Volume 28, Issue 6

The paper is concerned with polynomial transformations of a finite commutative local principal ideal of a ring (a finite commutative uniserial ring, a Galois–Eisenstein ring). It is shown that in the class of Galois–Eisenstein rings with equal cardinalities and nilpotency indexes over Galois rings there exist polynomial generators for which the period of the output sequence exceeds those of the output sequences of polynomial generators over other rings.

We study compositions (ordered partitions) of n . More particularly, our focus is on the bargraph representation of compositions which include or avoid squares of size s × s . We also extend the definition of a Durfee square (studied in integer partitions) to be the largest square which lies on the base of the bargraph representation of a composition (i.e., is ‘grounded’). Via generating functions and asymptotic analysis, we consider compositions of n whose Durfee squares are of size less than s × s . This is followed by a section on the total and average number of grounded s × s squares. We then count the number of Durfee squares in compositions of n .

We consider the synthesis problem of two-pole contact circuits implementing given Boolean functions and admitting short fault detection test with respect to contact breaks. For each n -place Boolean function, we found the smallest possible lengths of the single and complete fault detection tests. In particular, it is shown that such length are not greater than n .

We provide a new upper estimate for the modulus of the difference | Λ ∩ 𝓢| − vol n (𝓢)/det Λ , where 𝓢 ⊂ ℝ n is a set of volume vol n (𝓢) and Λ ⊂ ℝ n is a complete lattice with determinant det Λ . This result has an important practical application, for example, in estimating the number of integer solutions of an arbitrary system of linear and nonlinear inequalities.

The well-known Stone’s renewal theorem is refined for the case of arithmetic distributions having at least one exponentially decreasing tail. A very general version of the renewal theorem for arithmetic distributions with a semi-multiplicative bound of the residual term is proved.

Let a function f ( z ) be decomposed into a power series with nonnegative coefficients which converges in a circle of positive radius R . Let the distribution of the random variable ξ n , n ∈ {1, 2, …}, be defined by the formula P{ξn=N}=coeffznf(z)NN!coeffznexp⁡(f(z)),N=0,1,…$$\begin{array}{} \displaystyle P\{\xi_n=N\}=\frac{\mathrm{coeff}_{z^n}\left(\frac{\left(f(z)\right)^N}{N!}\right)}{\mathrm{coeff}_{z^n}\left(\exp(f(z))\right)},\,N=0,1,\ldots \end{array} $$ for some ∣ z ∣ < R (if the denominator is positive). Examples of appearance of such distributions in probabilistic combinatorics are given. Local theorems on asymptotical normality for distributions of ξ n are proved in two cases: a) if f ( z ) = (1 − z ) − λ , λ = const ∈ (0, 1] for ∣ z ∣ < 1, and b) if all positive coefficients of expansion f ( z ) in a power series are equal to 1 and the set A of their numbers has the form A={mr|m∈N},r=const,r∈{2,3,…}.$$\begin{array}{} \displaystyle A = \{m^r \, | \, m \in \mathbb{N} \}, \, \, r = \mathrm {const},\; r \in \{2,3,\ldots\}. \end{array} $$ A hypothetical general local limit normal theorem for random variables ξ n is stated. Some examples of validity of the statement of this theorem are given.