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mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor. For example, a homogeneous real-valued 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 α. The constant ''k'' is called the degree of homogeneity. More generally, if is a function between two vector spaces over a field ''F'', and ''k'' is an
integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, and −2048 are integers, while 9 ...
, then ''ƒ'' is said to be homogeneous of degree ''k'' if for all nonzero and . When the vector spaces involved are over the
real numbers Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...
, a slightly less general form of homogeneity is often used, requiring only that () 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 Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solv ...
. 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 ().


Examples


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 \left(x^2 + y^2\right) = t^2 f(x, y). For example, suppose ''x'' = 2, ''y'' = 4 and ''t'' = 5. Then * f(x, y) = 2^2 + 4^2 = 4 + 16 = 20 , and * f(5x, 5y) = 5^2 \left(2^2 + 4^2\right) = 25(20) = 500 .


Linear functions

Any linear map is homogeneous of degree 1 since by the definition of linearity :f(\alpha \mathbf) = \alpha f(\mathbf) for all and . Similarly, any multilinear function is homogeneous of degree ''n'' since by the definition of multilinearity : f(\alpha \mathbf_1, \ldots, \alpha \mathbf_n) = \alpha^n f(\mathbf_1, \ldots, \mathbf_n) for all and , , ..., . It follows that the ''n''-th differential of a function between two Banach spaces ''X'' and ''Y'' is homogeneous of degree ''n''.


Homogeneous polynomials

Monomials in ''n'' variables define homogeneous functions . 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^ x^5 y^2 z^3 = \alpha^ f(x, y, z). \, The degree is the sum of the exponents on the variables; in this example, . 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: :\left(x^k + y^k + z^k\right)^\frac


Min/max

For every set of weights w_1,\dots,w_n, the following functions are homogeneous of degree 1: * \min\left(\frac, \dots, \frac\right) (Leontief utilities) * \max\left(\frac, \dots, \frac\right)


Polarization

A multilinear function from the ''n''-th Cartesian product of ''V'' with itself to the underlying field ''F'' gives rise to a homogeneous function 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 (algebra), characteristic zero, then given a homogeneous polynomial ''ƒ'' of degree ''n'' on ''V'', the polarization of an algebraic form, polarization of ''ƒ'' is a multilinear function on the ''n''-th Cartesian product of ''V''. The polarization is defined by: : g(v_1, v_2, \dots, v_n) = \frac \frac\frac \cdots \fracf(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 over a field, 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 demonstrated with the following examples: f(5x) = \ln 5x = \ln 5 + f(x), f(10x) = \ln 10 + f(x), and f(15x) = \ln 15 + f(x). This is because 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 Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...
, the notion of positive homogeneity often plays a more important role than homogeneity in the above sense. Let (resp. ) be a vector space over a field \mathbb (resp. \mathbb), where \mathbb and \mathbb will usually be (or possibly just contain) the real numbers \mathbb or complex numbers \mathbb. Let be a map.Note in particular that if Y = \mathbb = \mathbb, then every \mathbb-valued function on is also \mathbb-valued. We defineFor a property such as real homogeneity to even be well-defined, the fields \mathbb and \mathbb must both contain the real numbers. We will of course automatically make whatever assumptions on \mathbb and \mathbb are necessary in order for the scalar products below to be well-defined. the following terminology:
  1. Strict positive homogeneity: for all and all real .
  2. Nonnegative homogeneity: for all and all real . * A non-negative real-valued functions with this property can be characterized as being a Minkowski functional. * This property is used in the definition of a sublinear function.
  3. Positive homogeneity: This is usually defined to mean "nonnegative homogeneity" but it is also frequently defined to instead mean "strict positive homogeneity". * This distinction is usuallyNote that sometimes 's codomain is the set of extended real numbers (which allows for ), in which case the multiplication will be undefined whenever . In this case, the conditions "" and "" may not necessarily be used interchangeably. irrelevant because for a function valued in a vector space or field, nonnegative homogeneity is the same as strict positive homogeneity: these notions are identical. See thisAssume that is strictly positively homogeneous and valued in a vector space or a field. Then so subtracting from both sides shows that . Writing , for all we have , which shows that is nonnegative homogeneous. footnote for a proof.
  4. Real homogeneity: for all and all real . * This property is used in the definition of a linear functional.
  5. Homogeneity: for all and all s \isin \mathbb. * It is emphasized that this definition depends on the scalar field \mathbb underlying the domain . * This property is used in the definition of linear functionals and linear maps.
  6. Semilinear map, Conjugate homogeneity: for all and all s \isin \mathbb. * If \mathbb = \mathbb then typically denotes the complex conjugate of . But more generally, could be the image of under some distinguished automorphism of \mathbb. * Along with Additive map, 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 equality with in which case we prefix that definition with the word "absolute" or "absolutely." For example,
  1. Absolute real homogeneity: for all and all real .
  2. Absolute homogeneity: for all and all s \isin \mathbb. * This property is used in the definition of a seminorm and a Norm (mathematics), norm.
If is a fixed real number then the above definitions can be further generalized by replacing the equality with (or with for conditions using the absolute value), in which case we say that the homogeneity is "of degree " (note in particular that all of the above definitions are "of degree "). For instance,
  1. Nonnegative homogeneity of degree : for all and all real .
  2. Real homogeneity of degree : for all and all real .
  3. Absolute real homogeneity of degree : for all and all real .
  4. Absolute homogeneity of degree : for all and all s \isin \mathbb.
A (nonzero) continuous function that is homogeneous of degree on \mathbb^n \backslash \lbrace 0 \rbrace extends continuously to \mathbb^n if and only if .


