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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Pathological (mathematics) | 3/3 | https://en.wikipedia.org/wiki/Pathological_(mathematics) | reference | science, encyclopedia | 2026-05-05T07:24:45.941558+00:00 | kb-cron |
The Cantor set is a subset of the interval
[
0
,
1
]
{\displaystyle [0,1]}
that has measure zero but is uncountable. The fat Cantor set is nowhere dense but has positive measure. The Fabius function is everywhere smooth but nowhere analytic. Volterra's function is differentiable with bounded derivative everywhere, but the derivative is not Riemann-integrable. The Peano space-filling curve is a continuous surjective function that maps the unit interval
[
0
,
1
]
{\displaystyle [0,1]}
onto
[
0
,
1
]
×
[
0
,
1
]
{\displaystyle [0,1]\times [0,1]}
. The Dirichlet function, which is the indicator function for rationals, is a bounded function that is not Riemann integrable. The Cantor function is a monotonic continuous surjective function that maps
[
0
,
1
]
{\displaystyle [0,1]}
onto
[
0
,
1
]
{\displaystyle [0,1]}
, but has zero derivative almost everywhere. The Minkowski question-mark function is continuous and strictly increasing but has zero derivative almost everywhere. Satisfaction classes containing "intuitively false" arithmetical statements can be constructed for countable, recursively saturated models of Peano arithmetic. The Osgood curve is a Jordan curve (unlike most space-filling curves) of positive area. An exotic sphere is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. 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. Over the course of history, they have led to more correct, more precise, and more powerful mathematics. For example, the Dirichlet function is Lebesgue integrable, and convolution with test functions is used to approximate any locally integrable function by smooth functions. Whether a behavior is pathological is by definition subject to personal intuition. Pathologies depend on context, training, and experience, and what is pathological to one researcher may very well be standard behavior to another. Pathological examples can show the importance of the assumptions in a theorem. For example, in statistics, the Cauchy distribution does not satisfy the central limit theorem, even though its symmetric bell-shape appears similar to many distributions which do; it fails the requirement to have a mean and standard deviation which exist and that are finite. Some of the best-known paradoxes, such as Banach–Tarski paradox and Hausdorff paradox, are based on the existence of non-measurable sets. Mathematicians, unless they take the minority position of denying the axiom of choice, are in general resigned to living with such sets.
== Computer science == In computer science, pathological has a slightly different sense with regard to the study of algorithms. Here, an input (or set of inputs) is said to be pathological if it causes atypical behavior from the algorithm, such as a violation of its average case complexity, or even its correctness. For example, hash tables generally have pathological inputs: sets of keys that collide on hash values. Quicksort normally has
O
(
n
log
n
)
{\displaystyle O(n\log {n})}
time complexity, but deteriorates to
O
(
n
2
)
{\displaystyle O(n^{2})}
when it is given input that triggers suboptimal behavior. 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. Also, the term in this sense is a matter of subjective judgment as with its other senses. 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).
== Exceptions ==
A similar but distinct phenomenon is that of exceptional objects (and exceptional isomorphisms), which occurs when there are a "small" number of exceptions to a general pattern (such as a finite set of exceptions to an otherwise infinite rule). By contrast, in cases of pathology, often most or almost all instances of a phenomenon are pathological (e.g., almost all real numbers are irrational). Subjectively, exceptional objects (such as the icosahedron or sporadic simple groups) are generally considered "beautiful", unexpected examples of a theory, while pathological phenomena are often considered "ugly", as the name implies. Accordingly, theories are usually expanded to include exceptional objects. For example, the exceptional Lie algebras are included in the theory of semisimple Lie algebras: the axioms are seen as good, the exceptional objects as unexpected but valid. By contrast, pathological examples are instead taken to point out a shortcoming in the axioms, requiring stronger axioms to rule them out. For example, requiring tameness of an embedding of a sphere in the Schönflies problem. 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.
== See also == Fractal curve List of mathematical jargon Runge's phenomenon Gibbs phenomenon Paradoxical set
== References ==
== Notes ==
== External links == Pathological Structures & Fractals – Extract of an article by Freeman Dyson, "Characterising Irregularity", Science, May 1978 This article incorporates material from pathological on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.