Definitions for homogeneous function
ho·mo·ge·neous func·tion

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1. homogeneous functionnoun

   homogeneous polynomial

2. homogeneous functionnoun

   the ratio of two homogeneous polynomials, such that the sum of the exponents in a term of the numerator is equal to the sum of the exponents in a term of the denominator.

3. homogeneous functionnoun

   a function f(x) which has the property that for any c, .

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1. Homogeneous function

   In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the degree; that is, if k is an integer, a function f of n variables is homogeneous of degree k if f ( s x 1 , … , s x n ) = s k f ( x 1 , … , x n ) {\displaystyle f(sx_{1},\ldots ,sx_{n})=s^{k}f(x_{1},\ldots ,x_{n})} for every x 1 , … , x n , {\displaystyle x_{1},\ldots ,x_{n},} and s ≠ 0. {\displaystyle s\neq 0.} For example, a homogeneous polynomial of degree k defines a homogeneous function of degree k. The above definition extends to functions whose domain and codomain are vector spaces over a field F: a function f : V → W {\displaystyle f:V\to W} between two F-vector spaces is homogeneous of degree k {\displaystyle k} if for all nonzero s ∈ F {\displaystyle s\in F} and v ∈ V . {\displaystyle v\in V.} This definition is often further generalized to functions whose domain is not V, but a cone in V, that is, a subset C of V such that v ∈ C {\displaystyle \mathbf {v} \in C} implies s v ∈ C {\displaystyle s\mathbf {v} \in C} for every nonzero scalar s. 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 s > 0 , {\displaystyle s>0,} and allowing any real number k 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.

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1. 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 : 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 for all nonzero α ∈ F and v ∈ V. 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 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, then an homogeneous function from S to W can still be defined by.

How to pronounce homogeneous function?

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Numerology

1. Chaldean Numerology

   The numerical value of homogeneous function in Chaldean Numerology is: 6

2. Pythagorean Numerology

   The numerical value of homogeneous function in Pythagorean Numerology is: 5

References

1. ^ Wiktionary
   https://en.wiktionary.org/wiki/Homogeneous_Function
2. ^ Wikipedia
   https://en.wikipedia.org/wiki/Homogeneous_Function
3. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=homogeneous function