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Spherical two-distance sets
A set S of unit vectors in n-dimensional Euclidean space is called spherical
two-distance set, if there are two numbers a and b, and inner products of
distinct vectors of S are either a or b. The largest cardinality g(n) of
spherical two-distance sets is not exceed n(n+3)/2. This upper bound is known
to be tight for n=2,6,22. The set of mid-points of the edges of a regular
simplex gives the lower bound L(n)=n(n+1)/2 for g(n.
In this paper using the so-called polynomial method it is proved that for
nonnegative a+b the largest cardinality of S is not greater than L(n). For the
case a+b<0 we propose upper bounds on |S| which are based on Delsarte's method.
Using this we show that g(n)=L(n) for 6<n<22, 23<n<40, and g(23)=276 or 277.Comment: 9 pages, (v2) several small changes and corrections suggested by
referees, accepted in Journal of Combinatorial Theory, Series
Sets and C^n; Quivers and A-D-E; Triality; Generalized Supersymmetry; and D4-D5-E6
The relation between Geisteswissenschaft and Naturwissenschaft has been
discussed by Munster in hep-th/9305104. The plan of this paper is to begin with
the empty set; use it to form sets and quivers (sets of points plus sets of
arrows between pairs of points); and then use them to make complex vector
spaces and to get the A-D-E Coxeter-Dynkin diagrams. The Dn Spin(2n) Lie
algebras have spinor representations to describe fermions. D4 Spin(8) triality
gives automorphisms among its vector and two half-spinor representations. D5
Spin(10) contains both Spin(8) and the complexification of the vector
representation of Spin(8). E6 contains both Spin(10) and the two half-spinor
representations of Spin(10), and therefore contains the adjoint representation
of Spin(8) and the complexifications of the vector and the two half-spinor
representations of Spin(8). E6 is the basis for construction of a fundamental
model of physics that is consistent with experiment (see hep-th/9302030,
hep-ph/9301210).Comment: 1+22 pages, THEP-93-5, LaTe
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