Since its original formulation, the theory of fuzzy sets has spawned a number of extensions where the role of membership values in the real unit interval $[0, 1]$ is handed over to more complex mathematical entities. Amongst the many existing extensions, two similar ones, the fuzzy multisets and the hesitant fuzzy sets, rely on collections of several distinct values to represent fuzzy membershi...

1 ECS University of Westminster, London, HA1 3TP, UK e-mail: [email protected] 2 Warrane College, University of New South Wales, Kensington, 1465, Australia e-mail: [email protected] 3 Department of Mathematics, Vellalar College for Women, Erode 638 009, Tamilnadu, India e-mail: [email protected] 4 Centre of Biomedical Engineering, Bulgarian Academy of Sciences P.O.Box 12, ...

Suppose that all groups of order n are defined on the same set G of cardinality n, and let the distance of two groups of order n be the number of pairs (a, b) ∈ G×G where the two group operations differ. Given a group G(◦) of order n, we find all groups of order n, up to isomorphism, that are closest to G(◦).

Given a 4-tuple of Boolean variables (a, b, c, d), logical proportions are modeled by a pair of equivalences relating similarity indicators (a∧ b and a ∧ b), or dissimilarity indicators (a ∧ b and a ∧ b) pertaining to the pair (a, b), to the ones associated with the pair (c, d). There are 120 semantically distinct logical proportions. One of them models the analogical proportion which correspon...

The rule of sum says that the number of ways to choose an element from one of two disjoints sets is the sum of the cardinalities of the sets. That is, if A and B are two finite sets with no members in common, then |A ∪B| = |A|+ |B|. The rule of product says that the number of ways to choose an ordered pair is the number of ways to choose the first element from A and the second element from B. T...

Dhage [1, 2, 3] introduced the concept of open balls in a D-metric space in two different ways and discussed at length the properties of the topologies generated by the family of all open balls of each kind. Here we observe that many of his results are either false or of doubtful validity. In some cases we give examples to show that either the results are false or that the proofs given by him a...

The problem of existence of closed knight’s tours in [n]d, where [n] = {0, 1, 2, . . . , n− 1}, was recently solved by Erde, Golénia, and Golénia. They raised the same question for a generalised, (a, b) knight, which is allowed to move along any two axes of [n]d by a and b unit lengths respectively. Given an even number a, we show that the [n]d grid admits an (a, 1) knight’s tour for sufficient...

Let σ(A) denote the number of automorphisms of a model A of power ω1 . We derive a necessary and sufficient condition in terms of trees for the existence of an A with ω1 < σ(A) < 2 ω1 . We study the sufficiency of some conditions for σ(A) = 21 . These conditions are analogous to conditions studied by D.Kueker in connection with countable models. The starting point of this paper was an attempt t...

Let m + n particles be thrown randomly, independently of each other into N cells, using the following two-stage procedure. 1. The first m particles are allocated equiprobably, that is, the probability of a particle falling into any particular cell is 1/N . Let the ith cell contain mi particles on completion. Then associate with this cell the probability ai = mi/m and withdraw the particles. 2. ...

Let c be a constant and (e1, f1), (e2, f2), . . . , (ecn, fcn) be a sequence of ordered pairs of edges from the complete graph Kn chosen uniformly and independently at random. We prove that there exists a constant c2 such that if c > c2 then whp every graph which contains at least one edge from each ordered pair (ei, fi) has a component of size Ω(n) and if c < c2 then whp there is a graph conta...