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The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. [3] Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important ...
Springer, T. A. (1977), Invariant Theory, New York: Springer, ISBN 0-387-08242-5 An older but still useful survey. Sturmfels, Bernd (1993), Algorithms in Invariant Theory, New York: Springer, ISBN 0-387-82445-6 A beautiful introduction to the theory of invariants of finite groups and techniques for computing them using Gröbner bases.
In mathematics, particularly in topology and knot theory, Arnold invariants are invariants introduced by Vladimir Arnold in 1994 [1] for studying the topology and geometry of plane curves. The three main invariants— J + {\displaystyle J^{+}} , J − {\displaystyle J^{-}} , and S t {\displaystyle St} —provide ways to classify and understand ...
In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor are the coefficients of the ...
In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation . For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping , and a difference of slopes is invariant under shear mapping .
In the classification of two-dimensional closed manifolds, Euler characteristic (or genus) and orientability are a complete set of invariants. Jordan normal form of a matrix is a complete invariant for matrices up to conjugation, but eigenvalues (with multiplicities) are not.
In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation. Specifically, for functions, a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an invariant set.
1. An invariant of the projective general linear group. 2. An invariant of a central extension of a group. protomorph A set of protomorphs is a set of seminvariants, such that any seminvariant is a polynomial in the protomorphs and the inverse of the first protomorph. (Elliott 1895, p.206)