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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.
Decomposition, defined as partitioning the edge set of a graph (with as many vertices as necessary accompanying the edges of each part of the partition), has a wide variety of questions. Often, the problem is to decompose a graph into subgraphs isomorphic to a fixed graph; for instance, decomposing a complete graph into Hamiltonian cycles.
A graph formed from a given graph by deleting one vertex, especially in the context of the reconstruction conjecture. See also deck, the multiset of all cards of a graph. carving width Carving width is a notion of graph width analogous to branchwidth, but using hierarchical clusterings of vertices instead of hierarchical clusterings of edges.
See the page on direction-preserving function for definitions. Continuous fixed-point theorems often require a convex set. The analogue of this property for discrete sets is an integrally-convex set. A fixed point of a discrete function f is defined exactly as for continuous functions: it is a point x for which f(x)=x.
A topological space has the fixed-point property if and only if its identity map is universal. A product of spaces with the fixed-point property in general fails to have the fixed-point property even if one of the spaces is the closed real interval. The FPP is a topological invariant, i.e. is preserved by any homeomorphism.
Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle ) is called the reference plane (sometimes fundamental plane ).
For example, the graph of y = x 2 − 4x + 7 can be obtained from the graph of y = x 2 by translating +2 units along the X axis and +3 units along Y axis. This is because the equation can also be written as y − 3 = (x − 2) 2. For many trigonometric functions, the parent function is usually a basic sin(x), cos(x), or tan(x).
The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. [2]By contrast, the Brouwer fixed-point theorem (1911) is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, [3] but it doesn ...