Band 20, Heft 1

We consider determinate functions with delay which are extensions of determinate functions and find some properties of these functions. The problem is posed to approximate continuous functions by functions with delay, and the assertion is proved that it is possible to approximate any continuous function with an arbitrary accuracy. Approximations for some functions are given, including the addition and multiplication functions which are minimal from the delay viewpoint.

It is shown that the inversion of a complex-valued power series can be realised asymptotically with complexity of 5/4 multiplications (if we compare the upper bounds). It is shown that the calculation of the square root requires asymptotically also no more than 5/4 multiplications, the computation of an exponential has the complexity equal to 13/6 multiplications, and raising to an arbitrary power requires 41/12 multiplications.

We define several abstract computing devices which allow us to characterise the ℰ 2 class of the Grzegorczyk hierarchy. For each of these devices, we estimate the time needed to compute functions of the class ℰ 2 .

We suggest a method to compute the linear complexity of binary periodic sequences formed on the basis of biquadratic and sextic residue classes through the use of expansion of the sequence period into a sum of squares of integers. The values of the sequence polynomial are computed with the use of cyclotomic numbers of orders four and six.

In this paper we suggest a method to find lower bounds for complexity of circuits of functional elements. The essence of the method consists of replacing the initial, maybe quite complex, basis containing multi-input elements by a more simple basis, say, of two-input elements, with subsequent use of known lower bounds for complexity of circuits in the simple basis. The efficiency of this method is illustrated on particular examples of finding the asymptotics and even precise values of complexities of realisation of special Boolean functions, monotone Boolean functions and functions with small number of ones.

This paper is devoted to questions concerning the complexity of solution of the problem on one-dimensional Boolean knapsack by the branch and bound method. For this complexity we obtain two upper bounds. We separate the special case of the knapsack problem where the complexity is polynomially bounded by the dimension of the problem. We also obtain an upper and lower bounds for the complexity of solution by the branch and bound method of the subset sum problem which is a special case of the knapsack problem.

We consider the near-rings generated by endomorphisms of some extra-special 2-groups. The most essential difference of a near-ring from a usual ring is the absence of the second distributivity. In this paper, we prove that the near-ring E ( G ) generated by endomorphisms of an extra-special 2-group G of order 2 2 n +1 has the order which divides 2 2 2 n +4 n 2 and that the near-ring E ( G ) of the extra-special 2-group G of type – of order 2 2 n +1 has the order divided by 2 2 2 n +4 n 2 –2 . In this case, for n = 1 and n = 2 the upper bound is attainable: the near-ring E ( G ) of the group D 8 has the order 2 8 , and the near-ring E ( G ) of an extra-special 2-group D 8 * Q 8 has the order 2 32 .