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A pair (,) thus provides a unique coupling between and so that can be found when is used as a key and can be found when is used as a key. Mathematically, a bidirectional map can be defined a bijection: between two different sets of keys and of equal cardinality, thus constituting an injective and surjective function:
Graphs of maps, especially those of one variable such as the logistic map, are key to understanding the behavior of the map. One of the uses of graphs is to illustrate fixed points, called points. Draw a line y = x (a 45° line) on the graph of the map. If there is a point where this 45° line intersects with the graph, that point is a fixed point.
remove a (,) pair from the collection, unmapping a given key from its value. The argument to this operation is the key. Lookup, find, or get find the value (if any) that is bound to a given key. The argument to this operation is the key, and the value is returned from the operation.
In mathematics, a chaotic map is a map (an evolution function) that exhibits some sort of chaotic behavior. Maps may be parameterized by a discrete-time or a continuous-time parameter. Discrete maps usually take the form of iterated functions. Chaotic maps often occur in the study of dynamical systems.
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
A pairing is called perfect if the above map is an isomorphism of R-modules and the other evaluation map ′: (,) is an isomorphism also. In nice cases, it suffices that just one of these be an isomorphism, e.g. when R is a field and M,N,L are finite dimensional vector spaces.
The all-pairs shortest path problem, in which we have to find shortest paths between every pair of vertices v, v' in the graph. These generalizations have significantly more efficient algorithms than the simplistic approach of running a single-pair shortest path algorithm on all relevant pairs of vertices.
An RR tachograph is a graph of the numerical value of the RR-interval versus time. In the context of RR tachography, a Poincaré plot is a graph of RR(n) on the x-axis versus RR(n + 1) (the succeeding RR interval) on the y-axis, i.e. one takes a sequence of intervals and plots each interval against the following interval. [3]