21:"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.",
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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
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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
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419:, Shujun Li, Chengqing Li, Kwok-Tung Lo, Guanrong Chen, IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 4, PAGES 1055-1063, APRIL 2008
407:, Franc Novak Bojan Hvala, Sandi Klavžar, "Proceedings of the conference on Design, automation and test in Europe", 1999, ISBN:1-58113-121-6
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Number theory isn't concerned solely discrete objects: Transcendental numbers, Diophantine approximation, p-adic analysis, function fields
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parallelisms, spreads, partial spreads, spreadsets, quasifields, collineations, automorphisms, autotopisms, translation planes, nets
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Computer science considers both discrete and continuous computational processes, and both discrete and continuous input/output:
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for studing continuous functions or analog signals, there are discrete transforms for discrete functions or digital signals.
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286:. If a discrete object doesn't have a continuous version it is just because noone has gotten round to continuifying it yet.
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336:. 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
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35:, Proceedings of the Sixth International Conference on Difference Equations, Aulbach, et al, ed., CRC Press, 2004.
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Discrete
Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations
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365:. Solutions to difference and functional equations can also be continuous functions. As well as the
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395:, Brian Chen and Gregory W. Wornell, IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 7, JULY 1998
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What is the most pure mathematics subject ? The queen of mathematics, number theory.
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484:, Wassim M. Haddad, Vijaysekhar Chellaboina, Qing Hui, Princeton University Press, 2010,
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Many concepts in analysis have discrete versions giving rise to discrete analysis. e.g.
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Complexity theory of continuous time computation using dynamical systems or other
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Novel architectures for P2P applications: the continuous-discrete approach
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Calculus of finite differences, discrete calculus or discrete analysis
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Cryptanalyzing an
Encryption Scheme Based on Blind Source Separation
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Discrete and
Continuous Mathematics, Pure and Applied Mathematics
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Analog Error-Correcting Codes Based on
Chaotic Dynamical Systems
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Other topics often categorized as part of discrete mathematics
25:, "One Mathematics", The Berliner Intelligencer, Berlin, 1998
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that the functions themselves are members of. A discrete
29:"Real analysis" is a degenerate case of discrete analysis
344:. A difference equation is the discrete equivalent of a
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What is the most applied ? Mathematical physics.
467:Substitution Dynamical Systems - Spectral Analysis
226:"metric graph theory" OR "geometric graph theory"
482:Nonnegative and Compartmental Dynamical Systems
289:Likewise, almost any continuous object can be
284:infinite proof trees with continuous branching
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244:continuous processes in operations research
69:continuous models of continuous computation
150:"analog Automata" OR "continuous Automata"
106:Information is encoded analogously in the
432:Combinatorics of Spreads and Parallelisms
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75:Complexity theory of numerical analysis
469:, Martine Queffélec, Springer, 2010,
454:, Daisuke Furihata, CRC Press, 2010,
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434:, Norman Johnson, CRC Press, 2010,
369:there are more general discrete or
164:"continuous-time" "process algebra"
357:is a unification of the theory of
266:Almost any discrete object can be
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47:Continuous computability theory:
180:Here are the Google results for
309:Discrete Calculus of Variations
77:(various approaches including
64:Continuous complexity theory:
1:
56:Algorithmic complexity theory
405:On Analog Signature Analysis
79:Information-based complexity
182:"Number theory and physics"
83:algebraic complexity theory
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300:
375:finite topological spaces
334:discrete dynamical system
324:f(n) is usually called a
204:infinitary combinatorics
116:analog signal processing
157:"Continuous Petri nets"
132:analog data compression
128:analog error correction
91:Including P!=NP over R
363:differential equations
199:analytic combinatorics
346:differential equation
242:Operations research:
371:finite metric spaces
359:difference equations
313:discrete mathematics
355:Time scale calculus
342:difference equation
338:recurrence relation
236:continuous lattices
51:Computable analysis
260:"continuous logic"
221:Analysis on graphs
120:analog electronics
351:harmonic analysis
303:finite difference
272:continuous graphs
252:Differential game
216:continuous graphs
136:analog encryption
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280:continuous proof
39:Computer science
33:Doron Zeilberger
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318:function spaces
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100:Hybrid computer
96:Analog computer
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212:Graph theory
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170:Number theory
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124:analog coding
122:. Aspects of
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23:László Lovász
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291:discretized
268:continuized
143:analog vlsi
381:References
328:a(n) (see
234:Lattices:
326:sequence
322:function
278:). Even
126:include
258:Logic:
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311:. See
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112:brains
426:Links
270:e.g.
114:, in
486:ISBN
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436:ISBN
373:and
340:or
110:of
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