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)
“Put shortly, these are the two views, then. One, that man is intrinsically good, spoilt by circumstance; and the other that he is intrinsically limited, but disciplined by order and tradition to something fairly decent. To the one party man’s nature is like a well, to the other like a bucket. The view which regards him like a well, a reservoir full of possibilities, I call the romantic; the one which regards him as a very finite and fixed creature, I call the classical.”
—Thomas Ernest Hulme (1883–1917)
“The mastery of one’s phonemes may be compared to the violinist’s mastery of fingering. The violin string lends itself to a continuous gradation of tones, but the musician learns the discrete intervals at which to stop the string in order to play the conventional notes. We sound our phonemes like poor violinists, approximating each time to a fancied norm, and we receive our neighbor’s renderings indulgently, mentally rectifying the more glaring inaccuracies.”
—W.V. Quine (b. 1908)
“Analysis as an instrument of enlightenment and civilization is good, in so far as it shatters absurd convictions, acts as a solvent upon natural prejudices, and undermines authority; good, in other words, in that it sets free, refines, humanizes, makes slaves ripe for freedom. But it is bad, very bad, in so far as it stands in the way of action, cannot shape the vital forces, maims life at its roots. Analysis can be a very unappetizing affair, as much so as death.”
—Thomas Mann (1875–1955)