Narrow Results

In this paper we give a new proof and a small improvement for the three-squares lemma for partial words with one hole. This recent result by Blanchet-Sadri et al. gives a lower bound for the length of the longest square in the situation where three squares begin at the same position of a one-hole word and the shortest square satisfies a primitivity condition.

We describe primitive and prime ideals in the C*-algebra C*(E) of a graph E satisfying Condition (K), together with the topologies on each of these spaces. In particular, we find that primitive ideals correspond to the set of maximal tails disjoint union the set of finite-return vertices, and that prime ideals correspond to the set of clusters of maximal tails disjoint union the set of finite-return vertices.

We prove that every group ring of a non-abelian locally free group which is the union of an ascending sequence of free groups is primitive. In particular, every group ring of a countable non-abelian locally free group is primitive. In addition, by making use of the result, we give a necessary and sufficient condition for group rings of ascending HNN extensions of free groups to be primitive, which extends the main result in [Group rings of proper ascending HNN extensions of countably infinite free groups are primitive, J. Algebra317 (2007) 581–592] to the general cardinality case.

The class of finitely presented algebras over a field $K$ with a set of generators $a_1,\dots ,a_n$ and defined by homogeneous relations of the form $a_1{a}_2\cdots a_n=a_{\sigma (1)}{a}_{\sigma (2)}\cdots a_{\sigma (n)}$, where $\sigma$ runs through a subset $H$ of the symmetric group ${Sym}_n$ of degree $n$, is investigated. Groups $H$ in which the cyclic group $〈(1,2,\dots ,n)〉$ is a normal subgroup of index $2$ are considered. Certain representations by permutations of the dihedral and semidihedral groups belong to this class of groups. A normal form for the elements of the underlying monoid $S_n(H)$ with the same presentation as the algebra is obtained. Properties of the algebra are derived, it follows that it is an automaton algebra in the sense of Ufnarovskij. The universal group $G_n$ of $S_n(H)$ is a unique product group, and it is the central localization of a cancellative subsemigroup of $S_n(H)$. This, together with previously obtained results on such semigroups and algebras, is used to show that the algebra $K[S_n(H)]$ is semiprimitive.

L systems generalize context-free grammars by incorporating parallel rewriting, and generate languages such as EDT0L and ET0L that are strictly contained in the class of indexed languages. In this paper, we show that many of the languages naturally appearing in group theory, and that were known to be indexed or context-sensitive, are in fact ET0L and in many cases EDT0L. For instance, the language of primitives and bases in the free group on two generators, the Bridson–Gilman normal forms for the fundamental groups of 3-manifolds or orbifolds, and the co-word problem of Grigorchuk’s group can be generated by L systems. To complement the result on primitives in rank 2 free groups, we show that the language of primitives, and primitive sets, in free groups of rank higher than two is context-sensitive. We also show the existence of EDT0L languages of intermediate growth.

Let M be a compact orientable 3-manifold, and A an essential annulus which cuts M into two 3-manifolds M₁ and M₂. We denote by g(M) the Heegaard genus of M. In this paper, we will give a lower bound to g(M) when M_i contains no bounded essential surfaces with large Euler's characteristic, where all boundary components of the essential surfaces lie in A.

A twisted torus knot is a knot obtained from a torus knot by twisting adjacent strands by full twists. The twisted torus knots lie in $F$, the genus 2 Heegaard surface for $S^3$. Primitive/primitive and primitive/Seifert knots lie in $F$ in a particular way. Dean gives sufficient conditions for the parameters of the twisted torus knots to ensure they are primitive/primitive or primitive/Seifert. Using Dean’s conditions, Doleshal shows that there are infinitely many twisted torus knots that are fibered and that there are twisted torus knots with distinct primitive/Seifert representatives with the same slope in $F$. In this paper, we extend Doleshal’s results to show there is a four parameter family of positive twisted torus knots. Additionally, we provide new examples of twisted torus knots with distinct representatives with the same surface slope in $F$.

This paper is a contribution to the investigation of flag-transitive primitive symmetric (v, k, λ) designs with λ at most 10. We prove that if 𝒟 = (𝒫, ℬ) is a nontrivial symmetric (v, k, λ) design, G ≤ Aut(𝒟) is flag-transitive primitive with Soc(G) = A_n for n ≥ 5, then 𝒟 is the unique symmetric (35,18,9) design which is the complement of the symmetric (35,17,8) design.

The paper which is based on combination of polyhedron theory studys the fundamental laws and the configuration characteristics of irregular triangular spherical shell. Then it realized the design and visual modeling of irregular spherical latticed silo roof on the computer. The silo roof is a kind of space assembly structure. Its construction is convenient so that it has short construction period and low cost. It also has attractive appearance. Above all it has broad application prospects.

This is a modified manuscript of my talk entitled "Open Questions on Fully Prime Rings" presented at the 6th CJK-ICRT. A brief survey of results on fully prime and related rings is given and it is followed by a few results and problems that may be helpful to investigate the main conjecture: under the assumption that the ring is right and left Noetherian, a fully prime ring is right primitive.