User:Bethnim

Discrete and Continuous Mathematics, Pure and Applied Mathematics

"The most prominent of these runs between Pure and Applied Mathematics. The controversy around Bourbaki focuses on Abstract vs. Concrete. The distinction between Structural Mathematics (whose main results are theorems and proofs) and Algorithmic Mathematics (whose results are algorithms and their analysis) can be traced back to ancient times. There is a deep division (or at least so it appears) between Continuous and Discrete Mathematics.", László Lovász, "One Mathematics", The Berliner Intelligencer, Berlin, 1998

"Real analysis" is a degenerate case of discrete analysis, Doron Zeilberger, Proceedings of the Sixth International Conference on Difference Equations, Aulbach, et al, ed., CRC Press, 2004.

Computer science

Computer science considers both discrete and continuous computational processes, and both discrete and continuous input/output:

• Continuous computability theory:
  • Computable analysis
  • Algorithmic complexity theory

• Continuous complexity theory:
  • Complexity theory of continuous time computation using dynamical systems or other continuous models of continuous computation.
  • Complexity theory of numerical analysis (various approaches including Information-based complexity, algebraic complexity theory)

Including P!=NP over R

• Analog computer, Hybrid computer

• Information is encoded analogously in the neural networks of brains, in analog signal processing, and analog electronics. Aspects of analog coding include analog error correction, analog data compression. analog encryption

• analog vlsi

• "analog Automata" OR "continuous Automata"

• "Continuous Petri nets"

• "continuous-time" "process algebra"

Number theory

What is the most pure mathematics subject ? The queen of mathematics, number theory.

What is the most applied ? Mathematical physics.

Here are the Google results for "Number theory and physics"

Number theory isn't concerned solely discrete objects: Transcendental numbers, Diophantine approximation, p-adic analysis, function fields

Other topics often categorized as part of discrete mathematics

• Combinatorics
  • analytic combinatorics
  • infinitary combinatorics

• Graph theory
  • continuous graphs
  • Analysis on graphs
  • "metric graph theory" OR "geometric graph theory"

• Lattices: continuous lattices

• Operations research: continuous processes in operations research

• Game theory: Differential game

• Logic: "continuous logic" (fuzzy logic)

Almost any discrete object can be continuized e.g. continuous graphs (Novel architectures for P2P applications: the continuous-discrete approach). Even continuous proof, infinite proof trees with continuous branching. If a discrete object doesn't have a continuous version it is just because noone has gotten round to continuifying it yet.

Likewise, almost any continuous object can be discretized.

Calculus of finite differences, discrete calculus or discrete analysis

Main article: finite difference

Many concepts in analysis have discrete versions giving rise to discrete analysis. e.g. Discrete Calculus of Variations. See discrete mathematics for more examples. So analysis shouldn't be contrasted with discrete. Analysis isn't just about limits or continuity in the traditional sense of real numbers, it is a collection of concepts and methods about functions - be they discrete or continuous, and the spaces that the functions act on, and the function spaces that the functions themselves are members of. A discrete function f(n) is usually called a sequence a(n) (see sequence space). A sequence could be a finite sequence from some data source or an infinite sequence from a discrete dynamical system. A discrete function could be defined explicitly by a list, or by a formula for f(n) or it could be given implicitly by a recurrence relation or difference equation. A difference equation is the discrete equivalent of a differential equation and can be used to approximate the latter or studied in its own right. Every question and method about differential equations has a discrete equivalent for difference equations. For instance where there are integral transforms in harmonic analysis for studing continuous functions or analog signals, there are discrete transforms for discrete functions or digital signals. Time scale calculus is a unification of the theory of difference equations with that of differential equations. Solutions to difference and functional equations can also be continuous functions. As well as the discrete metric there are more general discrete or finite metric spaces and finite topological spaces.

Links

• Combinatorics of Spreads and Parallelisms, Norman Johnson, CRC Press, 2010, ISBN 9781439819463
  • parallelisms, spreads, partial spreads, spreadsets, quasifields, collineations, automorphisms, autotopisms, translation planes, nets

• Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations, Daisuke Furihata, CRC Press, 2010, ISBN 9781420094459

• Substitution Dynamical Systems - Spectral Analysis, Martine Queffélec, Springer, 2010, ISBN 9783642112119

• Nonnegative and Compartmental Dynamical Systems, Wassim M. Haddad, Vijaysekhar Chellaboina, Qing Hui, Princeton University Press, 2010, ISBN 9780691144115