Extremaler

Extremaler is a term used in mathematics and related disciplines to denote objects that attain an extreme value of a given quantity under specified constraints. In many contexts, the standard noun form is extremal; “extremaler” may appear in translations or as a label for a class of optimal configurations.

In mathematics, extremaler occur in optimization and the calculus of variations. An extremal function is one that makes a functional stationary, often satisfying the Euler–Lagrange equation. Extremal problems ask how large or small a quantity can be under constraints, with solutions characterized by optimality conditions such as Lagrange multipliers.

In geometry and analysis, the concept extends to extremal length, a conformal invariant that measures the width of curve families; extremal metrics aim to optimize energy or length.

In combinatorics and graph theory, extremal graphs are those that maximize or minimize a property for a given number of vertices, subject to forbidden subgraphs. Turán’s theorem is a classic example, determining the maximum number of edges in an n-vertex graph without a complete subgraph of a fixed size.

In physics, extremal states refer to configurations with minimal energy or maximal action given constraints, often arising in variational formulations and thermodynamics.

Extremaler is thus a flexible label for optimal or limiting objects across disciplines, with common tools including calculus of variations, convex analysis, and extremal combinatorics.