$ for some constant $k$ and all real numbers $\alpha$ . The constant k is called the degree of homogeneity.

More generally, if ƒ : V → W 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

$f(\alpha \mathbf {v} )=\alpha ^{k}f(\mathbf {v} )$

(1)

for all nonzero α ∈ F and v ∈ V. 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

A homogeneous function is not necessarily continuous, as shown by this example. This is the function f defined by

f(x,y)=x

xy>0

f(x,y)=0

xy\leq 0

. This function is homogeneous of degree 1, i.e.

f(\alpha x,\alpha y)=\alpha f(x,y)

for any real numbers

\alpha ,x,y

. It is discontinuous at

y=0,x\neq 0

Example 1

The function $f(x,y)=x^{2}+y^{2}$ is homogeneous of degree 2
$f(tx,ty)=(tx)^{2}+(ty)^{2}=t^{2}(x^{2}+y^{2})=t^{2}f(x,y)$ ^[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

Any linear function ƒ : V → W is homogeneous of degree 1 since by the definition of linearity

f(\alpha \mathbf {v} )=\alpha f(\mathbf {v} )

for all α ∈ F and v ∈ V. Similarly, any multilinear function ƒ : V₁ × V₂ × ... V_n → W is homogeneous of degree n since by the definition of multilinearity

f(\alpha \mathbf {v} _{1},\ldots ,\alpha \mathbf {v} _{n})=\alpha ^{n}f(\mathbf {v} _{1},\ldots ,\mathbf {v} _{n})

for all α ∈ F and v₁ ∈ V₁, v₂ ∈ V₂, ..., v_n ∈ V_n. It follows that the n-th differential of a function ƒ : X → Y between two Banach spaces X and Y is homogeneous of degree n.

Homogeneous polynomials

Main article: Homogeneous polynomial

Monomials in n variables define homogeneous functions ƒ : Fⁿ → F. For example,

f(x,y,z)=x^{5}y^{2}z^{3}\,

is homogeneous of degree 10 since

f(\alpha x,\alpha y,\alpha z)=(\alpha x)^{5}(\alpha y)^{2}(\alpha z)^{3}=\alpha ^{10}x^{5}y^{2}z^{3}=\alpha ^{10}f(x,y,z).\,

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,

x^{5}+2x^{3}y^{2}+9xy^{4}\,

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:

(x^{k}+y^{k}+z^{k})^{1/k}

Min/Max

For every set of weights $w_{1},\dots ,w_{n}$ , the following functions are homogeneous of degree 1:

$\min(x_{1}/w_{1},\dots ,x_{n}/w_{n})$ (Leontief utilities)
$\max(x_{1}/w_{1},\dots ,x_{n}/w_{n})$

Polarization

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

f(v)=g(v,v,\dots ,v).

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 × ... V → F on the n-th Cartesian product of V. The polarization is defined by

g(v_{1},v_{2},\dots ,v_{n})={\frac {1}{n!}}{\frac {\partial }{\partial t_{1}}}{\frac {\partial }{\partial t_{2}}}\cdots {\frac {\partial }{\partial t_{n}}}f(t_{1}v_{1}+\cdots +t_{n}v_{n}).

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

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

Logarithms

The natural logarithm $f(x)=\ln x$ scales additively and so is not homogeneous.

This can be proved by noting that $f(5x)=\ln 5x=\ln 5+f(x)$ , $f(10x)=\ln 10+f(x)$ , and $f(15x)=\ln 15+f(x)$ . Therefore, there is no $k$ such that $f(\alpha \cdot x)=\alpha ^{k}\cdot f(x)$ .

Affine functions

Affine functions (the function $f(x)=x+5$ is an example) do not scale multiplicatively.

Positive homogeneity

In the special case of vector spaces over the real numbers, the notion 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

f(\alpha x)=\alpha ^{k}f(x)\,

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

\mathbf {x} \cdot \nabla f(\mathbf {x} )=kf(\mathbf {x} ).

This result follows at once by differentiating both sides of the equation ƒ(αy) = α^kƒ(y) with respect to α, applying the chain rule, and choosing α to be 1. The converse holds by integrating. Specifically, let $\textstyle g(\alpha )=f(\alpha \mathbf {x} )$ . Since $\textstyle \alpha \mathbf {x} \cdot \nabla f(\alpha \mathbf {x} )=kf(\alpha \mathbf {x} )$ ,

g'(\alpha )=\mathbf {x} \cdot \nabla f(\alpha \mathbf {x} )={\frac {k}{\alpha }}f(\alpha \mathbf {x} )={\frac {k}{\alpha }}g(\alpha ).

Thus, $\textstyle g'(\alpha )-{\frac {k}{\alpha }}g(\alpha )=0$ . This implies $\textstyle g(\alpha )=g(1)\alpha ^{k}$ . Therefore, $\textstyle f(\alpha \mathbf {x} )=g(\alpha )=\alpha ^{k}g(1)=\alpha ^{k}f(\mathbf {x} )$ : ƒ is positive homogeneous of degree k.

As a consequence, suppose that ƒ : Rⁿ → R is differentiable and homogeneous of degree k. Then its first-order partial derivatives $\partial f/\partial x_{i}$ are homogeneous of degree k − 1. The result follows from Euler's theorem by commuting the operator $\mathbf {x} \cdot \nabla$ 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

f'(x)-{\frac {k}{x}}f(x)=0

This equation may be solved using an integrating factor approach, with solution $\textstyle f(x)=cx^{k}$ for some constant real number c.

Homogeneous distributions

Main article: Homogeneous distribution

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

\int _{\mathbb {R} ^{n}}f(tx)\varphi (x)\,dx=t^{k}\int _{\mathbb {R} ^{n}}f(x)\varphi (x)\,dx

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

t^{-n}\int _{\mathbb {R} ^{n}}f(y)\varphi (y/t)\,dy=t^{k}\int _{\mathbb {R} ^{n}}f(y)\varphi (y)\,dy

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

t^{-n}\langle S,\varphi \circ \mu _{t}\rangle =t^{k}\langle S,\varphi \rangle

for all nonzero real t and all test functions $\varphi$ ;. Here the angle brackets denote the pairing between distributions and test functions, and μ_t : Rⁿ → Rⁿ is the mapping of scalar multiplication by the real number t.

Application to differential equations

Main article: Homogeneous differential equation

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

I(x,y){\frac {\mathrm {d} y}{\mathrm {d} x}}+J(x,y)=0,

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

x{\frac {\mathrm {d} v}{\mathrm {d} x}}=-{\frac {J(1,v)}{I(1,v)}}-v.

References

↑ wolfram.com

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

Hazewinkel, Michiel, ed. (2001), "Homogeneous function", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
"Homogeneous function". PlanetMath.

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