1936/02/01 by John von Neumann · 6 citations
Mathematics · Engineering · Computer Science · #Mathematics and Applications #Advanced Numerical Analysis Techniques #Computational Geometry and Mesh Generation
paper · doi:10.1073/pnas.22.2.101
openalex publication_date 1936/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
Construction of the Examples. 1.In the preceding note a system of geo- metrical axioms was formulated, which is satisfied (i) by the system L = Ln of all linear subspaces of any n-1-dimensional projective geometry Pn-,, n = 1, 2, ..(ii) by certain further systems L = L W For each system L satisfying these axioms a unique numerical dimension function D(a) (cf.Definition (12) in sec.8, loc.cit.) exists, its range beingThe object of the present note is to give effective examples of the new cases L = LX, and to discuss some of their properties.This note, too, will merely give results and outlines of proofs, the details being reserved for the subsequent publication mentioned in the preceding note.2. Let ,3 be a not-necessarily-commutative but associative division- algebra, and n = 1, 2,. By a left-ratio we mean n elements of 3, Ip .* X , nX not ti = ... = n = 0, combined to a symbol [L,: ... .: tn] with the understanding that [ti: ... : ] = [11 * . .7n] if and only if an element r * 0 of ,3 with -Ti = 71, .*, r = 77, exists. (This notion of equality is clearly reflexive, symmetric and transitive.)The left-ratios [%: . .: i are the elements of an n -1-dimensional projective space P -I = Pn- [8 ] .Any (finite) number of equations tlCtil + * * * + nCtfin = S i = 1) ... .my the aij, i = 1, .., m, j = 1, . . . n, being fixed elements of &3 describes a linear subspace a = a (aij; i = 1, . . rm, j = 1, . . . n) of Pn -1.The minimum m = 0, 1, 2, ... with which a given a can be characterized is its rank r(a).