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A double-sided shadow minimization problem in the Boolean cube layer is investigated in this paper. The problem is to minimize the size of the union of the lower and upper shadows of a k-uniform family of subsets of [n]. It is shown that if 3 ⋜ k ⋜ n−3, there is no total order such that all its initial segments have minimal double-sided shadow.

Polynomial normal forms for functions of κ-valued logics (for prime values of κ) are considered. A polynomial normal form modulo κ (p.n.f.) is the sum modulo κ of products of variables, or variables with one or several Post negations, taken with some coefficients. The length of a p.n.f. is the number of distinct terms that appear in the form with nonzero coefficients. If κ is a prime number, then each function of κ-valued logic may be represented by various p.n.f. The length of a function of κ-valued logic in the class of p.n.f. is the minimum length of a p.n.f. representing this function. The Shannon function Lp.n.f.κ (n) of the length of functions of κ-valued logic in the class of p.n.f. is the maximum length of a function in the class of p.n.f. among all n-ary functions of κ-valued logic. In thiswork the order of growth of the Shannon function Lp.n.fκ (n) is established (for prime values of κ): [ABSTRACT FROM AUTHOR]