Английская Википедия:Homogeneous function

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Шаблон:Short description Шаблон:More footnotes Шаблон:For In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar, then the function's value is multiplied by some power of this scalar; the power is called the degree of homogeneity, or simply the degree. That is, if Шаблон:Mvar is an integer, a function Шаблон:Mvar of Шаблон:Mvar variables is homogeneous of degree Шаблон:Mvar if

<math>f(sx_1,\ldots, sx_n)=s^k f(x_1,\ldots, x_n)</math>

for every <math>x_1, \ldots, x_n,</math> and <math>s\ne 0.</math>

For example, a homogeneous polynomial of degree Шаблон:Mvar defines a homogeneous function of degree Шаблон:Mvar.

The above definition extends to functions whose domain and codomain are vector spaces over a field Шаблон:Mvar: a function <math>f : V \to W</math> between two Шаблон:Mvar-vector spaces is homogeneous of degree <math>k</math> if Шаблон:NumBlk for all nonzero <math>s \in F</math> and <math>v \in V.</math> This definition is often further generalized to functions whose domain is not Шаблон:Mvar, but a cone in Шаблон:Mvar, that is, a subset Шаблон:Mvar of Шаблон:Mvar such that <math>\mathbf{v}\in C</math> implies <math>s \mathbf{v}\in C</math> for every nonzero scalar Шаблон:Mvar.

In the case of functions of several real variables and real vector spaces, a slightly more general form of homogeneity called positive homogeneity is often considered, by requiring only that the above identities hold for <math>s > 0,</math> and allowing any real number Шаблон:Mvar as a degree of homogeneity. Every homogeneous real function is positively homogeneous. The converse is not true, but is locally true in the sense that (for integer degrees) the two kinds of homogeneity cannot be distinguished by considering the behavior of a function near a given point.

A norm over a real vector space is an example of a positively homogeneous function that is not homogeneous. A special case is the absolute value of real numbers. The quotient of two homogeneous polynomials of the same degree gives an example of a homogeneous function of degree zero. This example is fundamental in the definition of projective schemes.

Definitions

The concept of a homogeneous function was originally introduced for functions of several real variables. With the definition of vector spaces at the end of 19th century, the concept has been naturally extended to functions between vector spaces, since a tuple of variable values can be considered as a coordinate vector. It is this more general point of view that is described in this article.

There are two commonly used definitions. The general one works for vector spaces over arbitrary fields, and is restricted to degrees of homogeneity that are integers.

The second one supposes to work over the field of real numbers, or, more generally, over an ordered field. This definition restricts to positive values the scaling factor that occurs in the definition, and is therefore called positive homogeneity, the qualificative positive being often omitted when there is no risk of confusion. Positive homogeneity leads to consider more functions as homogeneous. For example, the absolute value and all norms are positively homogeneous functions that are not homogeneous.

The restriction of the scaling factor to real positive values allows also considering homogeneous functions whose degree of homogeneity is any real number.

General homogeneity

Let Шаблон:Mvar and Шаблон:Mvar be two vector spaces over a field Шаблон:Mvar. A linear cone in Шаблон:Mvar is a subset Шаблон:Mvar of Шаблон:Mvar such that <math>sx\in C</math> for all <math>x\in C</math> and all nonzero <math>s\in F.</math>

A homogeneous function Шаблон:Mvar from Шаблон:Mvar to Шаблон:Mvar is a partial function from Шаблон:Mvar to Шаблон:Mvar that has a linear cone Шаблон:Mvar as its domain, and satisfies

<math>f(sx) = s^kf(x)</math>

for some integer Шаблон:Mvar, every <math>x\in C,</math> and every nonzero <math>s\in F.</math> The integer Шаблон:Mvar is called the degree of homogeneity, or simply the degree of Шаблон:Mvar.

A typical example of a homogeneous function of degree Шаблон:Mvar is the function defined by a homogeneous polynomial of degree Шаблон:Mvar. The rational function defined by the quotient of two homogeneous polynomials is a homogeneous function; its degree is the difference of the degrees of the numerator and the denominator; its cone of definition is the linear cone of the points where the value of denominator is not zero.

Homogeneous functions play a fundamental role in projective geometry since any homogeneous function Шаблон:Mvar from Шаблон:Mvar to Шаблон:Mvar defines a well-defined function between the projectivizations of Шаблон:Mvar and Шаблон:Mvar. The homogeneous rational functions of degree zero (those defined by the quotient of two homogeneous polynomial of the same degree) play an essential role in the Proj construction of projective schemes.

