Pathological (mathematics)
More than this, from the point of view of logic, it is these strange functions that are the most general; those that are met without being looked for no longer appear as more than a particular case, and they have only quite a little corner left them.
If you don't do so, the logicians might say, you will only reach exactness by stages.Since Poincaré, nowhere differentiable functions have been shown to appear in basic physical and biological processes such as Brownian motion and in applications such as the Black-Scholes model in finance.
[6] Like many other pathologies, the horned sphere in a sense plays on infinitely fine, recursively generated structure, which in the limit violates ordinary intuition.
[7] Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object—a function, a set, a space of one sort or another—is "well-behaved".
While the term has no fixed formal definition, it generally refers to the quality of satisfying a list of prevailing conditions, which might be dependent on context, mathematical interests, fashion, and taste.
To ensure that an object is "well-behaved", mathematicians introduce further axioms to narrow down the domain of study.
Some important historical examples of this are: At the time of their discovery, each of these was considered highly pathological; today, each has been assimilated into modern mathematical theory.
These examples prompt their observers to correct their beliefs or intuitions, and in some cases necessitate a reassessment of foundational definitions and concepts.
Mathematicians, unless they take the minority position of denying the axiom of choice, are in general resigned to living with such sets.
[citation needed] In computer science, pathological has a slightly different sense with regard to the study of algorithms.
The term is often used pejoratively, as a way of dismissing such inputs as being specially designed to break a routine that is otherwise sound in practice (compare with Byzantine).
On the other hand, awareness of pathological inputs is important, as they can be exploited to mount a denial-of-service attack on a computer system.
Given enough run time, a sufficiently large and diverse user community (or other factors), an input which may be dismissed as pathological could in fact occur (as seen in the first test flight of the Ariane 5).
In general, one may study the more general theory, including the pathologies, which may provide its own simplifications (the real numbers have properties very different from the rationals, and likewise continuous maps have very different properties from smooth ones), but also the narrower theory, from which the original examples were drawn.
