Sinvariants

Sinvariants is a term found in parts of mathematical literature to denote invariants that are constructed from sine-related expressions under a specified group of transformations. The term is not universally standardized and can vary by context, but it roughly refers to quantities built from sine functions that remain unchanged under symmetry operations.

In formal terms, let a group G act on a set X. A sinvariant is a function s: X -> R that can be expressed as a finite combination of sine terms of coordinates or parameters and satisfies s(g·x) = s(x) for all g in G. This emphasis on sine expressions distinguishes sinvariants from more general invariants that may use polynomials, exponentials, or other trigonometric functions.

Construction commonly relies on sine's periodicity and reflection or rotation symmetries. A typical method is to form a finite sum or product of sine terms in which the transformation reindexes or phase-shifts the arguments, causing the noninvariant parts to cancel out.

Examples include a planar figure with rotational symmetry of order n about the origin, where S(θ) = sum_{k=0}^{n-1} sin(θ + 2πk/n) is identically zero, hence invariant. In signal processing, a sinvariant may arise as a phase-invariant feature derived from sine components of a signal.

Applications are mostly theoretical, supporting the study of symmetries in geometry and harmonic analysis, and as a framing device for constructing invariants in problems where sine terms naturally appear.

Related concepts include trigonometric invariants and Fourier invariants, while the broader topic of invariants encompasses polynomial, geometric, and topological invariants.