How random are the digits of an algebraic irrational number in a given base? A common conjectured answer to this vague question is that these digits are ``really random.'' For example, define a normal number in base k 2 to be a real number x such that, for each integer d 1, each block of length d occurs in the k-ary expansion of x with frequency 1 k. A widely believed conjecture is that an alge...

An n-ary operation Q : Σ → Σ is called an n-ary quasigroup of order |Σ| if in the equation x0 = Q(x1, . . . , xn) knowledge of any n elements of x0, . . . , xn uniquely specifies the remaining one. Q is permutably reducible ifQ(x1, . . . , xn) = P ` R(xσ(1), . . . , xσ(k)), xσ(k+1), . . . , xσ(n) ́ where P and R are (n − k + 1)-ary and k-ary quasigroups, σ is a permutation, and 1 < k < n. An m-a...

An n-ary operation q : Σn → Σ is called an n-ary quasigroup of order |Σ| if in x0 = q(x1, . . . , xn) knowledge of any n elements of x0, . . . , xn uniquely specifies the remaining one. An n-ary quasigroup q is permutably reducible if q(x1, . . . , xn) = p(r(xσ(1), . . . , xσ(k)), xσ(k+1), . . . , xσ(n)) where p and r are (n− k + 1)-ary and k-ary quasigroups, σ is a permutation, and 1 < k < n. ...

The main tools in the theory of n-ary hyperstructures are the fundamental relations. The fundamental relation on an n-ary hypersemigroup is defined as the smallest equivalence relation so that the quotient would be the n-ary semigroup. In this paper we study neutral elements in n-ary hypersemigroups and introduce a new strongly compatible equivalence relation on an n-ary hypersemigroup so that ...

This paper introduces new type of m-ary systolic arrays called m-ary reversible systolic arrays which extends the new concept of two-valued reversible systolic arrays that was introduced in the first part of my article. The m-ary quantum systolic architectures’ computations of the new type of systolic arrays are also introduced. The new m-ary systolic arrays maintain the high level of regularit...

The k-ary n-cube is one of the popular topologies for interconnecting processors in multicomputers. This paper studies the difference in communication requirements between two Lee distance Gray codes when moving data from processors in normal radix k order to those in Gray code order in k-ary n-cube networks. Algorithms for k-ary to Gray code conversion, and vice versa, in k-ary n-cube networks...

A parallel algorithm for generating t-ary tree sequences in reverse B-order is presented. The algorithm generates t-ary trees by 0-1 sequences, and each 0-1 sequences is generated in constant average time O(1). The algorithm is executed on a CREW SM SIMD model, and is adaptive and cost-optimal. Prior to the discussion of the parallel algorithm a new sequential generation with O(1) average time ...