Volume 34, Issue 1

Let { Z n } be a weakly supercritical branching process in a random environment, and { S n } be its associated random walk. We consider a natural martingale W n = Z n exp(− S n ), where n ≥ 0. We prove two limit theorems for the random process W ⌊ nt ⌋ , where t ∈ [0, 1], which is considered either under the condition on the unfavourable environment {max 1≤ i ≤ n S i } or under the condition on the unfavourable environment { S n ≤ u }, where u is some positive constant.

Nonabelian 2-groups H containing a cyclic subgroup of index 2 are dihedral groups, generalized quaternion groups, quasidihedral groups and modular maximal-cyclic groups. Earlier the authors introduced the classes of piecewise quasiaffine transformations on an arbitrary nonabelian 2-group H with a cyclic subgroup of index 2. For the generalized group of quaternions of order 2 m we have obtained a complete classification of orthomorphisms, complete transformations and their left analogues in the class of piecewise quasiaffine transformations under consideration. This paper presents a similar classification for the remaining three groups (the dihedral group, the quasidihedral group and the modular maximal-cyclic group).

We study the multiplicative (nonscalar) complexity μ ( M d , n ) for the class M d , n of complex polynomials in n variables of degree at most d . It is shown that the relation μ ( M d , n ) ≍ n ⌈ d /2⌉ holds for any constant d ≥ 2. The lower bound for odd values of d in this relation is new. For the case of cubic polynomials we prove more accurate bounds n 2 /18 ≲ μ ( M 3, n ) ≲ n 2 /4.

In the classes EMF(n) $\begin{array}{} \displaystyle \operatorname{EM}_F^{(n)} \end{array}$ of extended matrices over rings of polynomials with idempotent variables, the following subclasses (hypercontact circuits) are defined: HCF(n) $\begin{array}{} \displaystyle \operatorname{HC}_F^{(n)} \end{array}$ (over an arbitrary field F ) and HCZ(n) $\begin{array}{} \displaystyle \operatorname{HC}_Z^{(n)} \end{array}$ (over the ring of integers), which algebraically extend the class of incident matrices of contact circuits (CC ( n ) ) and realize arbitrary n -place Boolean functions with contact complexity smaller than 32⋅2n/2. $\begin{array}{} \displaystyle 3\sqrt{2}\cdot2^{n/2}. \end{array}$ A lower estimate of the same order is obtained for the corresponding Shannon function in the class HCFq(n) $\begin{array}{} \displaystyle \operatorname{HC}_{F_q}^{(n)} \end{array}$ over an arbitrary finite field F q . For matrices from the class HCZ(n) $\begin{array}{} \displaystyle \operatorname{HC}_Z^{(n)} \end{array}$, we find a physical interpretation in the form of incident-linking matrices of contact-transformer circuits.

We consider one of the possible generalizations of sums of independent random variable to the case of operations on a finite set, namely quasigroup “sums” that use quasigroup operations on a given finite set instead of the addition operation. For quasigroup “sums” that contain n independent identically distributed random variables we prove that the rate of convergence of distributions to uniform distribution is exponential in n .