Integrability as a consequence of discrete holomorphicity: the Z_N model

It has recently been established that imposing the condition of discrete holomorphicity on a lattice parafermionic observable leads to the critical Boltzmann weights in a number of lattice models. Remarkably, the solutions of these linear equations also solve the Yang-Baxter equations. We extend this analysis for the Z_N model by explicitly considering the condition of discrete holomorphicity on two and three adjacent rhombi. For two rhombi this leads to a quadratic equation in the Boltzmann weights and for three rhombi a cubic equation. The two-rhombus equation implies the inversion relations. The star-triangle relation follows from the three-rhombus equation. We also show that these weights are self-dual as a consequence of discrete holomorphicity.Comment: 11 pages, 7 figures, some clarifications and a reference adde

Self-avoiding walks and connective constants

The connective constant $\mu(G)$ of a quasi-transitive graph $G$ is the asymptotic growth rate of the number of self-avoiding walks (SAWs) on $G$ from a given starting vertex. We survey several aspects of the relationship between the connective constant and the underlying graph $G$. $\bullet$ We present upper and lower bounds for $\mu$ in terms of the vertex-degree and girth of a transitive graph. $\bullet$ We discuss the question of whether $\mu\ge\phi$ for transitive cubic graphs (where $\phi$ denotes the golden mean), and we introduce the Fisher transformation for SAWs (that is, the replacement of vertices by triangles). $\bullet$ We present strict inequalities for the connective constants $\mu(G)$ of transitive graphs $G$, as $G$ varies. $\bullet$ As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a non-trivial group element is declared to be a further generator. $\bullet$ We describe so-called graph height functions within an account of "bridges" for quasi-transitive graphs, and indicate that the bridge constant equals the connective constant when the graph has a unimodular graph height function. $\bullet$ A partial answer is given to the question of the locality of connective constants, based around the existence of unimodular graph height functions. $\bullet$ Examples are presented of Cayley graphs of finitely presented groups that possess graph height functions (that are, in addition, harmonic and unimodular), and that do not. $\bullet$ The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with arXiv:1304.721

Trees for Food and Timber: are community interests in conflict with those of timber concessions in the Congo Basin?

Much of the Congo Basin is managed for timber from dozens of species. More than 60% also produce non-timber products, including foods. For five multiple use tree species in Cameroon, Democratic Republic of Congo (DRC) and Gabon (Entandrophragma cylindricum, Baillonella toxisperma, Erythrophleum suaveloens, Dacryodes buettneri and Gambeya lacourtiana), we studied gathering and consumption by communities, edible caterpillars hosted, the densities of trees around villages and in concessions and the impacts of timber harvesting . We also studied the consumption of forest foods and the nutritional values of fruits and seeds of various tree species. Villagers walked up to six km during day trips to collect fruits or caterpillars, gathering from concessions if the village was within or near it. When foods were gathered from trees smaller than the cutting diameter (which varied by country and species), there was no conflict with timber harvesting. However, the volume of edible caterpillars hosted increased with diameter and harvestable trees were the most productive. Caterpillars, tree fruits and seeds provide fats, vitamins and minerals that complement agricultural foods. Densities of B. toxisperma, valued for its edible oil, were higher around villages than in concessions. The proportion of commercial trees harvested for timber varied from less than 3% to more than 50%, depending on the species. Different species had different geneflow distances, meaning viable regeneration could be expected with residual adults at different maximum distances. E. cylindricum had more effective dispersal than E. suaveolens. The production of timber and nontimber products can be sustained from the same concessions, for different stakeholders, with appropriate practices and arrangements

Cluster pinch-point densities in polygons

In a statistical cluster or loop model such as percolation, or more generally the Potts models or O(n) models, a pinch point is a single bulk point where several distinct clusters or loops touch. In a polygon P harboring such a model in its interior and with 2N sides exhibiting free/fixed side-alternating boundary conditions, "boundary" clusters anchor to the fixed sides of P. At the critical point and in the continuum limit, the density (i.e., frequency of occurrence) of pinch-point events between s distinct boundary clusters at a bulk point w in P is proportional to _P. The w_i are the vertices of P, psi_1^c is a conformal field theory (CFT) corner one-leg operator, and Psi_s is a CFT bulk 2s-leg operator. In this article, we use the Coulomb gas formalism to construct explicit contour integral formulas for these correlation functions and thereby calculate the density of various pinch-point configurations at arbitrary points in the rectangle, in the hexagon, and for the case s=N, in the 2N-sided polygon at the system's critical point. Explicit formulas for these results are given in terms of algebraic functions or integrals of algebraic functions, particularly Lauricella functions. In critical percolation, the result for s=N=2 gives the density of red bonds between boundary clusters (in the continuum limit) inside a rectangle. We compare our results with high-precision simulations of critical percolation and Ising FK clusters in a rectangle of aspect ratio two and in a regular hexagon and find very good agreement.Comment: 31 pages, 1 appendix, 21 figures. In the second version of this article, we have improved the organization, figures, and references that appeared in the first versio

