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**Discrete** differential geometry is the study of **discrete** counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics and topological combinatorics.

A real-valued function is a **discrete** Morse function if it satisfies the following two properties:

Continuous or **discrete** variable - **Discrete** variable

In statistics, the probability distributions of **discrete** variables can be expressed in terms of probability mass functions.

A lattice in a locally compact topological group is a **discrete** subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of R n, this amounts to the usual geometric notion of a lattice, and both the algebraic structure of lattices and the geometry of the totality of all lattices are relatively well understood. Deep results of Borel, Harish-Chandra, Mostow, Tamagawa, M. S. Raghunathan, Margulis, Zimmer obtained from the 1950s through the 1970s provided examples and generalized much of the theory to the setting of nilpotent Lie groups and semisimple algebraic groups over a local field. In the 1990s, Bass and Lubotzky initiated the study of tree lattices, which remains an active research area.

When we refer to cochains as **discrete** (differential) forms, we refer to d as the exterior derivative. We also use the calculus notation for the values of the forms: :

Continuous or **discrete** variable - **Discrete** variable

In **discrete** time dynamics, the variable time is treated as discrete, and the equation of evolution of some variable over time is called a difference equation.

Continuous or **discrete** variable - **Discrete** variable

Methods of calculus do not readily lend themselves to problems involving **discrete** variables. Examples of problems involving **discrete** variables include integer programming.

Continuous or **discrete** variable - **Discrete** variable

In contrast, a **discrete** variable over a particular range of real values is one for which, for any value in the range that the variable is permitted to take on, there is a positive minimum distance to the nearest other permissible value. The number of permitted values is either finite or countably infinite. Common examples are variables that must be integers, non-negative integers, positive integers, or only the integers 0 and 1.

A **discrete** group is a group G equipped with the **discrete** topology. With this topology, G becomes a topological group. A **discrete** subgroup of a topological group G is a subgroup H whose relative topology is the **discrete** one. For example, the integers, Z, form a **discrete** subgroup of the reals, R (with the standard metric topology), but the rational numbers, Q, do not.

Stokes' theorem is a statement about the **discrete** differential forms on manifolds, which generalizes the fundamental theorem of **discrete** calculus for a partition of an interval: :

A function defined on an interval of the integers 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 a differential equations, but replace 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** transforms for **discrete** functions or digital signals. As well as the **discrete** metric there are more general **discrete** or finite metric spaces and finite topological spaces.

The Green's function of the **discrete** Schrödinger operator is given in the resolvent formalism by : where \delta_w is understood to be the Kronecker delta function on the graph: ; that is, it equals 1 if v=w and 0 otherwise.

then g(x) is called a **discrete** cubic spline.

Then H=\Delta+P is the **discrete** Schrödinger operator, an analog of the continuous Schrödinger operator.

It can be shown that the cardinalities in the two conditions cannot both be one simultaneously for a fixed cell \sigma, provided that \mathcal{X} is a regular CW complex. In this case, each cell can be paired with at most one exceptional cell : either a boundary cell with larger \mu value, or a co-boundary cell with smaller \mu value. The cells which have no pairs, i.e., whose function values are strictly higher than their boundary cells and strictly lower than their co-boundary cells are called critical cells. Thus, a **discrete** Morse function partitions the CW complex into three distinct cell collections:, where:

The fundamental result of **discrete** Morse theory establishes that the CW complex \mathcal{X} is isomorphic on the level of homology to a new complex \mathcal{A} consisting of only the critical cells. The paired cells in \mathcal{K} and \mathcal{Q} describe gradient paths between adjacent critical cells which can be used to obtain the boundary operator on \mathcal{A}. Some details of this construction are provided in the next section.

In algebraic geometry, the concept of a curve can be extended to **discrete** geometries by taking the spectra of polynomial rings over finite fields to be models of the affine spaces over that field, and letting 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 for K a field can be studied either as, a point, or as the spectrum of the 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.

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 dynamical systems.

**Discrete** time makes use of difference equations, also known as recurrence relations. An example, known as the logistic map or logistic equation, is

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.