EXPANSION OF THE BECKMAN-QUARLES THEOREM FOR RATIONAL SPACES ON MAPPINGS OF Q^d to Q^(d+1)
Keywords:
The Beckman’s Quarles theorem for rational spaces, unit-distance preserving, the clique number of the graph, isometric mappingAbstract
Let Rd and Qd denote the real and the rational d-dimensional space, respectively, equipped with the usual Euclidean metric. For a real number , a mapping where X is either Rd or Qd and is called - distance preserving implies , for all x,y in .
Let G (Qd,a) denote the graph that has Qd as its set of vertices, and where two vertices x and y are connected by edge if and only if . Thus, G (Qd, 1) is the unit distance graph. Let ω (G) denote the clique number of the graph G and let ω (d) denote ω (G (Qd, 1)).
The Beckman-Quarles theorem [1], (1953), states that every unit- distance-preserving mapping from Rd into Rd is an isometry, provided d ≥ 2.
The rational analogues of Beckman- Quarles theorem, means that, for certain dimensions d, every unit- distance preserving mapping from Qd into Qd is an isometry.
A few papers [2, 4, 6, 8, 9, 10, 11 and 12], were written about rational analogues of this theorem, i.e., treating, for some values of the property "Every unit- distance preserving mapping is an isometry". Joseph Zaks, W.Benz, Karin B Chilakamarri, Robert Connelly and others, has found some of these values of d, until the year of 2005, me, Wafiq Hebi [5], in my ph.D thesis, proved that the rational analogues of the Beckman-Quarles theorem is true for all the dimensions ≥5.
The purpose of this paper is to show that in some special dimension d, every unit distance-preserving mapping is an isometry.
In addition, which are the dimension for which ?.
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