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For four points in order around a circle, Ptolemy's inequality becomes an equality, known as Ptolemy's theorem: ¯ ¯ + ¯ ¯ = ¯ ¯. In the inversion-based proof of Ptolemy's inequality, transforming four co-circular points by an inversion centered at one of them causes the other three to become collinear, so the triangle equality for these three points (from which Ptolemy's inequality may ...
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
On the other hand, the existence of an infinite class field tower can give upper bounds on the values of α(ρ, σ). For example, the infinite class field tower over Q(√-m) with m = 3·5·7·11·19 produces fields of arbitrarily large degree with root discriminant 2 √ m ≈ 296.276, [28] so α(0,1) < 296.276.
Given a topological space X, the path-connected components of X are naturally encoded in its fundamental groupoid; the observation is that p and q are in the same path-connected component of X if and only if the collection of equivalence classes of continuous paths from p to q is nonempty.
There is a later (2017) second printing. Though usually there is little difference between the first and subsequent printings, in this case the second printing not only deletes from page 53 the Exercises 36, 40, 41, and 42 of Chapter 2 but also offers a (slightly, but still substantially) different presentation of part (ii) of Exercise 17.8.
[17] [20] If, as usual in modern mathematics, the axiom of choice is assumed, then f is surjective if and only if there exists a function : such that =, that is, if f has a right inverse. [20] The axiom of choice is needed, because, if f is surjective, one defines g by g ( y ) = x , {\displaystyle g(y)=x,} where x {\displaystyle x} is an ...
Image source: The Motley Fool. Highwoods Properties (NYSE: HIW) Q4 2024 Earnings Call Feb 12, 2025, 11:00 a.m. ET. Contents: Prepared Remarks. Questions and Answers. Call Participants
A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...