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I. Circumspheres in Hilbert space, II. Automatic handling of finite-dimensional, nonassociative algebras

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Part I. A nonempty, bounded subset X of a Hilbert space has a unique circumsphere S(X). Its center, c(X), belongs to the closure of the convex hull of X. Its radius r(X) never exceeds d/SQRT.(2) where d is the diameter of X. There is a closed bounded set X for which c(X) does not belong to the convex hull of X, X (INTERSECT) S(X) is empty, and r(X) = d/SQRT.(2);A set X is nonredundant if x lies outside S(X-x) for all x (ELEM) X. The closure of any bounded set in R('n) contains a nonredundant subset Y with S(Y) = S(X). Also. (VBAR)Y(VBAR) (LESSTHEQ) n + 1, and c(Y) is the unique point in the convex hull of Y which is equidistant from each point of Y. This characterization leads to a new algorithm for finding circumspheres in R('n);Part II. A multilinear function defined on a nonassociative algebra X is completely determined by its table, the matrix of a linear transformation from an appropriately defined vector space into X. Operations on tables are defined corresponding to operations on multi-linear functions. This method was used to find identities in the free alternative algebra on three generators. Let A, B, and C be the generators and X the associator (A,B,C). The following identities along with their permuted images have a linear span of dimension 16 which contains all identities in which each generator occurs exactly twice. (1) - ((AB)X)C + ((BA)X)C + (A,B,X)C; (2) - ((XA)B)C + ((XB)A)C + (A,B,X)C; (3) ((AB)C)X - ((BA)C)X - C(A,B,X) - 2XX; (4) - ((AB)C)X + ((AC)B)X + ((CA)B)X - ((CB)A)X - (A,B,X)C + (B,C,X)A; (5) ((AX)B)C - ((AX)C)B - ((CX)A)B + ((CX)B)A + (A,B,X)C - (B,C,X)A; (6) ((CA)B)X - ((CB)A)X - ((CX)A)B + ((CX)B)A; (7) ((CX)A)B - ((CX)B)A + ((XC)A)B - ((XC)B)A - (A,B,X)C - C(A,B,X) - 2XX; (8) ((XB)A)C - ((XB)C)A + ((XC)A)B - ((XC)B)A - B(A,C,X) - C(A,B,X); (9) 3((XA)B)C - 3((XA)C)B - (A,B,X)C + (A,C,X)B - (B,C,X)A - 2A(B,C,X) - B(A,C,X) + C(A,B,X). As a byproduct, we generated an example of an alternative algebra of dimension 307.