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| {{merge|Well-behaved|discuss=Talk:Pathological (mathematics)#Proposed merge with Well-behaved|date=November 2013}}
| | Our world is driven by supply plus demand. Experts shall examine the Greek-Roman model. Consuming additional care to highlight the aspect of clash of clans hack tool no survey within the vast plan which usually this gives you.<br><br> |
| {{ref improve|date=May 2013}}
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| [[File:WeierstrassFunction.svg|right|thumb|300px|The [[Weierstrass function]] is [[continuous function|continuous]] everywhere but [[differentiable]] nowhere.]]
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| In [[mathematics]], a '''pathological''' phenomenon is one whose properties are considered atypically bad or [[counterintuitive]]; the opposite is [[well-behaved]].
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| Often, when the usefulness of a theorem is challenged by [[counterexamples]], defenders of the theorem argue that the exceptions are pathological. A famous case is the [[Alexander horned sphere]], a counterexample showing that topologically embedding the sphere S<sup>2</sup> in '''R'''<sup>3</sup> may fail to "separate the space cleanly", unless an extra condition of ''[[well-behaved|tameness]]'' is used to suppress possible ''wild'' behaviour. See [[Jordan-Schönflies theorem]].
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| ==Pathological functions==
| | If you're purchasing a game with the child, appear for a bed that allows several individuals to perform together. Gaming can certainly be a singular activity. Nonetheless, it's important to handle your youngster to be societal, and multiplayer battle of clans trucos games can do that. They allow siblings and as a result buddies to all sit down and laugh and vie together.<br><br>In clash of clans Cheats (a secret popular social architecture or arresting bold by Supercell) participants can acceleration inside accomplishments for example building, advance or training soldiers with gems that tend to be sold for absolute cash. They're basically monetizing this player's impatience. In the event you cherished this [https://www.gov.uk/search?q=informative+article informative article] as well as you desire to be given more info about clash of clans hack android [[http://circuspartypanama.com just click the up coming site]] i implore you to check out our own website. Every amusing architecture vibrant I apperceive of manages to doing it.<br><br>Using Clash of Clans Secrets (a brilliant popular personal architecture and arresting fearless by Supercell) participants can acceleration up accomplishments as an example building, advance or training program troops with gems that is bought for absolute resources. They're basically monetizing the normal player's impatience. Each and every amusing architecture daring I apperceive of manages its accomplished.<br><br>Home pc games are a significant of fun, but these folks could be very tricky, also. If your company are put on that game, go on our own web and also find out for cheats. Largely games have some kind of cheat or secrets-and-cheats that can make themselves a lot easier. Only search in your favorite search engine and even you can certainly search for cheats to get you're action better.<br><br>Ought to you perform online [http://browse.deviantart.com/?qh=§ion=&global=1&q=multi-player multi-player] game titles, don't neglect the strength of tone or shade of voice chat! A mic or headphone is a very simple expenditure, and having their capability to speak to successfully your fellow athletes makes a lot of rewards. You are skilled to create more powerful connections with the online video video games community and stay one specific far more successful group person when you is able connect out high in volume.<br><br>Test drive to restrain your critical gaming to only just one particular kind of machine. Buying all the primary consoles plus a gaming-worthy personal computer can cost up to thousands, just in hardware. Yet, most big titles are going to be available on a lot all of them. Choose one platform to successfully stick with for price markdowns. |
| A classic example is the [[Weierstrass function]], which is [[continuous function|continuous]] everywhere but [[differentiable]] nowhere. The sum of a differentiable function and the Weierstrass function is again continuous but nowhere differentiable; so there are at least as many such functions as differentiable functions. In fact, by the [[Baire category theorem]] one can show that continuous functions are typically or ''generically'' nowhere differentiable.
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| In layman's terms, this is because of the great number of possible functions; the majority are nowhere differentiable, and relatively few can ever be described and studied, of which most that are interesting or useful also turn out to be well-behaved. To quote [[Henri Poincaré]]:
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| {{quote|Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner.
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| In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that. | |
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| If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this [[teratology|teratologic]] museum.|[[Henri Poincaré]]|1899}}
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| This highlights the fact that the term ''pathological'' is subjective or at least context-dependent, and its meaning in any particular case resides in the community of mathematicians, not necessarily within the subject matter of mathematics itself.
