Regularity

Regularity is the quality or condition of being regular. In everyday use, it denotes orderly patterns, predictable repetition, or conformity to a standard. The term is employed across disciplines to describe both structural properties of objects and the behavior of systems or processes.

In mathematics, regularity refers to the smoothness or well-behaved nature of functions, shapes, or solutions to equations. A function may be described as continuous, differentiable, or analytic, with higher levels of regularity indicating greater smoothness. Regularity theory, especially in the study of partial differential equations, investigates conditions under which weak or rough solutions gain additional smoothness. Sobolev spaces and Hölder or C^k norms provide formal means to quantify regularity. Elliptic and parabolic regularity theorems link the regularity of data and domains to that of the resulting solutions.

In geometry and topology, regularity describes symmetry and uniformity. A regular polygon has equal sides and angles, and a regular polyhedron has congruent faces and identical vertex figures. These concepts extend to higher dimensions and to various symmetry contexts, where regularity signals a high degree of uniformity.

In statistics and applied sciences, regularity conditions are assumptions required for the validity of estimators and test procedures. Such conditions address aspects like identifiability, moment existence, and independence, ensuring desirable asymptotic or finite-sample properties.

Other uses include ordinary patterns in language, music, and natural phenomena, where regularity denotes recurring or predictable structure. The term thus encompasses a broad notion of standardization, order, and smoothness across disciplines.