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For an integer n>=2, let gpf(x) denote the greatest prime factor of n, i.e., the number p_k in the factorization n=p_1^(a_1)...p_k^(a_k), with p_i<p_j for i<j. For n=2, 3, ...

Let A be an n×n real square matrix with n>=2 such that |sum_(i=1)^nsum_(j=1)^na_(ij)s_it_j|<=1 (1) for all real numbers s_1, s_2, ..., s_n and t_1, t_2, ..., t_n such that ...

A gyrobicupola is a bicupola in which the bases are in opposite orientations.

A birotunda in which the bases are in opposite orientations.

A cupolarotunda in which the bases are in opposite orientations.

Johnson solid J_(24).

Johnson solid J_(47).

Johnson solid J_(23).

Johnson solid J_(44).

The vectors +/-a_1, ..., +/-a_n in a three-space form a normalized eutactic star iff Tx=x for all x in the three-space.