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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.
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).
The dimension and homology groups of a topological object are invariant under homeomorphism. [5] The number of fixed points of a dynamical system is invariant under many mathematical operations. Euclidean distance is invariant under orthogonal transformations.
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 .
Any eigenvector for T spans a 1-dimensional invariant subspace, and vice-versa. In particular, a nonzero invariant vector (i.e. a fixed point of T) spans an invariant subspace of dimension 1. As a consequence of the fundamental theorem of algebra, every linear operator on a nonzero finite-dimensional complex vector space has an eigenvector ...
If μ is greater than 1 the system has two fixed points, one at 0, and the other at μ/(μ + 1). Both fixed points are unstable, i.e. a value of x close to either fixed point will move away from it, rather than towards it. For example, when μ is 1.5 there is a fixed point at x = 0.6 (since 1.5(1 − 0.6) = 0.6) but starting at x = 0.61 we get
The point z cannot lie in D, because, by passing to the limit, z would have to be a fixed point. The result for the case of fixed points implies that the maps f k {\displaystyle f_{k}} leave invariant all Euclidean disks whose hyperbolic center is located at z k {\displaystyle z_{k}} .
An invariant point is defined as a representation of an invariant system (0 degrees of freedom by Gibbs' phase rule) by a point on a phase diagram. A univariant line thus represents a univariant system with 1 degree of freedom. Two univariant lines can then define a divariant area with 2 degrees of freedom.
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