Does science also prefer a ternary pattern?

Nothing is as difficult to see as the obvious (B. Malinowski)

Through the importance of the number three in our culture and the strange preference for a ternary pattern of our nature one can perceive how and why number theory degraded to numerology. The strong preference of our minds for simple patterns can be read as the key to understanding not only the development of numerology, but also why scientists were and are prone to give their discoveries or creations a structure that is based on simple patterns and, among them, especially the ternary pattern. If some numbers seem to appear quite often throughout the fabric of reality, like zero, one, two, and three, reality has, nevertheless, been preferentially analysed in a way to enhance its ternary character. The present excursion through some scientific domains where the ternary pattern has been used tries to offer an answer to the fascination that some numbers, and patterns based on them, exerted and exerts, even today, on human minds. Throughout the overall models created by the human mind, be they natural or spiritual, the models based on a ‘unary’ and on a ‘dualistic’ pattern are surely important, but the model based on a ternary pattern ended up assuming an archetypical character, which is quite intriguing, and deserves to be deepened. The paper is a useful resource for projects for students and background material for teachers of mathematics touching, as it does, on biology, chemistry and physics.     

The number of binary cube-free words of length up to 47 and their numerical analysis

An analog of the work of Noonan and Zeilberger on square-free ternary words given for the case of cube-free binary words. As in their work, the Goulden-Jackso Cluster Method is used to derive a rigorous upper bound, as well as a non-rigorot estimate, for the limit of thenth roots of the terms. The Maple implementatin of the work is available from this paper's website www.math.temple. edu/`anne/cut free.html.     

The Nonexistence of Ternary [50,5,32] Codes

It is unknown (cf. Hill and Newton [8] or Hamada [3]) whether or not there exists a ternary [50,5,32] code meeting the Griesmer bound. The purpose of this paper is to prove the nonexistence of ternary [50,5,32] codes. Since there exists a ternary [51,5,32] code, this implies that n3(5,32) = 51, where n3(k,d) denotes the smallest value of n for which there exists a ternary [n,k,d] code.     

The Nonexistence of Ternary [38, 6, 23] Codes

It has been shown by Bogdanova and Boukliev [1] that there exist a ternary [38,5,24] code and a ternary [37,5,23] code. But it is unknown whether or not there exist a ternary [39,6,24] code and a ternary [38,6,23] code. The purpose of this paper is to prove that (1) there is no ternary [39,6,24] code and (2) there is no ternary [38,6,23] code using the nonexistence of ternary [39,6,24] codes. Since it is known (cf. Brouwer and Sloane [2] and Hamada and Watamori [14]) that (i) n₃(6,23) = 38> or 39 and d₃(38,6) = 22 or 23 and (ii) n₃(6,24) = 39 or 40 and d₃(39,6) = 23 or 24, this implies that n₃(6,23) = 39, d₃(38,6) = 22, n₃(6,24) = 40 and d₃(39,6) = 23, where n3<>(k,d) and d<>3(n,k) denote the smallest value of n and the largest value of d, respectively, for which there exists an [n,k,d] code over the Galois field GF(3).     

Testing primitivity on partial words

Primitive words, or strings over a finite alphabet that cannot be written as a power of another string, play an important role in numerous research areas including formal language theory, coding theory, and combinatorics on words. Testing whether or not a word is primitive can be done in linear time in the length of the word. Indeed, a word is primitive if and only if it is not an inside factor of its square. In this paper, we describe a linear time algorithm to test primitivity on partial words which are strings that may contain a number of “do not know” symbols. Our algorithm is based on the combinatorial result that under some condition, a partial word is primitive if and only if it is not compatible with an inside factor of its square. The concept of special, related to commutativity on partial words, is foundational in the design of our algorithm. A World Wide Web server interface at http://www.uncg.edu/mat/primitive/ has been established for automated use of the program.     

Ternary semigroups

A ternary semigroup is a nonempty set together with a ternary multiplication which is associative. Analogous to the theory of semigroups, a regularity condition on a ternary semigroup is introduced and the properties of regular ternary semigroups are studied. Associated with a ternary semigroup, a semigroup called the semigroup cover is constructed and its properties are investigated.     

On the projectivity of B-ovals

In this paper, a method of coordinatizing a B-oval is introduced which gives rise to a ternary ring structure called anoval ternary ring (or OTR). By examining these OTR's, we can deduce sufficient conditions for a B-oval to be projective. The connections between OTR's and the Hall ternary rings, which arise from the coordinatization of a projective plane, are also examined.     

Combinatorics on partial word correlations

Partial words are strings over a finite alphabet that may contain a number of “do not know” symbols. In this paper, we consider the period and weak period sets of partial words of length n over a finite alphabet, and study the combinatorics of specific representations of them, called correlations, which are binary and ternary vectors of length n indicating the periods and weak periods. We characterize precisely which vectors represent the period and weak period sets of partial words and prove that all valid correlations may be taken over the binary alphabet. We show that the sets of all such vectors of a given length form distributive lattices under suitably defined partial orderings. We show that there is a well-defined minimal set of generators for any binary correlation of length n and demonstrate that these generating sets are the primitive subsets of {1,2,…,n−1}. We also investigate the number of partial word correlations of length n. Finally, we compute the population size, that is, the number of partial words sharing a given correlation, and obtain recurrences to compute it. Our results generalize those of Guibas, Odlyzko, Rivals and Rahmann.     