Positive homogeneity

When working over the real numbers, or more generally over an ordered field, it is commonly convenient to consider positive homogeneity, the definition being exactly the same as that in the preceding section, with "nonzero Шаблон:Mvar" replaced by "Шаблон:Math" in the definitions of a linear cone and a homogeneous function.

This change allow considering (positively) homogeneous functions with any real number as their degrees, since exponentiation with a positive real base is well defined.

Even in the case of integer degrees, there are many useful functions that are positively homogeneous without being homogeneous. This is, in particular, the case of the absolute value function and norms, which are all positively homogeneous of degree Шаблон:Math. They are not homogeneous since <math>|-x|=|x|\neq -|x|</math> if <math>x\neq 0.</math> This remains true in the complex case, since the field of the complex numbers <math>\C</math> and every complex vector space can be considered as real vector spaces.

Euler's homogeneous function theorem is a characterization of positively homogeneous differentiable functions, which may be considered as the fundamental theorem on homogeneous functions.

Examples

Файл:HomogeneousDiscontinuousFunction.gif
A homogeneous function is not necessarily continuous, as shown by this example. This is the function <math>f</math> defined by <math>f(x,y) = x</math> if <math>xy > 0</math> and <math>f(x, y) = 0</math> if <math>xy \leq 0.</math> This function is homogeneous of degree 1, that is, <math>f(s x, s y) = s f(x,y)</math> for any real numbers <math>s, x, y.</math> It is discontinuous at <math>y = 0, x \neq 0.</math>

Simple example

The function <math>f(x, y) = x^2 + y^2</math> is homogeneous of degree 2: <math display="block">f(tx, ty) = (tx)^2 + (ty)^2 = t^2 \left(x^2 + y^2\right) = t^2 f(x, y).</math>

Absolute value and norms

The absolute value of a real number is a positively homogeneous function of degree Шаблон:Math, which is not homogeneous, since <math>|sx|=s|x|</math> if <math>s>0,</math> and <math>|sx|=-s|x|</math> if <math>s<0.</math>

The absolute value of a complex number is a positively homogeneous function of degree <math>1</math> over the real numbers (that is, when considering the complex numbers as a vector space over the real numbers). It is not homogeneous, over the real numbers as well as over the complex numbers.

More generally, every norm and seminorm is a positively homogeneous function of degree Шаблон:Math which is not a homogeneous function. As for the absolute value, if the norm or semi-norm is defined on a vector space over the complex numbers, this vector space has to be considered as vector space over the real number for applying the definition of a positively homogeneous function.

Linear functions

Any linear map <math>f : V \to W</math> between vector spaces over a field Шаблон:Mvar is homogeneous of degree 1, by the definition of linearity: <math display="block">f(\alpha \mathbf{v}) = \alpha f(\mathbf{v})</math> for all <math>\alpha \in {F}</math> and <math>v \in V.</math>

Similarly, any multilinear function <math>f : V_1 \times V_2 \times \cdots V_n \to W</math> is homogeneous of degree <math>n,</math> by the definition of multilinearity: <math display="block">f\left(\alpha \mathbf{v}_1, \ldots, \alpha \mathbf{v}_n\right) = \alpha^n f(\mathbf{v}_1, \ldots, \mathbf{v}_n)</math> for all <math>\alpha \in {F}</math> and <math>v_1 \in V_1, v_2 \in V_2, \ldots, v_n \in V_n.</math>

Homogeneous polynomials

Шаблон:Main article Monomials in <math>n</math> variables define homogeneous functions <math>f : \mathbb{F}^n \to \mathbb{F}.</math> For example, <math display="block">f(x, y, z) = x^5 y^2 z^3 \,</math> is homogeneous of degree 10 since <math display="block">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). \,</math> The degree is the sum of the exponents on the variables; in this example, <math>10 = 5 + 2 + 3.</math>

A homogeneous polynomial is a polynomial made up of a sum of monomials of the same degree. For example, <math display="block">x^5 + 2x^3 y^2 + 9xy^4</math> is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions.

