Abstract and Generic Rigidity in the Plane

AbstractWe consider the concept of abstract two-dimensional rigidity and provide necessary and sufficient conditions for a matroid to be an abstract rigidity matroid of a complete graph. This characterization is a natural extension of the characterization of graphic matroids due to Graver or Sachs. We also give an example of an abstract rigidity matroid which is not infinitesimal

Blanuša double

A snark is a non-trivial cubic graph admitting no Tait coloring. We examine the structure of the two known snarks on 18 vertices, the Blanuša graph and the Blanuša double. By showing that one is of genus 1, the other of genus 2, we obtain maps on the torus and double torus which are not 4-colorable. The Blanuša graphs appear also to be a counter example for the conjecture that the orientable genus of a dot product of n Petersen graphs is n-1 (Tinsley and Watkins, 1985). We also prove that the 6 known snarks of order 20 are all of genus 2

Algorithms for 3D rigidity analysis and a first order percolation transition

A fast computer algorithm, the pebble game, has been used successfully to study rigidity percolation on 2D elastic networks, as well as on a special class of 3D networks, the bond-bending networks. Application of the pebble game approach to general 3D networks has been hindered by the fact that the underlying mathematical theory is, strictly speaking, invalid in this case. We construct an approximate pebble game algorithm for general 3D networks, as well as a slower but exact algorithm, the relaxation algorithm, that we use for testing the new pebble game. Based on the results of these tests and additional considerations, we argue that in the particular case of randomly diluted central-force networks on BCC and FCC lattices, the pebble game is essentially exact. Using the pebble game, we observe an extremely sharp jump in the largest rigid cluster size in bond-diluted central-force networks in 3D, with the percolating cluster appearing and taking up most of the network after a single bond addition. This strongly suggests a first order rigidity percolation transition, which is in contrast to the second order transitions found previously for the 2D central-force and 3D bond-bending networks. While a first order rigidity transition has been observed for Bethe lattices and networks with ``chemical order'', this is the first time it has been seen for a regular randomly diluted network. In the case of site dilution, the transition is also first order for BCC, but results for FCC suggest a second order transition. Even in bond-diluted lattices, while the transition appears massively first order in the order parameter (the percolating cluster size), it is continuous in the elastic moduli. This, and the apparent non-universality, make this phase transition highly unusual.Comment: 28 pages, 19 figure

The orbit rigidity matrix of a symmetric framework

A number of recent papers have studied when symmetry causes frameworks on a graph to become infinitesimally flexible, or stressed, and when it has no impact. A number of other recent papers have studied special classes of frameworks on generically rigid graphs which are finite mechanisms. Here we introduce a new tool, the orbit matrix, which connects these two areas and provides a matrix representation for fully symmetric infinitesimal flexes, and fully symmetric stresses of symmetric frameworks. The orbit matrix is a true analog of the standard rigidity matrix for general frameworks, and its analysis gives important insights into questions about the flexibility and rigidity of classes of symmetric frameworks, in all dimensions. With this narrower focus on fully symmetric infinitesimal motions, comes the power to predict symmetry-preserving finite mechanisms - giving a simplified analysis which covers a wide range of the known mechanisms, and generalizes the classes of known mechanisms. This initial exploration of the properties of the orbit matrix also opens up a number of new questions and possible extensions of the previous results, including transfer of symmetry based results from Euclidean space to spherical, hyperbolic, and some other metrics with shared symmetry groups and underlying projective geometry.Comment: 41 pages, 12 figure

Internal states of model isotropic granular packings. I. Assembling process, geometry and contact networks

This is the first paper of a series of three, reporting on numerical simulation studies of geometric and mechanical properties of static assemblies of spherical beads under an isotropic pressure. Frictionless systems assemble in the unique random close packing (RCP) state in the low pressure limit if the compression process is fast enough, slower processes inducing traces of crystallization, and exhibit specific properties directly related to isostaticity of the force-carrying structure. The different structures of frictional packings assembled by various methods cannot be classified by the sole density. While lubricated systems approach RCP densities and coordination number z^*~=6 on the backbone in the rigid limit, an idealized "vibration" procedure results in equally dense configurations with z^*~=4.5. Near neighbor correlations on various scales are computed and compared to available laboratory data, although z^* values remain experimentally inaccessible. Low coordination packings have many rattlers (more than 10% of the grains carry no force), which should be accounted for on studying position correlations, and a small proportion of harmless "floppy modes" associated with divalent grains. Frictional packings, however slowly assembled under low pressure, retain a finite level of force indeterminacy, except in the limit of infinite friction.Comment: 29 pages. Published in Physical Review

