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I would like to collect a list of mathematical concepts that have been named after mathematicians, which are now used in lowercase form (such as "abelian"). This question is partly motivated by my former supervisor, who joked (something like):

You know you've made it as a mathematician when they start using your name in lowercase.

This is a community wiki, please:

  1. One mathematician per answer. Please update already-existing answers if that mathematician already exists.
  2. Please give a brief description of the concept and the person (ideally with links to wikipedia).
  3. This is not about whether or not you agree with the capitalisation.

Expect your answer to be edited otherwise.

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    $\begingroup$ i.e., mathematical eponyms people no longer bother to capitalize $\endgroup$ – J. M. isn't a mathematician Sep 13 '10 at 13:21
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    $\begingroup$ Pah! such usages are never acceptable! :-) $\endgroup$ – Robin Chapman Sep 13 '10 at 14:10
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    $\begingroup$ This might be only a language issue, but in italian there's a grammar rule that says that even if the adjective is taken from a person's name it must not be capitalized, since it's not a proper name anymore... :) thought it might be interesting $\endgroup$ – Andy Sep 13 '10 at 14:41
  • $\begingroup$ I once thought of asking this exact question... Great minds think alike $\endgroup$ – Seamus Sep 13 '10 at 16:20
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    $\begingroup$ @RobinChapman: Not only should one spell Abelian with a capital, one should also try to respect the Norwegian pronunciation (which includes but is not limited to not stressing the "e"). $\endgroup$ – Marc van Leeuwen Sep 17 '14 at 15:04

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Like "cartesian", "algorithm" is not only a lowercased but a derivative form of the name of a mathematician (al-Khwarismi).

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    $\begingroup$ Which letter of "algorithm" should be capitalized? :-) $\endgroup$ – Robin Chapman Sep 13 '10 at 16:29
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    $\begingroup$ For the uninitiated: the "the" (al-Mukkah, al-jabr) is but a mere modifier in Arabic; so Robin's question is valid. $\endgroup$ – J. M. isn't a mathematician Sep 13 '10 at 21:22
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For something even more common, "boolean" (e.g. "boolean type", "boolean algebra") is increasingly not being capitalized as well. George Boole formalized the manipulation of logical statements, and thus his name crops up frequently in connection to logic.

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cartesian. Many things are described as cartesian, after René Descartes. In philosophy things attributed to Descartes tend to keep the capital when described as "Cartesian", but in mathematics, I've seen "cartesian product" lower case a few times.

What's weird is that this isn't more prevalent, given that Descartes is older than Abel by almost 200 years, and I conjecture that cartesian products, cartesian planes and the like come up at least as much as abelian groups... And since the adjectival form of "Descartes" isn't just the name+ending, it's strange how common the capitalised form has remained...

Google books reveals that lowercase "cartesian plane" was used in the following books (for example):

  • Algebra Through Practice: Volume 2, Matrices and Vector Spaces, By T. S. Blyth, E. F. Robertson
  • Cracking the MCAT with CD-ROM, By James L. Flowers
  • Symmetry And Spectroscopy Of Molecules, By K Veera Reddy
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Many things have been named gaussian after Carl Friedrick Gauss: gaussian distribution, gaussian integers, etc.

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I've seen "jacobian" in lower case in a few books.

For the uninitiated: it refers to either a matrix of partial derivatives, or its determinant. This is named after Carl Gustav Jacob Jacobi.

Sources using lowercase "jacobi":

  • Claus Bendtsen, Preliminary Experiences with Extrapolation Methods for Parallel Solution of Differential, in Parallel scientific computing, 1994.
  • Foundations of Modern Analysis, By J. Dieudonne.
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David Rusin compiled a list of mathematicians whose names appear in the 1991 MSC. (That's the "mathematics subject classification".)

