Proportionella

Proportionella is a theoretical construct used in mathematics and related disciplines to describe systems in which a set of quantities scales proportionally with a single parameter. The idea centers on maintaining fixed proportions among components as the overall magnitude changes, so that the structure of the system remains constant under scaling.

Formal definition: Let q = (q1, q2, ..., qn) be a vector of nonnegative quantities. The system is proportionella if there exists a nonnegative vector s = (s1, s2, ..., sn) and a scalar t ≥ 0 such that qi = si t for all i. The vector s is called the proportion profile, and t is the global scale. Equivalently, all pairwise ratios qi/qj are constant across the system, equal to si/sj.

Properties: Proportionella is invariant under uniform scaling of the global parameter t; the ratios qi/qj are fixed and independent of t. The model is homogeneous of degree 1, meaning that doubling t doubles every qi. The concept emphasizes a fixed composition across all components, with the precise scale determined by t.

Applications: The notion helps in resource allocation, manufacturing, and economics where maintaining a stable product mix or input composition across different scales is desirable. In data analysis, proportionella can underlie normalization schemes that preserve relative structure. In computer graphics and modeling, it supports uniform scaling of multi-component objects while preserving proportions.

Relation to other notions: Proportionella generalizes direct proportionality to multi-quantity systems with a fixed composition. It is related to ideas of homogeneity and scaling, and contrasts with models where components vary independently.

See also: proportionality, proportional reasoning, scaling, homogeneity, normalization.