Given a homogeneous polynomial of degree <math>k</math> with real coefficients that takes only positive values, one gets a positively homogeneous function of degree <math>k/d</math> by raising it to the power <math>1 / d.</math> So for example, the following function is positively homogeneous of degree 1 but not homogeneous: <math display="block">\left(x^2 + y^2 + z^2\right)^\frac{1}{2}.</math>

Min/max

For every set of weights <math>w_1,\dots,w_n,</math> the following functions are positively homogeneous of degree 1, but not homogeneous:

  • <math>\min\left(\frac{x_1}{w_1}, \dots, \frac{x_n}{w_n}\right)</math> (Leontief utilities)
  • <math>\max\left(\frac{x_1}{w_1}, \dots, \frac{x_n}{w_n}\right)</math>

Rational functions

Rational functions formed as the ratio of two Шаблон:Em polynomials are homogeneous functions in their domain, that is, off of the linear cone formed by the zeros of the denominator. Thus, if <math>f</math> is homogeneous of degree <math>m</math> and <math>g</math> is homogeneous of degree <math>n,</math> then <math>f / g</math> is homogeneous of degree <math>m - n</math> away from the zeros of <math>g.</math>

Non-examples

The homogeneous real functions of a single variable have the form <math>x\mapsto cx^k</math> for some constant Шаблон:Mvar. So, the affine function <math>x\mapsto x+5,</math> the natural logarithm <math>x\mapsto \ln(x),</math> and the exponential function <math>x\mapsto e^x</math> are not homogeneous.

Euler's theorem

Roughly speaking, Euler's homogeneous function theorem asserts that the positively homogeneous functions of a given degree are exactly the solution of a specific partial differential equation. More precisely:

Шаблон:Math theorem

Шаблон:Math proof

As a consequence, if <math>f : \R^n \to \R</math> is continuously differentiable and homogeneous of degree <math>k,</math> its first-order partial derivatives <math>\partial f/\partial x_i</math> are homogeneous of degree <math>k - 1.</math> This results from Euler's theorem by differentiating the partial differential equation with respect to one variable.

In the case of a function of a single real variable (<math>n = 1</math>), the theorem implies that a continuously differentiable and positively homogeneous function of degree Шаблон:Mvar has the form <math>f(x)=c_+ x^k</math> for <math>x>0</math> and <math>f(x)=c_- x^k</math> for <math>x<0.</math> The constants <math>c_+</math> and <math>c_-</math> are not necessarily the same, as it is the case for the absolute value.

Application to differential equations

Шаблон:Main article The substitution <math>v = y / x</math> converts the ordinary differential equation <math display="block">I(x, y)\frac{\mathrm{d}y}{\mathrm{d}x} + J(x,y) = 0,</math> where <math>I</math> and <math>J</math> are homogeneous functions of the same degree, into the separable differential equation <math display="block">x \frac{\mathrm{d}v}{\mathrm{d}x} = - \frac{J(1,v)}{I(1,v)} - v.</math>

Generalizations

Homogeneity under a monoid action

The definitions given above are all specialized cases of the following more general notion of homogeneity in which <math>X</math> can be any set (rather than a vector space) and the real numbers can be replaced by the more general notion of a monoid.

Let <math>M</math> be a monoid with identity element <math>1 \in M,</math> let <math>X</math> and <math>Y</math> be sets, and suppose that on both <math>X</math> and <math>Y</math> there are defined monoid actions of <math>M.</math> Let <math>k</math> be a non-negative integer and let <math>f : X \to Y</math> be a map. Then <math>f</math> is said to be Шаблон:Em if for every <math>x \in X</math> and <math>m \in M,</math> <math display="block">f(mx) = m^k f(x).</math> If in addition there is a function <math>M \to M,</math> denoted by <math>m \mapsto |m|,</math> called an Шаблон:Em then <math>f</math> is said to be Шаблон:Em if for every <math>x \in X</math> and <math>m \in M,</math> <math display="block">f(mx) = |m|^k f(x).</math>

A function is Шаблон:Em (resp. Шаблон:Em) if it is homogeneous of degree <math>1</math> over <math>M</math> (resp. absolutely homogeneous of degree <math>1</math> over <math>M</math>).

More generally, it is possible for the symbols <math>m^k</math> to be defined for <math>m \in M</math> with <math>k</math> being something other than an integer (for example, if <math>M</math> is the real numbers and <math>k</math> is a non-zero real number then <math>m^k</math> is defined even though <math>k</math> is not an integer). If this is the case then <math>f</math> will be called Шаблон:Em if the same equality holds: <math display="block">f(mx) = m^k f(x) \quad \text{ for every } x \in X \text{ and } m \in M.</math>

The notion of being Шаблон:Em is generalized similarly.

