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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:
In computer science, an associative array, map, symbol table, or dictionary is an abstract data type that stores a collection of (key, value) pairs, such that each possible key appears at most once in the collection. In mathematical terms, an associative array is a function with finite domain. [1] It supports 'lookup', 'remove', and 'insert ...
An associative array stores a set of (key, value) pairs and allows insertion, deletion, and lookup (search), with the constraint of unique keys. In the hash table implementation of associative arrays, an array A {\displaystyle A} of length m {\displaystyle m} is partially filled with n {\displaystyle n} elements, where m ≥ n {\displaystyle m ...
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.
This set of intervals is the Julia set of the map – that is, it is the smallest invariant subset of the real line under this map. If μ is greater than the square root of 2, these intervals merge, and the Julia set is the whole interval from μ − μ 2 /2 to μ/2 (see bifurcation diagram).
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.
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.
Each quadrant contains at most one entry, either a key-value pair (leaf quadrant) or a key-subnode pair. For a key-subnode pair, the key represents the center of the subnode. The key is also the common prefix (bit-representation) of all keys in the subnode and its child subnodes. Each node has at least two entries, otherwise it is merged with ...