Ternary self-orthogonal codes of dual distance three and ternary quantum codes of distance three

Ternary self-orthogonal codes with dual distance three and ternary quantum codes of distance three constructed from ternary self-orthogonal codes are discussed in this paper. Firstly, for given code length n ≥ 8, a ternary [n, k]₃ self-orthogonal code with minimal dimension k and dual distance three is constructed. Secondly, for each n ≥ 8, two nested ternary self-orthogonal codes with dual distance two and three are constructed, and consequently ternary quantum code of length n and distance three is constructed via Steane construction. Almost all of these quantum codes constructed via Steane construction are optimal or near optimal, and some of these quantum codes are better than those known before.     

Ternary Derivations,Stability and Physical Aspects

Ternary algebras and modules are vector spaces on which products of three factors are defined. In this paper, we present several physical applications of ternary structures. Some recent results on the stability of ternary derivations are reviewed. Using a fixed point method, we also establish the generalized Hyers–Ulam–Rassias stability of ternary derivations from a normed ternary algebra into a Banach tri-module associated to the generalized Jensen functional equations and prove a superstability result. The author was in part supported by a grant from IPM (No. 85390031).     

辛三代数的Frattini子代数

本文研究了辛三代数的Frattini子代数和基本辛三代数的问题.利用Frattini子代数和基本辛三代数的性质,得到了辛三代数的非嵌入定理,从而推广了李三系中关于Frattini子系的结果.     

Ternary analogues of Lie and Malcev algebras

We consider two analogues of associativity for ternary algebras: total and partial associativity. Using the corresponding ternary associators, we define ternary analogues of alternative and assosymmetric algebras. On any ternary algebra the alternating sum [a, b, c] = abc − acb − bac + bca + cab − cba (the ternary analogue of the Lie bracket) defines a structure of an anticommutative ternary algebra. We determine the polynomial identities of degree ?7 satisfied by this operation in totally and partially associative, alternative, and assosymmetric ternary algebras. These identities define varieties of ternary algebras which can be regarded as ternary analogues of Lie and Malcev algebras. Our methods involve computational linear algebra based on the representation theory of the symmetric group.     

The Banzhaf power index for ternary bicooperative games

In this paper we analyze ternary bicooperative games, which are a refinement of the concept of a ternary voting game introduced by Felsenthal and Machover. Furthermore, majority voting rules based on the difference of votes are simple bicooperative games. First, we define the concepts of the defender and detractor swings for a player. Next, we introduce the Banzhaf power index and the normalized Banzhaf power index. The main result of the paper is an axiomatization of the Banzhaf power index for the class of ternary bicooperative games. Moreover, we study ternary bicooperative games with two lists of weights and compute the Banzhaf power index using generating functions.     

Some properties of hyperideals in ternary semihypergroups

Ternary semihypergroups are algebraic structures with one ternary associative hyperoperation. In this paper we give some properties of left (right) and lateral hyperideals in ternary semihypergroups. We introduce the notion of left simple, lateral simple, left (0-)simple and lateral 0-simple ternary semihypergroups and characterize the minimality and maximality of left (right) and lateral hyperideals in ternary semihypergroups. The relationship between them is investigated in ternary semihypergroups extending and generalizing the analogues results for ternary semigroups.     

Construction of Ternary H v -groups and Ternary P-hyperoperations

In this paper, we study the concept of ternary H _v -groups and some their properties. We give some examples of ternary H _v -groups. Also, we consider the fundamental relation β* on a ternary H _v -group and prove that β* is a compatible relation on a ternary H _v -group. In addition, we define the P-hyperoperation, and then, we construct a new ternary H _v -group.     

On the structure of finite elation Laguerre planes

Quasi-orderings are introduced in arbitrary affine planes and ternary rings as a common generalization of half-orderings and semiorderings. Quasi-ordered affine planes are described algebraically by quasi-ordered ternary rings. A lot of proper quasi-orderings are constructed in ternary rings. These algebraic examples show that all Lenz-Barlotti-classes, as far as they are known to be non-empty, are containing affine planes with proper quasi-orderings.Dedicated to Professor Dr. H. Lenz on the occasion of his 75th birthday     

On Identities of a Ternary Quaternion Algebra

This article studies a simple 4-dimensional ternary algebra 𝒜 which appears analogously to the quaternions from the Lie algebra 𝔰𝔩(2). We describe the heights 1 and 2 identities, and the derivations of 𝒜. Based on 𝒜, some ternary enveloping algebras for ternary Filippov algebras are constructed.     

Fuzzy ideals of ternary semigroups

In this paper, we introduce fuzzy ideals of ternary semigroups and study their related properties. Here we define fuzzy left (right, lateral) ideals of ternary semigroups and characterize regular and intra-regular ternary semigroups by using the concept of fuzzy ideals of ternary semigroups.     

辛代数的不变双线性型

考察了辛代数和与它相联系的李三系的双线性型之间的关系,并证明了辛代数的反对称不变双线性型可以唯一扩张到与它相联系的李三系中.作为这种关系的一个应用,得到了二次辛代数是单辛代数的一个充要条件,并证明二次辛代数的唯一分解定理.