Breadth-first search and the Andrews-Curtis conjecture

Andrews and Curtis conjectured in 1965 that every balanced presentation of the trivial group can be transformed into a standard presentation by a finite sequence of elementary transformations. Recent computational work by Miasnikov and Myasnikov on this problem has been based on genetic algorithms. We show that a computational attack based on a breadth-first search of the tree of equivalent presentations is also viable, and seems to outperform that based on genetic algorithms. It allows us to extract shorter proofs (in some cases, provably shortest) and to consider the length thirteen case for two generators. We prove that, up to equivalence, there is a unique minimum potential counterexample.

Testing integrability with a single bit of quantum information

We show that deterministic quantum computing with a single bit can determine whether the classical limit of a quantum system is chaotic or integrable using O(N) physical resources, where N is the dimension of the Hilbert space of the system under study. This is a square-root improvement over all known classical procedures. Our study relies strictly on the random matrix conjecture. We also present numerical results for the nonlinear kicked top.

Floral structure, breeding system and fruit-set in the threatened sub-shrub Tetratheca juncea Smith (Tremandraceae)

Tetratheca juncea Smith (Tremandraceae) has undergone a range contraction of approx. 50 km in the last 100 years and is now listed as a vulnerable sub-shrub restricted to the central and north coast regions of New South Wales, Australia. There are approx. 250 populations in a 110 km north-south distribution and populations are usually small with fewer than 50 plants/clumps. The reproductive ecology of the species was studied to determine why seed-set is reportedly rare. Flowers are bisexual, odourless and nectarless. Flowers are presented dependentally and there are eight stamens recurved around the pistil. Anthers are poricidal, contain viable pollen and basally contain a deep-red tapetal fluid that is slightly oily. Thus flowers are presented for buzz pollinators, although none were observed at flowers during our study. The species was found to be facultatively xenogamous with only one in 50 glasshouse flowers setting seed autogamously, i.e. without pollinator assistance. Field studies revealed fertile fruit in 24 populations but production varied significantly across sites from exceedingly low (0.6 fruits per plant clump) to low (17 fruits per plant clump). Fruit-set ranged from 0 to 65%, suggesting that pollen vectors exist or that autogamy levels in the field are variable and higher than glasshouse results. Fruit production did not vary with population size, although in three of the five populations in the south-west region more than twice as much fruit was produced as in populations elsewhere. A moderately strong relationship between foliage volume and fruit : flower ratios suggests that bigger plants may be more attractive than smaller plants to pollinators. A review of Tetratheca pollination ecology revealed that several species are poorly fecund and pollinators are rare. The habitat requirements for Tetratheca, a genus of many rare and threatened species, is discussed. (C) 2003 Annals of Botany Company.

Quantum-gate characterization in an extended Hilbert space

We describe an approach for characterizing the process performed by a quantum gate using quantum process tomography, by first modeling the gate in an extended Hilbert space, which includes nonqubit degrees of freedom. To prevent unphysical processes from being predicted, present quantum process tomography procedures incorporate mathematical constraints, which make no assumptions as to the actual physical nature of the system being described. By contrast, the procedure presented here assumes a particular class of physical processes, and enforces physicality by fitting the data to this model. This allows quantum process tomography to be performed using a smaller experimental data set, and produces parameters with a direct physical interpretation. The approach is demonstrated by example of mode matching in an all-optical controlled-NOT gate. The techniques described are general and could be applied to other optical circuits or quantum computing architectures.

A new system of variational inclusions with (H,eta)-monotone operators in Hilbert spaces

In this paper, we introduce and study a new system of variational inclusions involving (H, eta)-monotone operators in Hilbert space. Using the resolvent operator associated with (H, eta)monotone operators, we prove the existence and uniqueness of solutions for this new system of variational inclusions. We also construct a new algorithm for approximating the solution of this system and discuss the convergence of the sequence of iterates generated by the algorithm. (c) 2005 Elsevier Ltd. All rights reserved.

Chloroplast and total genomic diversity in the endemic Costa Rican tree Lonchocarpus costaricensis (J.D. Smith) Pittier (Papilionaceae).

In Mesoamerica, tropical dry forest is a highly threatened habitat, and species endemic to this environment are under extreme pressure. The tree species, Lonchocarpus costaricensis is endemic to the dry northwest of Costa Rica and southwest Nicaragua. It is a locally important species but, as land has been cleared for agriculture, populations have experienced considerable reduction and fragmentation. To assess current levels and distribution of genetic diversity in the species, a combination of chloroplast-specific (cpDNA) and whole genome DNA markers (amplified fragment length polymorphism, AFLP) were used to fingerprint 121 individual trees in 6 populations. Two cpDNA haplotypes were identified, distributed among populations such that populations at the extremes of the distribution showed lowest diversity. A large number (487) of AFLP markers were obtained and indicated that diversity levels were highest in the two coastal populations (Cobano, Matapalo, H = 0.23, 0.28 respectively). Population differentiation was low overall, F-ST = 0.12, although Matapalo was strongly differentiated from all other populations (F-ST = 0.16-0.22), apart from Cobano (F., = 0.11). Spatial genetic structure was present in both datasets at different scales: cpDNA was structured at a range-wide distribution scale, whilst AFLP data revealed genetic neighbourhoods on a population scale. In general, the habitat degradation of recent times appears not to have yet impacted diversity levels in mature populations. However, although no data on seed or saplings were collected, it seems likely that reproductive mechanisms in the species will have been affected by land clearance. It is recommended that efforts should be made to conserve the extant genetic resource base and further research undertaken to investigate diversity levels in the progeny generation.

The Fa,b,c conjecture, I

In 1977 a five-part conjecture was made about a family of groups related to trivalent graphs and one part of the conjecture was proved. The conjecture completely determines all finite members of the family. Here we prove another part of the conjecture and foreshadow a paper which completes the proof of the other three parts.

The Fa,b,c conjecture is true, II

In 1977 a five-part conjecture was made about a family of groups related to trivalent graphs and subsequently two parts of the conjecture were proved. The conjecture completely determines all finite members of the family. Here we complete the proof of the conjecture by giving proofs for the remaining three parts. (c) 2006 Elsevier Inc. All rights reserved.