Generalizations

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


Monoids and monoid actions

A monoid is a pair consisting of a set and an associative operator where there is some element in called an identity element, which we will denote by , such that for all . :Notation: If is a monoid with identity element and if , then we will let , , , and more generally for any positive integers , let be the product of instances of ; that is, . :Notation: It is common practice (e.g. such as in algebra or calculus) to denote the multiplication operation of a monoid by juxtaposition, meaning that we may write rather than . This allows us to not even have to assign a symbol to a monoid's multiplication operation. Moreover, when we use this juxtaposition notation then we will automatically assume that the monoid's identity element is denoted by . Let be a monoid with identity element whose operation is denoted by juxtaposition and let be a set. A monoid action of on is a map , which we will also denote by juxtaposition, such that and for all and all .


Homogeneity

Let be a monoid with identity element , let and be sets, and suppose that on both and there are defined monoid actions of . Let be a non-negative integer and let be a map. Then we say that is homogeneous of degree over if for every and , ::. If in addition there is a function , denoted by , called an ''absolute value'' then we say that is absolutely homogeneous of degree over if for every and , ::. If we say that a function is homogeneous over (resp. absolutely homogeneous over ) then we mean that it is homogeneous of degree over (resp. absolutely homogeneous of degree over ). More generally, note that it is possible for the symbols to be defined for with being something other than an integer (e.g. if is the real numbers and is a non-zero real number then is defined even though is not an integer). In this case, we say that is homogeneous of degree over if the same equality holds: : for every and . The notion of being absolutely homogeneous of degree over is generalized similarly.


Euler's homogeneous function theorem

Continuously differentiable positively homogeneous functions are characterized by the following theorem: This result follows at once by differentiating both sides of the equation with respect to , applying the chain rule, and choosing to be . The converse is proved by integrating. Specifically, let \textstyle g(\alpha) = f(\alpha \mathbf). Since \textstyle \alpha \mathbf \cdot \nabla f(\alpha \mathbf)= k f(\alpha \mathbf), : g'(\alpha) = \mathbf \cdot \nabla f(\alpha \mathbf) = \frac f(\alpha \mathbf) = \frac g(\alpha). Thus, \textstyle g'(\alpha) - \frac g(\alpha) = 0. This implies \textstyle g(\alpha) = g(1) \alpha^k. Therefore, \textstyle f(\alpha \mathbf) = g(\alpha) = \alpha^k g(1) = \alpha^k f(\mathbf): is positively homogeneous of degree . As a consequence, suppose that f : \mathbb^n \rarr \mathbb is Differentiable function, differentiable and homogeneous of degree . Then its first-order partial derivatives \partial f/\partial x_i are homogeneous of degree . The result follows from Euler's theorem by commuting the operator \mathbf\cdot\nabla with the partial derivative. One can specialize the theorem to the case of a function of a single real variable (), in which case the function satisfies the ordinary differential equation : f'(x) - \frac f(x) = 0. This equation may be solved using an integrating factor approach, with solution \textstyle f(x) = c x^k, where .


Homogeneous distributions

A continuous function ƒ on \mathbb^n is homogeneous of degree if and only if :\int_ f(tx)\varphi(x)\, dx = t^k \int_ f(x)\varphi(x)\, dx for all compactly supported test functions \varphi; and nonzero real . Equivalently, making a integration by substitution, change of variable , ƒ is homogeneous of degree if and only if :t^\int_ f(y)\varphi\left(\frac\right)\, dy = t^k \int_ f(y)\varphi(y)\, dy for all ''t'' and all test functions \varphi. The last display makes it possible to define homogeneity of distribution (mathematics), distributions. A distribution is homogeneous of degree if :t^\langle S, \varphi\circ\mu_t\rangle = t^k\langle S,\varphi\rangle for all nonzero real and all test functions \varphi. Here the angle brackets denote the pairing between distributions and test functions, and \mu_t : \mathbb^n \rarr \mathbb^n is the mapping of scalar division by the real number .


Application to differential equations

: The substitution converts the ordinary differential equation : I(x, y)\frac + J(x,y) = 0, where and are homogeneous functions of the same degree, into the Separation of variables, separable differential equation :x \frac=-\frac-v.


See also

* Production function#Homogeneous and homothetic production functions, Production function * Triangle center function * Weierstrass elliptic function


Notes


References

*


External links

* * {{MathWorld, title=Euler's Homogeneous Function Theorem, urlname=EulersHomogeneousFunctionTheorem, author=Eric Weisstein Linear algebra Differential operators Types of functions Leonhard Euler