theorems.（シュトゥームの定理とは）とは - 英語 - 多言語

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theorems.

Sturm's theorem

(mathematics) A theorem expressing the number of distinct real roots of p located in an interval in terms of the number of changes of signs of the values of the Sturm sequence at the bounds of the interval. Applied to the interval of all the real numbers, it gives the total number of real roots of p.

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Sturm's theorem determines the number of distinct real roots of a polynomial within an interval by counting the sign changes of the Sturm sequence at the interval's endpoints.

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Sperner's theorem

固有名詞

(set theory) A theorem that describes the largest possible families of finite sets none of which contain any other sets in the family.

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In our combinatorics seminar, we used Sperner's theorem to determine the maximum size of a family of subsets of an n-element set in which no set contains another.

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Thue's theorem

固有名詞

(mathematics) One of the earlier theorems leading to Roth's theorem (a Diophantine approximation to algebraic numbers), establishing an exponent d/2+1+ε.

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Thue's theorem was a crucial stepping stone toward Roth's theorem, introducing an exponent d in Diophantine approximation.

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Maharam's theorem

固有名詞

(mathematics) A result about the decomposability of measure spaces, playing an important role in the theory of Banach spaces. In brief, it states that every complete measure space is decomposable into "non-atomic parts" (copies of products of the unit interval [0,1] on the reals), and "purely atomic parts", using the counting measure on some discrete space.

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Maharam's theorem shows that any complete measure space can be decomposed into non-atomic parts (copies of products of the unit interval [0,1]) and purely atomic parts with the counting measure, a fact that plays an important role in the theory of Banach spaces.

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free will theorem

固有名詞

A theorem stating that, if we have free will in the sense that our choices are not a function of the past, then, subject to certain assumptions, so must some elementary particles.

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The free will theorem challenges our intuitions by suggesting that if our choices are not functions of the past, then, under certain assumptions, some elementary particles must be as well.

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Pasch's theorem

固有名詞

(geometry) A result in plane geometry which cannot be derived from Euclid's postulates. It states that, given points a, b, c, and d on a line, if it is known that the points are ordered as (a, b, c) and (b, c, d), then it is also true that (a, b, d).

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Roth's theorem

固有名詞

(mathematics) A fundamental result in Diophantine approximation to algebraic numbers, stating that these numbers cannot have many rational number approximations that are 'very good' (variously defined through history).

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Roth's theorem implies that algebraic numbers cannot have infinitely many exceptionally close rational approximations.

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Hjelmslev's theorem

固有名詞

(geometry) a theorem stating that if a set of points P_i lying on a line is isometrically mapped to points P_i' on a different line, then the midpoints of the segments P_iP_i' also lie on a line.