(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.

(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).

(geometry) Pythagorean theorem

(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).

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

(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.

(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.

(game theory) A theorem about finite two-person games of perfect information in which the players move alternately and chance does not affect the decision-making process. It states that if the game cannot end in a draw, then one of the two players must have a winning strategy, i.e. force a win.

Abbreviation of Kochen-Specker theorem.

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.