rotnoduler

RoTNoduler is a term used to describe a mathematical construct designed to model systems that combine a modular operation with rotation symmetry. Informally, a rotnoduler consists of a set M equipped with a binary operation and an action of the circle group that expresses how elements rotate, while the operation remains compatible with that rotation.

Formally, a rotnoduler is a triple (M, ⋆, α) where M is a set with a binary operation ⋆: M × M → M, making M into an algebraic structure, and α: T × M → M is a group action of the circle group T = {e^{iθ} : θ ∈ [0, 2π)} on M such that α_θ(x ⋆ y) = α_θ(x) ⋆ α_θ(y) for all θ ∈ T and x, y ∈ M. The action α encodes rotation of elements, and the compatibility condition ensures the rotation structure is preserved by the algebraic operation.

Examples include: M = R^2 with standard addition as ⋆ and α_θ(x) = R_θ x, where R_θ is the 2D rotation by angle θ. This yields a rotnoduler that captures continuous rotational symmetry. More generally, any real vector space V with ⋆ as vector addition and α_θ given by a fixed family of linear rotations provides a rotnoduler when the actions form a group homomorphism from T into Aut(V).

Properties often studied in rotnodulers include fixed-point substructures M^T = {x ∈ M : α_θ(x) = x for all θ} and decompositions into rotation-invariant and non-invariant components. Averaging techniques over the group can yield projections onto invariant subspaces.

Applications of rotnoduler concepts appear in areas such as symmetric modeling, crystallography, computer graphics, and dynamical systems, wherever a combination of modular composition and rotation symmetry is relevant. The term is used in some literature as a descriptive label for these rotation-equivariant algebraic structures. See also: group actions, equivariant algebra, rotational symmetry.