Calculus of Finite Differences, Discrete Calculus or Discrete Analysis
As the above section on topological spaces makes clear, analysis isn't just about continuity in the traditional sense of real numbers. Analysis is fundamentally about functions, 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). 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 studying continuous functions or analog 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.
Read more about this topic: Mathematical Analysis
Famous quotes containing the words calculus of, calculus, finite, discrete and/or analysis:
“I try to make a rough music, a dance of the mind, a calculus of the emotions, a driving beat of praise out of the pain and mystery that surround me and become me. My poems are meant to make your mind get up and shout.”
—Judith Johnson Sherwin (b. 1936)
“I try to make a rough music, a dance of the mind, a calculus of the emotions, a driving beat of praise out of the pain and mystery that surround me and become me. My poems are meant to make your mind get up and shout.”
—Judith Johnson Sherwin (b. 1936)
“We know then the existence and nature of the finite, because we also are finite and have extension. We know the existence of the infinite and are ignorant of its nature, because it has extension like us, but not limits like us. But we know neither the existence nor the nature of God, because he has neither extension nor limits.”
—Blaise Pascal (1623–1662)
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—W.V. Quine (b. 1908)
“A commodity appears at first sight an extremely obvious, trivial thing. But its analysis brings out that it is a very strange thing, abounding in metaphysical subtleties and theological niceties.”
—Karl Marx (1818–1883)