Proporionalitii

Proporionalitii is a theoretical construct in multivariate mathematics used to describe generalized proportional relationships between inputs and outputs. In this framework, a vector of inputs X = (x1, ..., xn) is mapped to a vector of outputs Y = (y1, ..., ym) by a map F that is homogeneous of degree one, meaning that for every scalar c > 0, F(c X) = c F(X). This expresses the principle that scaling all inputs by the same factor scales all outputs by the same factor, extending the classical notion of proportionality beyond a single quantity pair.

Formally, proporionalitii refers to any function F: R^n -> R^m that is positively homogeneous of degree one. Linear maps (Y = K X) are a primary example, but nonlinear homogeneous maps are also allowed, such as y1 = sqrt(x1^2 + x2^2) and y2 = x1. In this setting, proportional relationships are governed by the shared homogeneity property rather than fixed ratios alone.

Variants include strict proportionality where each output depends on a single scaled input, piecewise proportionality where the active pattern changes by context, and context-dependent proportionality where external parameters select among several homogeneous maps. A simple example is Y = (2 x1, 3 x2), a linear proportional mapping, while Y = (sqrt(x1^2 + x2^2), x1 + x2) is nonlinear but homogeneous.

Applications appear in scale-invariant modeling, dimensional analysis, and normalization methods in statistics and economics. Proporionalitii is not a standard term in mainstream mathematics; it is more commonly described via the broader theory of homogeneous functions and scale invariance. The concept provides a compact language for discussing how outputs respond to uniform input scaling.