There are 357. He claims Abel's name is the only one among these which appears noncapitalized in the MSC. The other people who have been named in this question (Noether, Descartes, Euclid, Karoubi, Gauss, Boole, Jacobi) all appear in the MSC.

Of course, the 1991 MSC is nearly twenty years old, and is likely to reflect more-formal-than-average usage.

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  • $\begingroup$ Ta, I noticed that MathSciNet seems to almost always capitalise every name except for "abelian". $\endgroup$ – Douglas S. Stones Sep 14 '10 at 0:37
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The wronskian, named after Józef Hoene-Wroński, sometimes appears uncapitalised (though it doesn't on Wikipedia, it does in several of the first hits on Google Scholar).

The wronskian of a set of $n$ functions, as a function of $x$, is defined as the determinant of the matrix whose $(i,j)$'th entry is the $(i-1)$'th derivative of the $j$'th function, evaluated at $x$.

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  • $\begingroup$ On the other hand, I have yet to see "Casoratian" (the corresponding concept for difference equations) uncapitalized. $\endgroup$ – J. M. isn't a mathematician Sep 13 '10 at 21:24
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Here they talk about abelian and tauberian theorems after (Niels-Henrik Abel and Alfred Tauber), however it is also common to use Abelian theorem and Tauberian theorem. Also, in the last paper, by Lennart Carleson, we can read about "this thread" -- that is about people that becomes adjectives.

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Noetherian rings:

http://en.wikipedia.org/wiki/Noetherian_ring

in honor of Emmy Noether:

http://en.wikipedia.org/wiki/Emmy_Noether

Example sources:

  • Lectures on modules and rings, By Tsit-Yuen Lam
  • Commutative algebra, By Oscar Zariski, Pierre Samuel, Irvin Sol Cohen
  • Model theory and modules, By Mike Prest

(plus many others)

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  • $\begingroup$ Don't forget the Artinian ring, while we are here. $\endgroup$ – AD. Sep 13 '10 at 20:29
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I have found the term "karoubianization" in the literature (although google only finds for me instances where it is capitalized...). It is a synonym for "pseudo-abelianization" or "idempotent-completion", and refers to the process whereby one enlarges a category universally so that all its idempotents split. The word refers to Max Karoubi.

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  • $\begingroup$ This is more commonly called the 'Karoubi closure' of the category. $\endgroup$ – Greg Muller Dec 8 '10 at 20:11
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The algebraic structure formed by a set with a commutative binary operation is described as abelian. If G is a group, then the quotient group G/[G,G] (where [G,G] is the commutator subgroup) is the abelianization of G.

The Niels Henrik Abel wikipedia page suggests that he pretty much invented group theory. [[citation needed]]

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Well, both newton and gauss are units which are often written in lower case, e is Euler's number, and (of course) abelian groups is generally lowercased. However, it's very standard to keep upper-case Cartesian, Jacobian, Noetherian, Artinian, Newtonian, Gaussian, Eulerian, and other such constructions, with the occasional exception of boolean, when used as a noun rather than an adjective (as in "this number is a boolean").

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    $\begingroup$ Names of SI units, whether derived from names of people or not, are always written in lowercase as a matter of policy. $\endgroup$ – user856 Jun 7 '12 at 7:10
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One that hasn't been mentioned yet is "banachizable" (see e.g. Horvath, "Topological Vector Spaces and Distributions", or http://www.springerlink.com/content/n2058x72p1h733q0/ It refers to a topological vector space that can be made into a Banach space by giving it a suitable norm.

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Euclid's name has been attached to euclidean space and the euclidean algorithm.

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    $\begingroup$ I don't get it -- the references you cite write "Euclidean space" and "Euclidean algorithm", which is standard. $\endgroup$ – Pete L. Clark Sep 13 '10 at 15:17
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    $\begingroup$ Walter Rudin used "euclidean space" in his Real and Complex Analysis (p. 34). $\endgroup$ – Per Manne Nov 5 '13 at 18:25

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