Distributions (generalized functions)

Шаблон:Main article A continuous function <math>f</math> on <math>\R^n</math> is homogeneous of degree <math>k</math> if and only if <math display="block">\int_{\R^n} f(tx) \varphi(x)\, dx = t^k \int_{\R^n} f(x)\varphi(x)\, dx</math> for all compactly supported test functions <math>\varphi</math>; and nonzero real <math>t.</math> Equivalently, making a change of variable <math>y = tx,</math> <math>f</math> is homogeneous of degree <math>k</math> if and only if <math display="block">t^{-n}\int_{\R^n} f(y)\varphi\left(\frac{y}{t}\right)\, dy = t^k \int_{\R^n} f(y)\varphi(y)\, dy</math> for all <math>t</math> and all test functions <math>\varphi.</math> The last display makes it possible to define homogeneity of distributions. A distribution <math>S</math> is homogeneous of degree <math>k</math> if <math display="block">t^{-n} \langle S, \varphi \circ \mu_t \rangle = t^k \langle S, \varphi \rangle</math> for all nonzero real <math>t</math> and all test functions <math>\varphi.</math> Here the angle brackets denote the pairing between distributions and test functions, and <math>\mu_t : \R^n \to \R^n</math> is the mapping of scalar division by the real number <math>t.</math>

Glossary of name variants

Шаблон:Or section Let <math>f : X \to Y</math> be a map between two vector spaces over a field <math>\mathbb{F}</math> (usually the real numbers <math>\R</math> or complex numbers <math>\Complex</math>). If <math>S</math> is a set of scalars, such as <math>\Z,</math> <math>[0, \infty),</math> or <math>\Reals</math> for example, then <math>f</math> is said to be Шаблон:Em if <math display=inline>f(s x) = s f(x)</math> for every <math>x \in X</math> and scalar <math>s \in S.</math> For instance, every additive map between vector spaces is Шаблон:Em <math>S := \Q</math> although it [[Cauchy's functional equation|might not be Шаблон:Em]] <math>S := \R.</math>

The following commonly encountered special cases and variations of this definition have their own terminology:

  1. (Шаблон:Em) Шаблон:Em:Шаблон:Sfn <math>f(rx) = r f(x)</math> for all <math>x \in X</math> and all Шаблон:Em real <math>r > 0.</math>
  2. Шаблон:Em: <math>f(rx) = r f(x)</math> for all <math>x \in X</math> and all real <math>r.</math>
  3. Шаблон:Em:Шаблон:Sfn <math>f(sx) = s f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>
  4. Шаблон:Em:Шаблон:Sfn <math>f(sx) = \overline{s} f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>
    • If <math>\mathbb{F} = \Complex</math> then <math>\overline{s}</math> typically denotes the complex conjugate of <math>s</math>. But more generally, as with semilinear maps for example, <math>\overline{s}</math> could be the image of <math>s</math> under some distinguished automorphism of <math>\mathbb{F}.</math>
    • Along with additivity, this property is assumed in the definition of an antilinear map. It is also assumed that one of the two coordinates of a sesquilinear form has this property (such as the inner product of a Hilbert space).

All of the above definitions can be generalized by replacing the condition <math>f(rx) = r f(x)</math> with <math>f(rx) = |r| f(x),</math> in which case that definition is prefixed with the word Шаблон:Nowrap or Шаблон:Nowrap For example,

  1. Шаблон:Em:Шаблон:Sfn <math>f(sx) = |s| f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>
    • This property is used in the definition of a seminorm and a norm.

If <math>k</math> is a fixed real number then the above definitions can be further generalized by replacing the condition <math>f(rx) = r f(x)</math> with <math>f(rx) = r^k f(x)</math> (and similarly, by replacing <math>f(rx) = |r| f(x)</math> with <math>f(rx) = |r|^k f(x)</math> for conditions using the absolute value, etc.), in which case the homogeneity is said to be Шаблон:Nowrap (where in particular, all of the above definitions are Шаблон:Nowrap). For instance,

  1. Шаблон:Em: <math>f(rx) = r^k f(x)</math> for all <math>x \in X</math> and all real <math>r.</math>
  2. Шаблон:Em: <math>f(sx) = s^k f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>
  3. Шаблон:Em: <math>f(rx) = |r|^k f(x)</math> for all <math>x \in X</math> and all real <math>r.</math>
  4. Шаблон:Em: <math>f(sx) = |s|^k f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>

A nonzero continuous function that is homogeneous of degree <math>k</math> on <math>\R^n \backslash \lbrace 0 \rbrace</math> extends continuously to <math>\R^n</math> if and only if <math>k > 0.</math>

See also

Notes

Шаблон:Reflist

Proofs

Шаблон:Reflist

References

Шаблон:Reflist

Sources

Шаблон:Sfn whitelist

External links

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