Exploring Species Limits in Two Closely Related Chinese Oaks

Background. The species status of two closely related Chinese oaks, Quercus liaotungensis and Q. mongolica, has been called into question. The objective of this study was to investigate the species status and to estimate the degree of introgression between the two taxa using different approaches. [br/] Methodology/Principal Findings. Using SSR (simple sequence repeat) and AFLP (amplified fragment length polymorphism) markers, we found that interspecific genetic differentiation is significant and higher than the differentiation among populations within taxa. Bayesian clusters, principal coordinate analysis and population genetic distance trees all classified the oaks into two main groups consistent with the morphological differentiation of the two taxa rather than with geographic locations using both types of markers. Nevertheless, a few individuals in Northeast China and many individuals in North China have hybrid ancestry according to Bayesian assignment. One SSR locus and five AFLPs are significant outliers against neutral expectations in the interspecific FST simulation analysis, suggesting a role for divergent selection in differentiating species.[br/] Main Conclusions/Significance. All results based on SSRs and AFLPs reached the same conclusion: Q. liaotungensis and Q. mongolica maintain distinct gene pools in most areas of sympatry. They should therefore be considered as discrete taxonomic units. Yet, the degree of introgression varies between the two species in different contact zones, which might be caused by different population history or by local environmental factors

Advancing Eucalyptus genomics: identification and sequencing of lignin biosynthesis genes from deep-coverage BAC libraries

<p>Abstract</p> <p>Background</p> <p><it>Eucalyptus </it>species are among the most planted hardwoods in the world because of their rapid growth, adaptability and valuable wood properties. The development and integration of genomic resources into breeding practice will be increasingly important in the decades to come. Bacterial artificial chromosome (BAC) libraries are key genomic tools that enable positional cloning of important traits, synteny evaluation, and the development of genome framework physical maps for genetic linkage and genome sequencing.</p> <p>Results</p> <p>We describe the construction and characterization of two deep-coverage BAC libraries EG_Ba and EG_Bb obtained from nuclear DNA fragments of <it>E. grandis </it>(clone BRASUZ1) digested with <it>Hind</it>III and <it>BstY</it>I, respectively. Genome coverages of 17 and 15 haploid genome equivalents were estimated for EG_Ba and EG_Bb, respectively. Both libraries contained large inserts, with average sizes ranging from 135 Kb (Eg_Bb) to 157 Kb (Eg_Ba), very low extra-nuclear genome contamination providing a probability of finding a single copy gene ≥ 99.99%. Libraries were screened for the presence of several genes of interest <it>via </it>hybridizations to high-density BAC filters followed by PCR validation. Five selected BAC clones were sequenced and assembled using the Roche GS FLX technology providing the whole sequence of the <it>E. grandis </it>chloroplast genome, and complete genomic sequences of important lignin biosynthesis genes.</p> <p>Conclusions</p> <p>The two <it>E. grandis </it>BAC libraries described in this study represent an important milestone for the advancement of <it>Eucalyptus </it>genomics and forest tree research. These BAC resources have a highly redundant genome coverage (> 15×), contain large average inserts and have a very low percentage of clones with organellar DNA or empty vectors. These publicly available BAC libraries are thus suitable for a broad range of applications in genetic and genomic research in <it>Eucalyptus </it>and possibly in related species of <it>Myrtaceae</it>, including genome sequencing, gene isolation, functional and comparative genomics. Because they have been constructed using the same tree (<it>E. grandis </it>BRASUZ1) whose full genome is being sequenced, they should prove instrumental for assembly and gap filling of the upcoming <it>Eucalyptus </it>reference genome sequence.</p

Genetic Structure Among 50 Species of the Northeastern Pacific Rocky Intertidal Community

Comparing many species' population genetic patterns across the same seascape can identify species with different levels of structure, and suggest hypotheses about the processes that cause such variation for species in the same ecosystem. This comparative approach helps focus on geographic barriers and selective or demographic processes that define genetic connectivity on an ecosystem scale, the understanding of which is particularly important for large-scale management efforts. Moreover, a multispecies dataset has great statistical advantages over single-species studies, lending explanatory power in an effort to uncover the mechanisms driving population structure. Here, we analyze a 50-species dataset of Pacific nearshore invertebrates with the aim of discovering the most influential structuring factors along the Pacific coast of North America. We collected cytochrome c oxidase I (COI) mtDNA data from populations of 34 species of marine invertebrates sampled coarsely at four coastal locations in California, Oregon, and Alaska, and added published data from 16 additional species. All nine species with non-pelagic development have strong genetic structure. For the 41 species with pelagic development, 13 show significant genetic differentiation, nine of which show striking FST levels of 0.1–0.6. Finer scale geographic investigations show unexpected regional patterns of genetic change near Cape Mendocino in northern California for five of the six species tested. The region between Oregon and Alaska is a second focus of intraspecific genetic change, showing differentiation in half the species tested. Across regions, strong genetic subdivision occurs more often than expected in mid-to-high intertidal species, a result that may reflect reduced gene flow due to natural selection along coastal environmental gradients. Finally, the results highlight the importance of making primary research accessible to policymakers, as unexpected barriers to marine dispersal break the coast into separate demographic zones that may require their own management plans