Volume 14, Issue 5

Some results on decomposition of boolean functions are generalised to the case of discrete k -valued functions. Autotopy and isostrophy groups and parameters of functions with non-trivial decompositions are considered.

Let ε > 0 be a fixed real number, ⊂ R s be a full rank lattice with determinant Δ ∈ Q . We call this lattice ε -regular if λ 1 () > Δ 1/ s ( h (Δ)) – ε , where λ 1 () is the length of the shortest nonzero vector of and h (Δ) is the maximum of absolute values of the numerator and the denominator of the irreducible rational fraction for Δ. In this paper, we consider two full rank lattices in the space R s : the lattice ℒ( a , W ) connected with the linear congruent sequence ( x N ), x N +1 = ax N (mod W ), N =1 ,2, . . . , (1) and the lattice ℒ ∗ ( a , W ) dual to ℒ( a , W ). There is a conjecture which states that for any natural number s , any real number 0 < ε < ε 0 ( s ), and any natural number W > W 0 ( s , ε ), the lattices ℒ( a , W ) and ℒ ∗ ( a , W ) are ε -regular for all a = 0 ,1, . . . , W – 1 excluding some set of numbers a of cardinality at most W 1– ε . In the case s = 3, A. M. Frieze, J. Hestad, R. Kannan, J. C. Lagarias, and A. Shamir in a paper published in 1988 proved a more weak assertion (in their estimate the number of exceptional values a is at most W 1– ε /2 ). Using the methods of this paper, it is not difficult to prove the conjecture for s = 1 and s = 2. In our paper, we prove the conjecture for s = 4. With the help of our methods we improve the result of the paper mentioned above and prove the conjecture for s = 3. Our result can be applied to the reconstruction of a linear congruent sequence (1) if the high-order bits of its first s elements are given.

We consider the equation ( x 1 + . . . + x n ) m = ax 1 . . . x n , where a is a nonzero element of the finite field F q , n ≥ 2, and m is a positive integer. Explicit formulas for the number of solutions of this equation in under the condition d ∈ {1, 2, 3, 6}, where d = gcd( m – n, q – 1), are found. Moreover, we obtain formulas for the number of solutions for arbitrary d > 2 if there exists positive integer l such that d | ( p l + 1), where p is the characteristic of F q .

We study the question on average and typical values of sums of pairwise Hamming distances for subsets of vertices of the n -dimensional unit cube. We suggest an approach to the problem of evaluation of average and typical values of arbitrary functionals defined on subsets of a finite set as the sum of values assigned to ordered pairs of elements of this set; general formulas for this case are obtained. We find average and typical values of sums of pairwise distances in the case of all subsets of vertices of the n -dimensional unit cube and of sums of pairwise distances for subsets of vertices of fixed cardinality.

We consider a vector (multicriteria) problem of Boolean programming where sub-criteria are projections of linear functions onto R + . We give a bound for variation of coefficients of such functions in the metric l 1 which preserves strict efficiency of the solution.

We consider a sequence of independent trials, under the hypothesis H 0 each of them is a realisation of a certain polynomial scheme. We introduce a family of multivariate χ 2 -statistics where the outcomes are equipped with weights depending on the ordinal of the trial. Earlier results on families of multivariate χ 2 -statistic are extended to this family.

In this paper, we present the list of minimal sets of systems of parastrophs for all 109 classes of isomorphic loops and 22 classes of isotopic quasigroups of order 6.