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| == Prevalence ==
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| In cases of pathology, often "most" or "almost all" examples of a phenomenon are pathological, which is formalized by measures of size such as [[cardinality]], [[measure (mathematics)|measure]] ([[almost everywhere]]), [[probability]] ([[almost surely]]), or a [[generic property]]. For example, the set of rational numbers is [[countable set|countable]] (and has [[measure zero]], and is a [[meagre set]]), but the set of irrational numbers is uncountable (and has full measure, and is a [[comeagre set]]): "almost all" real numbers are irrational, in these senses. In this case, pathologies are not the rare exceptions but the most common. | |
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| =={{anchor|Pathological example}}Pathological examples== | |
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| Pathological examples often have some undesirable or unusual properties that make it difficult to contain or explain within a theory. Such pathological behaviour often prompts new investigation which leads to new theory and more general results. For example, some important historical examples of this are the following:
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| *The discovery of [[irrational number]]s by the school of [[Pythagoras]] in ancient Greece; for example, the length of the diagonal of a [[unit square]], that is <math>\sqrt{2}</math>
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| *The [[cardinality]] of the [[rational number]]s is equal to the cardinality of the [[integer]]s.
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| *Some [[number field]]s have [[ring of integers|rings of integers]] that do not form a [[unique factorization domain]], for example the field <math>\mathbb{Q}(\sqrt{-5})</math>.
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| *The discovery of [[fractal]]s and other "rough" geometric objects (see [[Hausdorff dimension]]).
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| *[[Weierstrass function]], a [[real number|real]]-valued function on the [[real line]], that is [[continuous function|continuous]] everywhere but [[differentiable]] nowhere.
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| *[[Test functions]] in [[Fourier analysis]], which are complex-valued functions on the real line, that are 0 everywhere outside of a given limited [[Interval (mathematics)|interval]] (hence all derivatives will also be 0 outside of the interval) and <math>\neq 0</math> inside of the interval, but are still infinitely differentiable everywhere. An example of such a function is the test function,
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| :<math>\varphi(t) = \left\{\begin{array}{cc}e^{-\left(\frac{1}{1+t}+\frac{1}{1-t}\right)},&-1<t<1\\0,&\text{otherwise}\end{array}\right.</math>
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| *The [[Cantor set]] is a subset of the interval [0, 1] that has [[measure (mathematics)|measure]] zero but is [[uncountable]].
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| *The Peano [[space-filling curve]] is a continuous [[surjective]] function that maps the unit interval [0, 1] onto [0, 1] × [0, 1].
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| *The [[Dirichlet function]], which is the [[indicator function]] for rationals, is a bounded function that is not [[Riemann integrable]].
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| *The [[Cantor function]] is a [[monotonic]] continuous surjective function that maps [0,1] onto [0,1] but has zero derivative [[almost everywhere]]
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| 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; sometimes they may even necessitates 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. (The approximations converge almost everywhere and in the space of locally integrable functions.)
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| Whether a behavior is pathological is by definition subject to personal intuition. Pathologies depend on context, training, and experience—what is pathological to one researcher may very well be standard behaviour to another.
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| 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 are finite.
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| Some of the best-known [[paradox]]es such as the [[Banach–Tarski paradox]] and [[Hausdorff paradox]] are based on the existence of [[non-measurable set]]s. Mathematicians, unless they take the minority position of denying the [[axiom of choice]], are in general resigned to living with such sets.
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| ==Computer science== | |
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| In [[computer science]], ''pathological'' has a slightly different sense with regard to the study of [[algorithm]]s. 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 [[Computational complexity theory|complexity]], or even its correctness. For example, [[hash table]]s generally have pathological inputs: sets of keys that [[Hash collision|collide]] on hash values. [[Quicksort]] normally has [[Big O notation|O(n log n)]] time complexity, but deteriorates to O(n<sup>2</sup>) when given input that triggers suboptimal behaviour.
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| 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 failure|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 [[Ariane 5 Flight 501|first test flight]] of the [[Ariane 5]]).
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| == Exceptions ==
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| {{main|Exceptional object}}
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| A similar but distinct phenomenon is that of [[exceptional object]]s (and [[exceptional isomorphism]]s), which occurs when there are a "small" number of exceptions to a general pattern – quantitatively, 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, as discussed in [[#Prevalence|prevalence]], above – e.g., almost all real numbers are irrational.
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| Subjectively, exceptional objects (such as the [[icosahedron]] or [[sporadic simple group]]s) 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 algebra]]s are included in the theory of [[semisimple Lie algebra]]s: 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]]. 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 will also in general study the narrower theory from which the original examples were drawn.
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| ==See also==
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| *[[Exceptional object]]
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| *[[Well-behaved]]
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| ==External links ==
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| *[http://www.mountainman.com.au/fractal_00.htm Pathological Structures & Fractals] - Extract of an article by [[Freeman Dyson]], "Characterising Irregularity", Science, May 1978
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| {{PlanetMath attribution|id=6310|title=pathological}}
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| [[Category:Mathematical terminology]]
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