Discrete Mathematics and Its Applications

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Computational geometry applies algorithms to geometrical problems and representations of geometrical objects, while computer image analysis applies them to representations of images. Information theory also includes continuous topics such as: analog signals, analog coding, analog encryption.

Discrete Mathematics and Its Applications - Goodreads Discrete Mathematics and Its Applications - Goodreads

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. Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials. Difference equations are similar to 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. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions).

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. As well as discrete metric spaces, there are more general discrete topological spaces, finite metric spaces, finite topological spaces.

Discrete Mathematics and Its Applications Sixth Edition 2006 Discrete Mathematics and Its Applications Sixth Edition 2006

Logical formulas are discrete structures, as are proofs, which form finite trees [10] or, more generally, directed acyclic graph structures [11] [12] (with each inference step combining one or more premise branches to give a single conclusion).Indeed, contemporary work in descriptive set theory makes extensive use of traditional continuous mathematics. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems.