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Let be a metric space with distance function .Let be a set of indices and let () be a tuple (indexed collection) of nonempty subsets (the sites) in the space .The Voronoi cell, or Voronoi region, , associated with the site is the set of all points in whose distance to is not greater than their distance to the other sites , where is any index different from .
A word equation is a formal equality:= = between a pair of words and , each over an alphabet comprising both constants (c.f. ) and unknowns (c.f. ). [1] An assignment of constant words to the unknowns of is said to solve if it maps both sides of to identical words.
If each unknown appears at most twice, then a word equation is called quadratic; in a quadratic word equation the graph obtained by repeatedly applying Levi's lemma is finite, so it is decidable if a quadratic word equation has a solution. [2] A more general method for solving word equations is Makanin's algorithm. [3] [4]
It disregards word order (and thus most of syntax or grammar) but captures multiplicity. The bag-of-words model is commonly used in methods of document classification where, for example, the (frequency of) occurrence of each word is used as a feature for training a classifier. [1] It has also been used for computer vision. [2]
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. [ 21 ] [ 22 ] [ 23 ] Differential equations play a prominent role in engineering , physics , economics , biology , and other disciplines.
A difference equation is an equation where the unknown is a function f that occurs in the equation through f(x), f(x−1), ..., f(x−k), for some whole integer k called the order of the equation. If x is restricted to be an integer, a difference equation is the same as a recurrence relation
The Fourier transform is useful in applied mathematics, particularly physics and signal processing. It is another integral operator; it is useful mainly because it converts a function on one (temporal) domain to a function on another (frequency) domain, in a way effectively invertible. No information is lost, as there is an inverse transform ...
In a mathematical programming model, if the objective functions and constraints are represented entirely by linear equations, then the model is regarded as a linear model. If one or more of the objective functions or constraints are represented with a nonlinear equation, then the model is known as a nonlinear model.