Efficient Identification of k-Closed Strings

A closed string contains a proper factor occurring as both a prefix and a suffix but not elsewhere in the string. Closed strings were introduced by Fici (WORDS 2011) as objects of combinatorial interest. This paper addresses a new problem by extending the closed string problem to the [Formula: see text]-closed string problem, for which a level of approximation is permitted up to a number of Hamming distance errors, set by the parameter [Formula: see text]. We address the problem of deciding whether or not a given string of length [Formula: see text] over an integer alphabet is [Formula: see text]-closed and additionally specifying the border resulting in the string being [Formula: see text]-closed. Specifically, we present an [Formula: see text]-time and [Formula: see text]-space algorithm to achieve this along with the pseudocode of an implementation and proof-of-concept experimental results.

Words with many Palindrome Pair Factors

Motivated by a conjecture of Frid, Puzynina, and Zamboni, we investigate infinite words with the property that for infinitely many $n$, every length-$n$ factor is a product of two palindromes. We show that every Sturmian word has this property, but this does not characterize the class of Sturmian words. We also show that the Thue—Morse word does not have this property. We investigate finite words with the maximal number of distinct palindrome pair factors and characterize the binary words that are not palindrome pairs but have the property that every proper factor is a palindrome pair.

Admissibility of Propensity Evidence in Paedophilia Cases

<p>Application of the propensity evidence rule in paedophilia cases starkly highlights the tension between the two competing principles underlying the rule. Paedophilia is an area where the danger of prejudice is at its highest. At the same time the probative significance of propensity evidence may be great. In their habitual offending behaviour, paedophiles seek to establish opportunity and overcome the child’s resistance to the sexual acts, typically through a long and drawn out grooming process in a manner calculated to minimise the risk of detection. Evidence of similar misconduct of the accused may be highly probative in establishing this pattern of paedophilia behaviour, and a proper factor to be taken account of in balancing probative value against risk of prejudice. In this context the Hoch/Pfennig test for admissibility is unduly strict. A more satisfactory balance between probative value and danger of prejudice may be achieved under the provisions of the uniform Evidence Acts and the pre-Hoch/Pfennig common law contained in s 398A of the Crimes Act 1958 (Vic).</p>

ON SOME TYPES OF NOETHERIAN MODULES

We describe FC-hypercentral locally soluble groups G such that a groups algebra DG over a Dedekind domain D has a noetherian module A with faithful action of G such that A is subdirectly reducible and every proper factor module of A is finitely generated over D. It is proved, in particular, that G has an abelian normal subgroup of finite index.

On the probability of generating a minimal d-generated group

We consider finite groups with the property that any proper factor can be generated by a smaller number of elements than the group itself. We study some problems related with the probability of generating these groups with a given number of elements.

Groups with constraints for proper factor-groups and the modules associated with them

We cover essential elements of researches on jn-X-groups, in particular, the theoretic-modular part. We explain some new results about jn-X-groups. We describe the set of questions that are related to further researches on jn-X-groups.

Groups whose proper quotients are hypercentral

Groups, all proper factor-groups of which are hypercentral of finite torsion-free rank, are studied in this article.