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Homogeneous function

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Title: Homogeneous function  
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Homogeneous function

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 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).

Contents

  • Examples 1
    • Linear functions 1.1
    • Homogeneous polynomials 1.2
    • Polarization 1.3
    • Rational functions 1.4
  • Non-examples 2
    • Logarithms 2.1
    • Affine functions 2.2
  • Positive homogeneity 3
  • Homogeneous distributions 4
  • Application to differential equations 5
  • See also 6
  • References 7
  • External links 8

Examples

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, x \neq 0.

Linear functions

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

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

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:

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 × ... 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 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 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

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 μt : RnRn is the mapping of scalar multiplication by the real number t.

Application to differential equations

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.

See also

References

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

External links

  • Hazewinkel, Michiel, ed. (2001), "Homogeneous function",  
  • Homogeneous function at PlanetMath.org.
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