Ratko Tosic

Enumeration and classification of coronoid hydrocarbons: Part V. Primitive coronoids

Some topological properties of primitive coronoids are discussed. A method of generating such systems from the corona hole is described in detail. The results from computer-aided enumerations of primitive coronoids are given for h (the number of hexagons ) up to 25. The results account for 1 075 554 nonisomorphic systems. The distributions into symmetry groups are specified. The forms of all the primitive coronoids for h ⩽ 15 are depicted and supplied with K numbers ( Kekule structure counts ). Finally the average K values and related quantities for the systems under consideration are reported.

On set-valued functions and Boolean collections

The notion of a Boolean collection of set is introduced, and several combinatorial aspects of these collections are exploited. These collections of set appear to play a role in the approximation of non-Boolean set-valued functions by Boolean functions and, therefore, are relevant in the study of biocircuits and in the study of circuits based on frequency multiplexing, where set-valued functions are used.<<ETX>>

Functional completeness and weak completeness in set logic

The functional completeness problems in r-valued set logic, which is the logic of functions mapping n-tuples of subsets into subset over r values, is discussed. It is shown that r-valued set logic is isomorphic to 2/sup r/-valued logic, meaning that the well-known completeness criteria in multiple-valued Post algebras apply to set-valued logic. Since Boolean functions are convenient choice as building blocks in the design of set logic functions, the notion of weak completeness of a set is introduced; i.e., a set is weak complete if it becomes complete once all Boolean functions are added to the set. A full description of weak complete sets, weak maximal sets, weak bases, and weak Sheffer functions is given for the case of two-valued set logic.<<ETX>>

An optimal search procedure

Abstract Consider the following problem. There are exactly two defective (unknown) elements in the set X ={ x 1 , x 2 ,…, x n }, all possibilities occuring with equal probabilities. We want to identify the unknown (defective) elements by testing some subsets A of X , and for each such set A determining whether A contains any of them. The test on an individual subset A informs us that either all elements of the tested set A are good, or that at least one of them is defective (but we do not know which ones or how many). A set containing at least one defective element is said to be defective. Our aim is to minimize the maximal number of tests. For the optimal strategy, let the maximal test length be denoted by l 2 ( n ). We obtain the value of this function for an infinite sequence of values of n .

Five counterfeit coins

Abstract We consider the problem of ascertaining the minimum number of weighings which suffice to determine all counterfeit (heavier) coins in a set of n coins of the same appearance, given a balance scale and the information that there are exactly five heavier coins present. For an infinite set of n s we determine an upper bound for the maximum number of steps of an optimal procedure which differs by just two from the information-theoretical lower bound. We also consider a slightly modified problem, i.e. the case when we are given a certain number (not greater than 2 n -3) of additional coins for which we know that they are all good (not counterfeit). For that case, and arbitrary n , we determine an upper bound for the maximum number of steps of an optimal procedure which differs by just seven from the information-theoretical lower bound.

Generating and counting unbranched catacondensed benzenoids

The problem of “cell-growth” is a classical problem in When applied to hexagonal “animals”, it has relevance to the studies of benzenoid (polycyclic aromatic) hydrocarbons, especially to their enumeration.&l5 In the present paper we consider polyhex graphs. Here a polyhex graph, also called a benzenoid or hexagonal system, is defined as in Gutman’s review.8 A polyhex graph corresponds to a network obtained by arranging congruent regular hexagons in the plane so that two hexagons are either disjoint or possess exactly one common edge. It is a strictly planar system that is simply connected (Figure 1). The last restriction is released in the definition of coronoids (Figure 2). In the present work we consider a special class of benzenoid systems-unbranched catacondensed benzenoid systems (UCBS) A benzenoid system is catacondensed8 if it does not posses any internal vertex. All other systems are referred to as pericondensed. A catacondensed benzenoid system is said to be unbranched if each of its hexagons has at most two neighbors (Figure 3). All other catacondensed benzenoid systems are referred to as branched (Figure 4). Given a number of hexagons, how many benzenoid systems can be constructed? This general mathematical problem is still unsolved. In the present paper we report the new results of enumeration of UCBS, where the range of computation is extended with regard to the h value up to 20. Here h denotes the number of hexagons of a UCBS. Some previous results of enumeration of such systems are summarized in Brunvoll et al.,’ Table I, where the range of computation has been extended to 12 for symmetrical UCBS and to 11 for unsymmetrical UCBS. In the case of the benzenoid systems there is a bijection between the molecular graph and the dualistls obtained by the replacement of the hexagon centers with vertices. Two such vertices are w~ected by an edge if and only if they correspond to adjacent hexagons (Figure 5 ) . The dualist of a hexagonal system can be considered as a subgraph of a triangular lattice T obtained by tiling the plane by congruent regular triangles (Figure 6). It is easy to see that the dualist of a UCBS is a path in a triangular lattice (Figure 7).

Enumeration and classification of benzenoid hydrocarbons: Part XII. Catacondensed systems of regular trigonal symmetry

New principles in the enumeration of benzenoid systems are reported. They imply a class of benzenoids referred to as special catacondensed systems. The techniques are applied to catacondensed (branched) benzenoids of regular trigonal (D3h) symmetry. The results of enumeration up to h = 25 are given.

On spectra of many-valued logic symmetric functions

Many-valued logic symmetric functions appearing in various applications are investigated from the standpoint of determining the number of n-ary functions belonging to a considered set (called the spectrum of the set). Respective spectra are given of k-valued functions that are p-symmetric, self-dual, and self-dual p-symmetric, where p is a partition of (1,. . .,n). It is proved that there exist self-dual totally symmetric n-ary k-valued logic functions if and only if the greatest common divisor of k and n is equal to one. A test for detecting the self-dual symmetry property is described. Respective spectra are also given of k-valued symmetric functions that are threshold, multithreshold, monotone, and unate (for the monotone and unate functions k=3 only).<<ETX>>

The Computing Capacity of Three-Input Multiple-Valued One-Threshold Perceptrons

In this paper, an exact and general formula is derived for the number of linear partitions of a given point set V in three-dimensional space, depending on the configuration formed by the points of V. The set V can be a multi-set, that is it may contain points that coincide. Based on the formula, we obtain an efficient algorithm for counting the number of k-valued logic functions simulated by a three-input k-valued one-threshold perceptron.

Benzenoid chains with the unique clar formula

Abstract A characterization is given for a benzenoid chain to be a Clar chain, i.e. the benzenoid chain with the unique Clar formula. The number of non-isoarithmic Clar chains with h hexagons is determined.