homogeneitywhere

Homogeneitywhere is a generalized mathematical concept that extends the traditional idea of homogeneity by specifying a domain condition through a where clause. It describes when a function preserves a scaling relation only on a subset of its inputs, determined by a predicate W(x) that must be satisfied for the relation to hold.

Definition. Let f: R^n -> R be a function and let W be a predicate on R^n. The function f is homogeneous of degree r on the region where x satisfies W, if for every α > 0 and every x with W(x) true and W(αx) true, we have f(αx) = α^r f(x). The region defined by W is the domain where the homogeneityproperty is required to hold. If W is true for all x, homogeneitywhere reduces to ordinary homogeneity.

Relation to standard concepts. Homogeneitywhere generalizes ordinary homogeneity by allowing the scaling property to apply only to inputs that lie in a restricted region. When the region is scale-invariant (i.e., W(αx) holds whenever W(x) holds), the property is well-behaved under repeated scaling. If W is removed or becomes universal, the concept coincides with standard homogeneous functions. The approach is related to quasi-homogeneity and piecewise homogeneous models, but it emphasizes domain-specific applicability of the scaling law.

Examples. f(x) = ||x||^r on R^n is homogeneous of degree r on the full space, but a restricted example is f(x1, x2) = x1^2/x2 defined on the region where x2 ≠ 0 and x2 > 0; here, f(αx) = α f(x) for α > 0 whenever αx remains in the region defined by W.

Applications. Homogeneitywhere can be used in economics, physics, and statistics to model scaling laws that hold only under certain conditions or regimes, such as sector-specific production functions or constrained physical systems.

See also. Homogeneous function, Euler’s theorem, piecewise functions, scale invariance, quasi-homogeneity.