Homogeneous function

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In mathematics, a homogeneous function is 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

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

 

 

 

 

(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 more 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 an 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 f(x,y)=x if xy>0 or f(x,y)=0 if xy \leq 0. This function is homogeneous of order 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.

Linear functions[edit]

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

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

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

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 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,

f(x,y,z)=x^5y^2z^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^5y^2z^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 + 2 x^3 y^2 + 9 x y^4 \,

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

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 an homogeneous function ƒ : VF 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 an 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

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_1v_1+\cdots+t_nv_n).

These two constructions, one of an homogeneous polynomial from a multilinear form and the other of a multilinear form from an 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 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  \nexists \; k such that f(\alpha \cdot x) = \alpha^k \cdot f(x).

Affine functions[edit]

Affine functions (the function f(x) = x + 5 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

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

for all α > 0. Here k can be any complex 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 ƒy) = αkƒ(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})= k f(\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 : RnR 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.

Homogeneous distributions[edit]

A compactly supported 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 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

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[disambiguation needed]

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

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]