Start studying Urinalysis. Learn vocabulary, terms, and more with flashcards, games, and other study tools. 1.0-1.8 liters. Non-pathological conditions for. Jan 27, 2020 We suggest that the pathological changes that we report in splenic vasculature in PH are the result of a raised portal pressure and HC, similar to the effect of raised pulmonary artery pressure on pulmonary vasculature. Splenic artery in the presence of PH also shows pathological abnormalities.

The Weierstrass function is continuous everywhere but differentiable nowhere.

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In mathematics, a pathological object is one which possesses deviant, irregular or counterintuitive property, in such a way that distinguishes it from what is conceived as a typical object in the same category. The opposite of pathological is well-behaved.^[1]^[2]^[3]

In analysis[edit]

A classic example of a pathological structure is the Weierstrass function, which is continuous everywhere but differentiable nowhere.^[2] 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 generically nowhere differentiable.^[4]

In layman's terms, the majority of functions are nowhere differentiable, and relatively few can ever be described or studied. In general, most useful functions also have some sort of physical basis or practical application, which means that they cannot be pathological at the level of hard mathematics or logic; aside from certain limiting cases like the delta distribution, they tend to be quite well-behaved and intuitive. To quote Henri Poincaré:

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.

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.

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 teratologic museum.

— Henri Poincaré, 1899^[vague]

This highlights the fact that the term pathological (and correspondingly, the word well-behaved) is subjective, context-dependent, and subject to wearing off.^[1] Its meaning in any particular case resides in the community of mathematicians, and not necessarily within mathematics itself. Also, the quotation shows how mathematics often progresses via counter-examples to what is thought intuitive or expected. For instance, the 'lack of derivatives' mentioned is intimately connected with current study of magnetic reconnection events in solar plasma.^{[citation needed]}

In topology[edit]

One of the most notorious pathologies in topology is the Alexander horned sphere, a counterexample showing that topologically embedding the sphere S² in R³ may fail to separate the space cleanly. As a counter-example, it motivated the extra condition of tameness, which suppresses the kind of wild behavior the horned sphere exhibits.^[5]

Like many other pathologies, the horned sphere in a sense plays on infinitely fine, recursively generated structure, which in the limit violates ordinary intuition. In this case, the topology of an ever-descending chain of interlocking loops of continuous pieces of the sphere in the limit fully reflects that of the common sphere, and one would expect the outside of it, after an embedding, to work the same. Yet it does not: it fails to be simply connected.

For the underlying theory, see Jordan–Schönflies theorem.

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Well-behaved[edit]

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,^[6] 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. This has the benefit of making analysis easier, but produces a loss of generality of any conclusions reached. For example, non-Euclidean geometry were once considered to be ill-behaved, but have since become common objects of study from 19th century and onwards.^[7]

In both pure and applied mathematics (e.g., optimization, numerical integration, mathematical physics), well-behaved also means not violating any assumptions needed to successfully apply whatever analysis is being discussed.^[6]

The opposite case is usually labeled pathological. It is not unusual to have situations in which most cases (in terms of cardinality or measure) are pathological, but the pathological cases will not arise in practice—unless constructed deliberately.

The term 'well-behaved' is generally applied in an absolute sense—either something is well-behaved or it is not. For example:

In algorithmic inference, a well-behaved statistic is monotonic, well-defined, and sufficient.
In Bézout's theorem, two polynomials are well-behaved, and thus the formula given by the theorem for the number of their intersections is valid, if their polynomial greatest common divisor is a constant.
A meromorphic function is a ratio of two well-behaved functions, in the sense of those two functions being holomorphic.
The Karush–Kuhn–Tucker conditions are first-order necessary conditions for a solution in a well-behaved nonlinear programming problem to be optimal; a problem is referred to as well-behaved if some regularity conditions are satisfied.
In probability, events contained in the probability space's corresponding sigma-algebra are well-behaved, as are measurable functions.

Unusually, the term could also be applied in a comparative sense:

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In calculus:
- Analytic functions are better-behaved than general smooth functions.
- Smooth functions are better-behaved than general differentiable functions.
- Continuous differentiable functions are better-behaved than general continuous functions. The larger the number of times the function can be differentiated, the mo