“ transdiagnostic approaches,” which look at underlying features of conditions like depression and anxiety to derive a more accurate, and useful, picture, or " biotypes," which look for underlying varieties of mental illness based on computational models rather than clinical experience. 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. For example, hash tables generally have pathological inputs: sets of keys that collide on hash values. 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. Other examples include the Peano space-filling curve which maps the unit interval continuously onto ×, and the Cantor set which is a subset of the interval and has the pathological property that it is uncountable, yet its measure is zero.Ĭomputer science uses this term in a slightly different sense with regard to the study of algorithms. Mathematicians, unless they take the minority position of denying the axiom of choice, are in general resigned to living with such sets. The best-known paradoxes such as the Banach-Tarski paradox and Hausdorff paradox are based on the existence of non-measurable sets. 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. Pathological examples can show the importance of the assumptions in a theorem.
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