On slender 0L languages

AbstractWe give a complete proof of Theorem 3.1 in [2]. A pathological exception of Theorem 4.3 in [2] is exhibited and a condition to remove it is mentioned

Abstract numeration systems on bounded languages and multiplication by a constant

A set of integers is $S$-recognizable in an abstract numeration system $S$ if the language made up of the representations of its elements is accepted by a finite automaton. For abstract numeration systems built over bounded languages with at least three letters, we show that multiplication by an integer $\lambda\ge2$ does not preserve $S$-recognizability, meaning that there always exists a $S$-recognizable set $X$ such that $\lambda X$ is not $S$-recognizable. The main tool is a bijection between the representation of an integer over a bounded language and its decomposition as a sum of binomial coefficients with certain properties, the so-called combinatorial numeration system

Fishes of the King Edward and Carson Rivers with their Belaa and Ngarinyin names

During two separate sampling trips, in October to November 2004 and in June to July 2005, 42 sites on the King Edward River and Carson River and their tributaries were sampled for fish. Fish were captured using fine mesh seine nets and gill nets and were observed using mask and snorkel and from visual surveys. The results of this study have revealed that the number of freshwater fishes (species diversity) of the King Edward River is higher than has previously been recorded for a Western Australian river. Twenty-six freshwater fish species were recorded, which is three species higher than the much larger Fitzroy River in the southern Kimberley. The study also resulted in a number of range extensions, including Butler’s Grunter and Silver Cobbler to the west, and the Slender Gudgeon to the north and east. What appears to be an undescribed species of glassfish was captured. It differs morphologically from described species in head spines (or lack of), fin rays, as well as relative body measurements. Similarly, an undescribed ectoparasite, Argulus sp. (Crustacea: Branchiura), was found on the caudal (tail) fin lobes of one Black Bream (Jenkin’s Grunter) and three Spangled Perch. Interestingly, a considerable proportion of Black Bream, which is widespread throughout the system but essentially restricted to main channel sites, had ‘blubber-lips’. There were significant differences in the prevailing fish fauna of the different reaches of the King Edward River system. Thus fish associations in the upper King Edward River main channel were significantly different to those in the tributaries and the main channel of the Carson River. Similarly, the fauna of the Carson River, which was much more diverse than the King Edward River main channel and tributary sites, was characterised by many species that were not found in other parts of the river. The presence of barriers, in the form of waterfalls which do not permit upstream migrations of fishes are considered to be the main factor in limiting the distribution of many species. For example, many species are restricted to the lower sections of the Carson River, and include Bony Bream, Lesser Salmon Catfish, Silver Cobbler, Black Catfish, False-spined Catfish, Freshwater Longtom, Prince Regent Hardyhead, Mouth Almighty, Barred Grunter and Butler’s Grunter. It is hypothesised that these natural barriers were in place long before many of these latter species colonised the King Edward River. Some species tend to only be found within tributary sites, e.g. Kimberley Mogurnda, while others are most abundant in tributaries rather than main channel sites, e.g. Western Rainbowfish and Spangled Perch. Waterfalls are also seen as limiting the number of migratory marine/estuarine species that enter freshwaters. For example, only three species that require the marine/estuarine environment to complete their life-cycle (i.e. salt water) were captured in the freshwaters of the King Edward River system. This compares to 14 species that utilise the freshwaters of the Fitzroy River

Detecting palindromes, patterns, and borders in regular languages

Given a language L and a nondeterministic finite automaton M, we consider whether we can determine efficiently (in the size of M) if M accepts at least one word in L, or infinitely many words. Given that M accepts at least one word in L, we consider how long a shortest word can be. The languages L that we examine include the palindromes, the non-palindromes, the k-powers, the non-k-powers, the powers, the non-powers (also called primitive words), the words matching a general pattern, the bordered words, and the unbordered words.Comment: Full version of a paper submitted to LATA 2008. This is a new version with John Loftus added as a co-author and containing new results on unbordered word

Limits of Baumslag-Solitar groups and dimension estimates in the space of marked groups

We prove that the limits of Baumslag-Solitar groups which we previously studied are non-linear hopfian C*-simple groups with infinitely many twisted conjugacy classes. We exhibit infinite presentations for these groups, classify them up to group isomorphism, describe their automorphisms and discuss the word and conjugacy problems. Finally, we prove that the set of these groups has non-zero Hausforff dimension in the space of marked groups on two generators.Comment: 30 pages, no figures, englis