Homogeneous Function

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homogeneous function

[‚hä·mə′jē·nē·əs ′fəŋk·shən]
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.

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A homogeneous function of degree one is called linearly homogeneous.
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
Associated to homogeneous functions are power-log functions, which arise when taking the derivative with respect to the degree of homogeneity z.
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.
Given that all of the Heckscher-Ohlin-Samuelson-Jones pre-requisites are satisfied in this problem, that is, that both countries use the same linear homogeneous functions for abatement and sequestration as well as for production, that the level of pollution is global, and that pollution damage is identically multiplicative, the factor price equalization theorem must in fact hold.

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