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Folds can be regarded as consistently replacing the structural components of a data structure with functions and values. Lists, for example, are built up in many functional languages from two primitives: any list is either an empty list, commonly called nil ([]), or is constructed by prefixing an element in front of another list, creating what is called a cons node ( Cons(X1,Cons(X2,Cons ...
Map functions can be and often are defined in terms of a fold such as foldr, which means one can do a map-fold fusion: foldr f z . map g is equivalent to foldr (f . g) z. The implementation of map above on singly linked lists is not tail-recursive, so it may build up a lot of frames on the stack when called with a large list. Many languages ...
Two- and three-dimensional Poincaré plots show the stretching-and-folding structure of the logistic map. This stretching-and-folding does not just produce a gradual divergence of the sequences of iterates, but an exponential divergence (see Lyapunov exponents), evidenced also by the complexity and unpredictability of the chaotic logistic map ...
The map folding and stamp folding problems are related to a problem in the mathematics of origami of whether a square with a crease pattern can be folded to a flat figure. If a folding direction (either a mountain fold or a valley fold ) is assigned to each crease of a strip of stamps, it is possible to test whether the result can be folded ...
Each node in the map space is associated with a "weight" vector, which is the position of the node in the input space. While nodes in the map space stay fixed, training consists in moving weight vectors toward the input data (reducing a distance metric such as Euclidean distance) without spoiling the topology induced from the map space. After ...
For example, the Miura map fold is a rigid fold that has been used to deploy large solar panel arrays for space satellites. The napkin folding problem is the problem of whether a square or rectangle of paper can be folded so the perimeter of the flat figure is greater than that of the original square.
Polls showed Americans trusted the former Republican president more on the economy and immigration than Harris. In the final months of the presidential campaign, Trump did it his way: diverging ...
Given a dataset, "Map" consists of mapping arbitrary data to elements of a specific monoid. "Reduce" consists of folding those elements, so that in the end we produce just one element. For example, if we have a multiset, in a program it is represented as a map from elements to their numbers. Elements are called keys in this case.