Volume 27, Issue 1

We consider a periodic sequence {ck}k=0∞$\{c_k\}_{k=0}^\infty$ and investigate a numerical properties of an irrational number α=∑k=0∞ckk!.$\alpha = \sum_{k=0}^\infty \frac{c_k}{k!}.$ As an application of our results we present a simple transformation of periodic sequence {ck}k=0∞$\{c_k\}_{k=0}^\infty$ into aperiodic sequence.

A function of k -valued logic is called polynomial if it may be represented by a polynomial modulo k . For any composite number k we propose a uniquely defined canonical form of polynomials for polynomial functions of k -valued logic depending on an arbitrary number of variables. This canonical form is used to find, for any composite k , a formula for the number of n -place polynomial functions of k -valued logic. As a corollary, for any composite k we find the asymptotic behaviour of the logarithm of the number of n -place polynomial functions of k -valued logic.

We obtain new upper bounds for the size and the depth of formulae for some MOD-functions (that is, functions counting n bits modulo m ). In particular, the depth of counting n bits modulo 3 is bounded by 2.8 log 2 n + O (1) in the standard basis {Λ, V, — }; the size of counting modulo 5 is bounded by O ( n 3.22 ) in the same basis; the depth of counting modulo 7 is bounded by 2.93 log 2 n + O (1) in the basis of all binary Boolean functions.

For random equiprobable Boolean functions we investigate the distribution of the number of subfunctions which have a given number of variables and are close to the set of affine Boolean functions. It is shown, for example, that for Boolean functions of n variables the mean number of subfunctions having s ⩾ 3 + log 2 n variables and the Hamming distance to the set of affine functions smaller than 2 s −2 tends to 0 as n → ∞.

A Galois ring is said to be nontrivial if it is neither a field, nor a residue ring. Divisors and multiples of the minimal polynomial of the second p -adic coordinate sequence of a linear recurrent sequence of maximum period (MP LRS) over a nontrivial Galois ring of an odd characteristic are described, as well as their relation to the initial vector of the LRS. As a corollary, nontrivial upper and lower bounds for the rank of the second coordinate sequence of a MP LRS over a nontrivial Galois ring of an odd characteristic are obtained.

We obtain an asymptotic formula for the number S n of repetition-free Boolean functions of n variables in the basis {&, ∨, ⊕, ¬} for n → ∞ : S n ∼ cn −3/2 α n n !, where c ≈ 0.1998398363, α ≈ 7.549773429.

The paper is concerned with the problem of complexity of reversible circuits consisting of NOT, CNOT and 2-CNOT gates. For a reversible circuit implementing a map f:Z2n→ Z2n $f: \mathbb Z_2^n\, \to\, \mathbb Z_2^n$ we define the Shannon complexity function L ( n , q ) as a function of n and the number q of additional inputs in the circuit. We prove the lower estimate L(n,q)⩾ 2n(n− 2)3log2⁡ (n+q)− n3 $L(n,q)\, \geqslant\, \displaystyle\frac{2^n(n-2)}{3\log_2(n+q)} - \frac{n}{3}$ for the complexity of a reversible circuit and derive the upper estimate L ( n , 0) ⩽ 48 n 2 n (1 + o (1)) / log 2 n if there are no additional inputs. The asymptotic upper estimate for the complexity is shown to be L ( n , q 0 ) ≲ 2 n with q 0 ∽ n 2 n − o ( n ) additional inputs.