Carotenoid-Based Colours Reflect the Stress Response in the Common Lizard

Under chronic stress, carotenoid-based colouration has often been shown to fade. However, the ecological and physiological mechanisms that govern colouration still remain largely unknown. Colour changes may be directly induced by the stressor (for example through reduced carotenoid intake) or due to the activation of the physiological stress response (PSR, e.g. due to increased blood corticosterone concentrations). Here, we tested whether blood corticosterone concentration affected carotenoid-based colouration, and whether a trade-off between colouration and PSR existed. Using the common lizard (Lacerta vivipara), we correlatively and experimentally showed that elevated blood corticosterone levels are associated with increased redness of the lizard's belly. In this study, the effects of corticosterone did not depend on carotenoid ingestion, indicating the absence of a trade-off between colouration and PSR for carotenoids. While carotenoid ingestion increased blood carotenoid concentration, colouration was not modified. This suggests that carotenoid-based colouration of common lizards is not severely limited by dietary carotenoid intake. Together with earlier studies, these findings suggest that the common lizard's carotenoid-based colouration may be a composite trait, consisting of fixed (e.g. genetic) and environmentally elements, the latter reflecting the lizard's PSR

Dissecting the Shared Genetic Architecture of Suicide Attempt, Psychiatric Disorders, and Known Risk Factors

Background Suicide is a leading cause of death worldwide, and nonfatal suicide attempts, which occur far more frequently, are a major source of disability and social and economic burden. Both have substantial genetic etiology, which is partially shared and partially distinct from that of related psychiatric disorders. Methods We conducted a genome-wide association study (GWAS) of 29,782 suicide attempt (SA) cases and 519,961 controls in the International Suicide Genetics Consortium (ISGC). The GWAS of SA was conditioned on psychiatric disorders using GWAS summary statistics via multitrait-based conditional and joint analysis, to remove genetic effects on SA mediated by psychiatric disorders. We investigated the shared and divergent genetic architectures of SA, psychiatric disorders, and other known risk factors. Results Two loci reached genome-wide significance for SA: the major histocompatibility complex and an intergenic locus on chromosome 7, the latter of which remained associated with SA after conditioning on psychiatric disorders and replicated in an independent cohort from the Million Veteran Program. This locus has been implicated in risk-taking behavior, smoking, and insomnia. SA showed strong genetic correlation with psychiatric disorders, particularly major depression, and also with smoking, pain, risk-taking behavior, sleep disturbances, lower educational attainment, reproductive traits, lower socioeconomic status, and poorer general health. After conditioning on psychiatric disorders, the genetic correlations between SA and psychiatric disorders decreased, whereas those with nonpsychiatric traits remained largely unchanged. Conclusions Our results identify a risk locus that contributes more strongly to SA than other phenotypes and suggest a shared underlying biology between SA and known risk factors that is not mediated by psychiatric disorders.Peer reviewe

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Fish populations, gill net catches and gill net selectivity in the Zambezi and Chobe Rivers, Namibia, from 1997 to 2000

