Homogeneous Function


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homogeneous function

[‚hä·mə′jē·nē·əs ′fəŋk·shən]
(mathematics)
A real function ƒ(x1, x2, …, xn ) is homogeneous of degree r if ƒ(ax1, ax2, …, axn ) = a rƒ(x1, x2, …, xn ) for every real number a.

Homogeneous Function

 

a function of one or several variables that satisfies the following condition: when all independent variables of a function are simultaneously multiplied by the same (arbitrary) factor, the value of the function is multiplied by some power of this factor. In algebraic terms, a function f(x, y, …, u) is said to be homogeneous of degree n if for all values of x, y, …, u and for any λ

fx, λy, …, λu) = λnf(x, y, …, u)

For example, the functions x2 – 2y2, (xy – 3z)/(z2 + xy), and Homogeneous Function are homogeneous of degree 2, –1, and 4/3, respectively. Functions homogeneous of degree n are characterized by Euler’s theorem that asserts that if the differential of each independent variable is replaced with the variable itself in the expression for the complete differential

then we obtain the function f(x, y, …, u) multiplied by the degree of homogeneity:

Homogeneous functions are frequently encountered in geometric formulas. In the equation x = f(a, b, …, l), where a, b, …, l are the lengths of segments expressed in terms of the same unit, f must be a homogeneous function (of degree 1, 2, or 3, depending on whether x signifies length, area, or volume). For example, in the formula for the volume of a truncated cone

V is a homogeneous function of degree 3 in h, R, and r.

References in periodicals archive ?
In general, such a linearly homogeneous function F(x) is an FFF only if the true function V(x) is also homogeneous of degree one.
0] is an FFF for a linear homogeneous function satisfying (FLEX0) and (FLEX2) under the constraint (HOMO2) (but condition [FLEX1] need not be satisfied by [F.
Zhen-Hang Yang, defined and studied the monotonicity of Homogeneous function [{[[f([a.
In [50], the authors introduced the homogeneous function with two parameters r and s by,
and F([omega], [xi]) is a homogeneous function of degree -2, because the elements of the matrices [U.
A homogeneous function of degree one is called linearly homogeneous.
That theorem says that any linear homogeneous function can be written as the sum of its first partial derivatives each multiplied by the associated independent variable.
Access through dial-up and frame relay networks meant that companies and financial institutions were really only able to outsource highly recurring, homogeneous functions.
n] are the complete homogeneous functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] indexed by compositions of n.
Since the coefficients of N are homogeneous functions of degree 1, thus
Drawn from papers presented at the their September 2004 gathering in honor of Edgar Earle Enoch, contributors describe here their work on such subjects as Warfield's hom and tensor relations, HFDs and UFDs, a counter example for a question on pseudo-calculation rings, creating co-local subgroups of Abelian groups, associated primes of the local cohomology modules, modules and point set topological spaces, commutative ideal theory, torsionless linearly compact modules, the co-torsion dimension of modules and rings, isotype separable subgroups of homogeneous functions, pure invariance in torsion-free Abelian groups and compressible and related modules.
Following [10] we define the symmetric functions on X by the generating functions of the complete homogeneous functions [S.

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