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The + and invariants keep track of how curves change under these transformations and deformations. The + invariant increases by 2 when a direct self-tangency move creates new self-intersection points (and decreases by 2 when such points are eliminated), while decreases by 2 when an inverse self-tangency move creates new intersections (and increases by 2 when they are eliminated).
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.
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant , under the transformations from ...
[5] The number of fixed points of a dynamical system is invariant under many mathematical operations. Euclidean distance is invariant under orthogonal transformations. Area is invariant under linear maps which have determinant ±1 (see Equiareal map § Linear transformations).
Jordan normal form of a matrix is a complete invariant for matrices up to conjugation, but eigenvalues (with multiplicities) are not. Realizability of invariants [ edit ]
If Γ acts by isometries on a (separable) Banach space E, leaving a weakly closed convex subset C of the closed unit ball of E* invariant, then Γ has a fixed point in C. There is a left invariant norm-continuous functional μ on ℓ ∞ (Γ) with μ(1) = 1 (this requires the axiom of choice). There is a left invariant state μ on any left ...
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