Hay, C.J., Næsje, T.F., Kapirika, S. Koekemoer, J.H., Strand, R., Thorstad, E.B. & Hårsaker, K. 2002. Fish populations, gill net catches and gill net selectivity in the Zambezi and Chobe Rivers, Namibia, from 1997 to 2000. NINA Project Report 017: 1-88. Objective The objective of this report is to provide baseline information about the fish resources in the Zambezi and Chobe Rivers to form the biological foundation for recommendations for a sustainable management of the fisheries. Based on fish survey data from the period 1997-2000, the fish resources are described through studies of species diversity, relative importance of the different species, life history parameters, catch per unit effort and selectivity of gill nets. Methods Fish were collected in five areas (Katima Mulilo, Kalimbeza, Lake Lisikili, Impalila and Kabula Bula) with survey gill nets (multi-filament, 22–150 mm stretched mesh size) and ten other sampling methods, such as seine nets, cast nets, electrofishing apparatus and rotenone. These are collectively called ”other gears” in this report. The gill nets were used to survey open, deep water habitats (> 1 m) in the main stream near the shore and deep backwater areas with some aquatic vegetation. The other gears targeted mainly small species and juveniles of long-lived species in shallow, vegetated and rocky habitats. Nordic multi mesh sized mono-filament gill nets were included during the survey in 2000 to improve sampling in the deep-water habitats and of the smaller species. Furthermore, fish caught during a fishing competition were sampled in 2000 to include biological data from larger specimens. Data from sampling with Nordic nets and the fishing competition were only included in the life history analyses of selected species, and not in other analyses. The restrictive use of these data ensures comparable data sets with previously reported Okavango River surveys, where these methods were not used (Hay et al. 2000) Surveys were carried out three years in the spring during 1997-2000 and three years in the autumn during 1997-1999. A total of 66875 fish were sampled, 39852 in gill nets, 7005 in Nordic nets, 562 during the fishing competition and 19456 with other gears. The most important species in the survey catches were identified by using an index of relative importance (IRI), which is a measure of the relative abundance or commonness of the species based on number and weight of individuals in catches, as well as their frequency of occurrence. Seventeen of the most important species collected were selected for a more detailed analysis of life history and gill net selectivity. Results A total of 69 fish species were recorded during the surveys, in addition to unidentified Synodontis species. Due to difficulties with the taxonomic classification in the Synodontis spp. group, these species have been pooled, except the easily recognised Synodontis nigromaculatus. Seven Synodontis species have previously been listed for the Zambezi River, thus there may be up to six Synodontis species in the pooled Synodontis spp. group. The fish families represented with the highest number of species were the Cyprinidae and the Cichlidae, with 20 and 17 species, respectively. Six species were considered to be habitat specialists, which means their life history activities are confined to specific habitats, and that they required particular effort and equipment for collection. The habitat specialists recorded were Barbus codringtonii, Nannocharax macropterus, Leptoglanis cf dorae, Clariallabes platyprosopos, Chiloglanis fasciatus and Chiloglanis neumanni. Four of the species were difficult to find, whereas C. platyprosopos was common in its habitat. Low numbers of Barbus kerstenii and Clarias stappersii were caught. These species are, therefore, also considered rare in the Namibian part of the Zambezi/Chobe Rivers. Fourty species were caught in the gill nets (excluding Synodontis spp.). The ten most important species constituted 96 % of the total IRI. The two most important species (Brycinus lateralis and Schilbe intermedius) contributed to 73 % of the total IRI. The Characidae was the most important family in the gill net catches according to IRI (56 %), whereas the Cichlidae family constituted only a small part (2 %). Sixty-seven species were caught with the other gears (excluding Synodontis spp.). The ten most important species constituted 74 % of the total IRI. The two most important species (Tilapia sparmanni and Pharyngochromis acuticeps) contributed to 30 % of the total IRI. In contrast to the gill net catches, the Cichlidae was the most important family in catches with other gears, according to IRI (58 %). The species diversity was higher for the catches with other gears than with gill nets, which is attributed to the flexibility of the other gears, and that a much wider range of habitats was sampled. Thirty-six species were caught with gill nets at Kalimbeza, 33 species at both Kabula-Bula and Lake Lisikili, 28 species at Katima Mulilo and 24 species at Impalila (excluding Synodontis spp). Generally, ranking of the ten most important species in the gill net catches were corresponding at the different stations. When listing the ten most important species according to IRI at the five stations, only 15 species were represented in total. According to IRI, B. lateralis and S. intermedius dominated the gill net