Editing Discrete mathematics (section)

===Discrete analogues of continuous mathematics===
There are many concepts and theories in continuous mathematics which have discrete versions, such as [[discrete calculus]], [[discrete Fourier transform]]s, [[discrete geometry]], [[discrete logarithm]]s, [[discrete differential geometry]], [[discrete exterior calculus]], [[discrete Morse theory]], [[discrete optimization]], [[discrete probability theory]], [[discrete probability distribution]], [[difference equation]]s, [[discrete dynamical system]]s, and [[Shapley–Folkman lemma#Probability and measure theory|discrete vector&nbsp;measures]].

==== Calculus of finite differences, discrete analysis, and discrete calculus ====
In [[discrete calculus]] and the [[calculus of finite differences]], a [[function (mathematics)|function]] defined on an interval of the [[integer]]s is usually called a [[sequence]]. A sequence could be a finite sequence from a data source or an infinite sequence from a [[discrete dynamical system]]. Such a discrete function could be defined explicitly by a list (if its domain is finite), or by a formula for its general term, or it could be given implicitly by a [[recurrence relation]] or [[difference equation]]. Difference equations are similar to [[differential equation]]s, but replace [[derivative|differentiation]] by taking the difference between adjacent terms; they can be used to approximate differential equations or (more often) studied in their own right. Many questions and methods concerning differential equations have counterparts for difference equations. For instance, where there are [[integral transforms]] in [[harmonic analysis]] for studying continuous functions or analogue signals, there are [[discrete transform]]s for discrete functions or digital signals. As well as [[discrete metric space]]s, there are more general [[discrete topological space]]s, [[finite metric space]]s, [[finite topological space]]s.

The [[time scale calculus]] is a unification of the theory of [[difference equations]] with that of [[differential equations]], which has applications to fields requiring simultaneous modelling of discrete and continuous data. Another way of modeling such a situation is the notion of [[hybrid system|hybrid dynamical system]]s.

==== Discrete geometry ====
[[Discrete geometry]] and combinatorial geometry are about combinatorial properties of ''discrete collections'' of geometrical objects. A long-standing topic in discrete geometry is [[tessellation|tiling of the plane]].

In [[algebraic geometry]], the concept of a curve can be extended to discrete geometries by taking the [[Spectrum of a ring|spectra]] of [[polynomial ring]]s over [[finite field]]s to be models of the [[affine space]]s over that field, and letting [[Algebraic variety|subvarieties]] or spectra of other rings provide the curves that lie in that space. Although the space in which the curves appear has a finite number of points, the curves are not so much sets of points as analogues of curves in continuous settings. For example, every point of the form <math>V(x-c) \subset \operatorname{Spec} K[x] = \mathbb{A}^1</math> for <math>K</math> a field can be studied either as <math>\operatorname{Spec} K[x]/(x-c) \cong \operatorname{Spec} K</math>, a point, or as the spectrum <math>\operatorname{Spec} K[x]_{(x-c)}</math> of the [[Localization of a ring|local ring at (x-c)]], a point together with a neighborhood around it. Algebraic varieties also have a well-defined notion of [[tangent space]] called the [[Zariski tangent space]], making many features of calculus applicable even in finite settings.

==== Discrete modelling ====

In [[applied mathematics]], [[discrete modelling]] is the discrete analogue of [[continuous modelling]]. In discrete modelling, discrete formulae are fit to [[data]]. A common method in this form of modelling is to use [[recurrence relation]]. [[Discretization]] concerns the process of transferring continuous models and equations into discrete counterparts, often for the purposes of making calculations easier by using approximations. [[Numerical analysis]] provides an important example.