Marcus contextual languages consisting of primitive words

In this paper we prove that the language of all primitive (strongly primitive) words over a nontrivial alphabet can be generated by certain types of Marcus contextual grammars.     

Weinbaum factorizations of primitive words

C. M. Weinbaum [1] showed the following: Let w be a primitive word and a be letter in w. Then a conjugate of w can be written as uv such that a is a prefix and a suffix of u, but v neither starts nor ends with a, and u and v have a unique position in w as cyclic factors. The latter condition means that there is exactly one conjugate of w having u as a prefix and there is exactly one conjugate of w having v as a prefix. It is this condition which makes the result non-trivial. We give a simplified proof for Weinbaum’s result. Guided by this proof we exhibit quite different, but still simple, proofs for more general statements. For this purpose we introduce the notion of Weinbaum factor and Weinbaum factorization.     

CLASSIFICATION OF PRIMITIVE （J1,2）-ARC TRANSITIVE GRAPHS

In the author＇s Ph. D thesis, a non-quasiprimitive graph admitting a quasiprimitive automorphism group isomorphic to J1 was constructed ,where J1 is Janko simple group of order 175560. Is this the only one for J1？ In this paper all primitive （J1,2）-arc transitive graphs Г are given and that AutГ≌J1 is proved.     

Unbordered partial words

An unbordered word is a string over a finite alphabet such that none of its proper prefixes is one of its suffixes. In this paper, we extend the results on unbordered words to unbordered partial words. Partial words are strings that may have a number of “do not know” symbols. We extend a result of Ehrenfeucht and Silberger which states that if a word u can be written as a concatenation of nonempty prefixes of a word v, then u can be written as a unique concatenation of nonempty unbordered prefixes of v. We study the properties of the longest unbordered prefix of a partial word, investigate the relationship between the minimal weak period of a partial word and the maximal length of its unbordered factors, and also investigate some of the properties of an unbordered partial word and how they relate to its critical factorizations (if any).     

The primitive matrices of sandwich semigroups of generalized circulant Boolean matrices

Let Gn（C） be the sandwich semigroup of generalized circulant Boolean matrices with the sandwich matrix C and Gc（Jr~） the set of all primitive matrices in Gn（C）. In this paper, some necessary and sufficient conditions for A in the semigroup Gn（C） to be primitive are given. We also show that Gc（Jn） is a subsemigroup of Gn（C）.     

Alexander polynomials of doubly primitive knots

We give a formula for Alexander polynomials of doubly primitive knots. This also gives a practical algorithm to determine the genus of any doubly primitive knot.

     

The coefficients of primitive polynomials over finite fields

For , we prove that there always exists a primitive polynomial of degree over a finite field with the first and second coefficients prescribed in advance.

     

A primitive trinomial of degree 6972593

The only primitive trinomials of degree over are and its reciprocal.

     

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.     

Finite vertex primitive 2-arc regular graphs

A classification is given of finite graphs that are vertex primitive and 2-arc regular. The classification involves various new constructions of interesting 2-arc transitive graphs.     

Unbordered factors and Lyndon words

A primitive word w is a Lyndon word if w is minimal among all its conjugates with respect to some lexicographic order. A word w is bordered if there is a nonempty word u such that w=uvu for some word v. A right extension of a word w of length n is a word wu where all factors longer than n are bordered. A right extension wu of w is called trivial if there exists a positive integer k such that w^k=uv for some word v.We prove that Lyndon words have only trivial right extensions. Moreover, we give a conjecture which characterizes a property of every word w which has a nontrivial right extension of length 2|w|-2.     

ON THE EXPONENT SET OF PRIMITIVELOCALLY SEMICOMPLETE DIGRAPHS

A locally semicomplete digraph is a digraph D=(V,A) satisfying the following condi-tion for every vertex x∈V the D[O(x)] and D[I(x)] are semicomplete digraphs. In this paper,we get some properties of cycles and determine the exponent set of primitive locally semicompleted digraphs.     

围长为2的本原无限布尔方阵类的本原指数集

研究了围长为2的无限布尔方阵的本原性,通过无限有向图D（A）的直径给出了这类矩阵的本原指数的上确界,最后证明了直径小于等于d且围长为2的本原无限布尔方阵所构成的矩阵类的本原指数集为Ed^0={2,3,…,3d}．     

Finite locally primitive abelian Cayley graphs

Let Γ be a finite connected locally primitive Cayley graph of an abelian group.It is shown that one of the following holds:(1) Γ = Kn,Kn,n,Kn,n-nK2,Kn × · · · × Kn;(2) Γ is the standard double cover of Kn × · · · × Kn ;(3) Γ is a normal or a bi-normal Cayley graph of an elementary abelian or a meta-abelian 2-group.     

INJECTIVE MAPS ON PRIMITIVE SEQUENCES OVER Z/（p^e）

Let Z/(pe) be the integer residue ring modulo pe with p an odd prime and integer e ≥ 3. For a sequence (a) over Z/(pe), there is a unique p-adic decomposition (a) = (a)0 (a)1·p … (a)e-1 ·pe-1, where each (a)i can be regarded as a sequence over Z/(p), 0 ≤ i ≤ e - 1. Let f(x) be a primitive polynomial over Z/(pe) and G' (f(x), pe) the set of all primitive sequences generated by f(x) over Z/(pe). For μ(x) ∈ Z/(p)[x] with deg(μ(x)) ≥ 2 and gcd(1 deg(μ(x)),p- 1) = 1,set ψe-1 (x0, x1,…, xe-1) = xe-1·[ μ(xe-2) ηe-3 (x0, x1,…, xe-3)] ηe-2 (x0, x1,…, xe-2),which is a function of e variables over Z/(p). Then the compressing map ψe-1: G'(f(x),pe) → (Z/(p))∞,(a) (→)ψe-1((a)0, (a)1,… ,(a)e-1) is injective. That is, for (a), (b) ∈ G' (f(x), pe), (a) = (b) if and only if ψe - 1 ((a)0, (a)1,… , (a)e - 1) =ψe - 1 ((b)0,(b)1,… ,(b)e-1). As for the case of e = 2, similar result is also given. Furthermore, if functions ψe-1 and ψe-1 over Z/(p) are both of the above form and satisfy ψe-1((a)0,(a)1,… ,(a)e-1) = ψe-1((b)0,(b)1,… ,(b)e-1) for (a),(b) ∈ G'(f(x),pe), the relations between (a) and (b), ψe-1 and ψe-1 are discussed.     

Shortest directing words of nondeterministic directable automata

In this paper, we will survey several results on the shortest directing words of various types of nondeterministic directable automata.¹