catches at all stations, with the exception of the Lake Lisikili, where Petrocephalus catostoma contributed more in number and weight than S. intermedius. Species diversity in the gill net catches measured as the Shannon diversity index differed among stations, with the highest diversity in the Lake Lisikili and the lowest at Katima Mulilo. The year round presence of vegetation and lenthic conditions may have contributed to the high species diversity in the Lake Lisikili. All the other stations included main stream habitats that usually yielded a lower catch and less variability in species. Hydrocynus vittatus was absent at Kabula-Bula in the Chobe River, both in gill net catches and in catches with other gears. The backwater habitat at Kabula-Bula is considered less favourable for H. vittatus. Among the ten most important species according to IRI, nine species were on the list both during high and low water. B. lateralis dominated the gill net catches during both periods. Water level had little effect on the species diversity in the gill net catches. However, three species had a marked decrease in the IRI from the high to the low water period, whereas six species had a marked increase. The body length of the fish caught was up to 92 cm. The modal length of fish caught in gill nets was 8.0-8.9 cm, whereas for fish caught with the other gears 3.0-3.9 cm. Thus, larger fish were caught with gill nets than with other gears, and this was true both for the species combined and for individual species. Twenty of the species caught had a maximum body length of 6 cm or smaller. Of the selected species, twelve species had a minimum length at maturity smaller than 10 cm, two species between 11 and 20 cm and two species larger than 20 cm. The minimum length at maturity was larger than or similar to the smallest fish caught with gill nets in all the selected species, except for both sexes of M. acutidens and males of P. catostoma. The length at 50 % maturity was larger than the minimum length of fish caught with gill nets for all the species of which 50 % maturity could be determined. The 17 species selected for a more detailed data analysis, contributed to 93 % of the biomass of fish caught with gill nets and 56 % of the biomass of fish caught with other gears (one of the selected species was never caught in gill nets). These species represented a large variation in biology, distribution and sizes. Measured as numbers of fish caught per setting, the smaller gill net mesh sizes were the most effective in catching these species. For nine of the species, catch per unit effort in numbers was highest for the 22 or 28 mm mesh size, and for three of the species the 35 mm mesh size. Only two species were most effectively caught in the larger mesh sizes (57 and 73 mm). Measured as weight per setting, larger mesh sizes were nina Project Report 017 more effective; six species were most effectively caught in the 22-28 mm mesh size, five species in the 35-45 mm mesh size and five species in the 57-150 mm mesh size. For all species combined, the 28 mm mesh size was the most effective measured both as numbers of fish caught and weight per setting. The Lake Lisikili station showed the highest catch per unit effort, both in terms of number of fish caught and weight. The lake resembles a large backwater habitat, especially during flood, which may increase the productivity of the area. The lowest catch per unit effort in both number and weight was at Katima Mulilo, where the main stream habitat dominates. Main stream habitats are usually less productive than backwater and floodplain habitats. The results did not show an unambiguous relationship between the catch per unit effort, habitat (mainstream versus backwater) and water level (low water versus high water). Statistical analyses were carried out in all cases where comparable data for all mesh sizes existed, separating the effects of station, habitat and water level. Furthermore, comparisons were made for small mesh sizes (22 to 35 mm) and large mesh sizes (45 to 73 mm) separately, and for catch per unit effort measured in numbers and weight separately. Backwaters had in all cases a significantly higher catch per unit effort than the mainstream - or no differences between backwaters and the mainstream were found. Regarding high and low water, no particular pattern could be seen. Conclusions The results from the surveys in the Zambezi/Chobe Rivers were compared with previous studies in the Okavango River (Hay et al. 2000). Generally, the fish fauna in the Zambezi/Chobe and Okavango Rivers showed great similarities, and there is a considerable overlap in the distribution of species between the rivers. The complex and diverse nature of the fish fauna in the Namibian part of the Upper Zambezi has been revealed through the present surveys. However, detailed knowledge on the biology and behaviour of most of the species are still lacking. Basic information on life history, reproduction, movements, habitat preferences and habitat utilisation of target species is needed to regulate the fishery among the different countries and exploitation methods, and to evaluate the possible benefits of nature reserves and sanctuaries. The Upper Zambezi is presently still relatively undisturbed by human impacts. For that reason alone, this system should be better studied to provide a baseline for future manipulations