By Alfred North Whitehead
An Unabridged, Unaltered Printing Of quantity I of III: half I - MATHEMATICAL good judgment - the speculation Of Deduction - idea Of obvious Variables - sessions And family - common sense And relatives - items And Sums Of sessions - half II - PROLEGOMENA TO CARDINAL mathematics - Unit periods And - Sub-Classes, Sub-Relations, And Relative kinds - One-Many, Many-One, And One-One family members - choices - Inductive kin
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Example text
We begin by noting that if h: Rn+1 → Rn is an affine map, then h−1 (0) either is empty, or it is an affine subspace of dimension at least 1. Now let σ be an (n+1)-dimensional simplex and h an affine map σ → Rn . We say that h is generic if h−1 (0) intersects no face of σ of dimension smaller than n. In such case, h−1 (0) either is empty, or it is a segment lying in the interior of σ, with endpoints lying in the interior of two (distinct) n-faces of σ: σ h−1 (0) If we represent an affine map h: σ → Rn by the (n+2)-tuple of values at the vertices of σ, all such maps constitute a real vector space of dimension n(n+2).
If all coordinates of y are nonzero, then both x and −x lie in Fn+1 . (LS-c) =⇒ (BU2a) We need an auxiliary result: There exists a covering of S n−1 by closed sets F1 , . . , Fn+1 such that no Fi contains a pair of antipodal points (to see this, we consider an n-simplex in Rn containing 0 in its interior, and we project the facets centrally from 0 on S n−1 ). Then if a continuous antipodal mapping f : S n → S n−1 existed, the sets f −1 (F1 ), . . , f −1 (Fn+1 ) would contradict (LS-c). (LS-c) =⇒ (LS-o) follows from the fact that for every open cover U1 , .
The proof shown above follows Freund and Todd [FT81]. They were aiming at an algorithmic proof. Such algorithms are of great interest and have actually been used for numeric computation of zeros of functions. ∗ (A quantitative metric version of the Borsuk–Ulam theorem; Dubins and Schwarz [DS81]) (a) Let δ(n) = 2(n+1)/n denote the edge length of a regular simplex inscribed in the unit ball B n . Prove that any simplex that contains 0 and has all vertices on S n−1 has an edge of length at least δ(n).
Principia Mathematica by Alfred North Whitehead
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