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Trigonometric polynomials are usually defined on the lattice of integers.We consider the larger class of weight and root lattices with crystallographic symmetry.This article gives a new approach to minimize trigonometric polynomials, which are invariant under the associated reflection group.The invariance assumption allows us to rewrite the objective function in terms of generalized Chebyshev polynomials. The new objective function is defined on a compact basic semi-algebraic set, so that we can benefit from the rich theory of polynomial optimization.We present an algorithm to compute the minimum: Based on the Hol-Scherer Positivstellensatz, we impose matrix-sums of squares conditions on the objective function in the Chebyshev basis.The degree of the sums of squares is weighted, defined by the root system. Increasing the degree yields a converging Lasserre-type hierarchy of lower bounds.This builds a bridge between trigonometric and polynomial optimization, allowing us to compare with existing techniques.The chromatic number of a set avoiding graph in the Euclidean space is defined through an optimal coloring.It can be computed via a spectral bound by minimizing a trigonometric polynomial. If the to be avoided set has crystallographic symmetry, our method has a natural application.Specifically, we compute spectral bounds for the first time for boundaries of symmetric polytopes.For several cases, the problem has such a simplified form that we can give analytical proofs for sharp spectral bounds.In other cases, we certify the sharpness numerically.

We develop an algorithm for computing the cohomology of complements of toric arrangements. In the case a finite group $$\Gamma $$ Γ is acting on the arrangement, the algorithm determines the cohomology groups as representations of $$\Gamma $$ Γ . As an important application, we determine the cohomology groups of the complements of the toric arrangements associated with root systems of exceptional type as representations of the corresponding Weyl groups.

Understanding the relationship between root systems, soil macropore networks, and soil hydraulic properties is important to better assess ecosystem health. In this study, treatments were performed in forested wetland soils with different vegetation densities, i.e., large (LWa) and small communities (LWb) of reed (Phragmites australis (Cav.) Trin. ex Steud.). At each plot, three undisturbed PVC cylinders (10 cm in diameter and 50 cm in height) were obtained, and X-ray microtomography (μCT) scanning was used to determine the root and macropore architectures. Results showed that the values of total root length and total root volume at LWa were significantly larger than those at LWb (p < 0.05). Imaged macroporosity, macropore volume, macropore length density, macropore node density, macropore branch density, mean macropore surface area, mean macropore diameter, and mean macropore volume at LWa were significantly larger than those at LWb (p < 0.05), whereas mean macropore length, mean macropore branch length, and mean macropore tortuosity at LWb were larger than those at LWa. Total root length and total root volume were positively correlated with soil saturated hydraulic conductivity. Imaged macroporosity, macropore volume, macropore length density, macropore node density, macropore branch density, mean macropore surface area, mean macropore diameter, and mean macropore volume were positively correlated with soil saturated hydraulic conductivity, whereas mean macropore length, mean macropore branch length, and mean macropore tortuosity were negatively correlated with soil saturated hydraulic conductivity. In conclusion, root systems and soil macropore networks constitute a complex synthesis inside soil environments, and together affect soil hydrological responses.

The hydrological connectivity below the soil surface can influence the forest structure and function, especially soil and plant productivity. However, few studies have determined the changes in the hydrological connectivity below the soil surface with increasing soil depth and have quantified the effects of root systems on the hydrological connectivity in forest ecosystems. In this study, we evaluated the index of the hydrological connectivity (IHC) below the soil surface using a field dye tracing method and compared the difference in the index of hydrological connectivity in two subtropical forest stands (i.e., pine trees [SS] and bamboo [ZL]). We analyzed the interactions between the parameters of root system architecture and the index of hydrological connectivity. Back propagation (BP) neural networks were used to quantify which parameter can contribute the most relative importance to the changes of the IHC. The results revealed that the maximum value of the index of hydrological connectivity occurs at the soil surface, and it exhibits a non-linear decreasing trend with increasing soil depth. The parameters of root system architecture (root length, root projected area, root surface area, root volume, and root biomass) were rich in the top soil layers (0–20 cm) in the two sites. Those parameters were positively correlated with the IHC and the root length had the largest positive influence on the hydrological connectivity. Furthermore, we found that root system architecture with different root diameters had different degrees of influence on the index of hydrological connectivity. The very fine root systems (0 < D < 1 mm) had the greatest effect on the hydrological connectivity (p < 0.01). The results of this study provide more information for the assessment of the hydrological connectivity below the soil surface and a better understanding of the effects of root systems in soil hydrology within the rhizosphere.

The identification of genotypes more tolerant to water deficit and more efficient in the use of nutrients little available in the soil is an important low-cost strategy to promote sustainable agriculture in marginalized regions. In this sense, a study was carried out in the SITIS Phenotyping Platform of the Embrapa Arroz e Feijão, with the objective of evaluating the drought tolerance of upland rice genotypes tolerant to phosphorus (P) deficiency. The experimental design was performed in a randomized complete block with split-split plots and two replications. In the plots were established two water regimes (with and without water deficit), in the subplots two soil phosphorus contents (25 and 200 mg dm-3) and in sub-subplots 48 upland rice genotypes. The water deficit had more impact on grain yield than the phosphorus supply. The genotypes AB 062037, AB 062041, AB 062138, Arroz Mato Grosso, BRA 02601, BRA 052045, CNA 4098, CNA 6187, Guapa, Guaporé and Rio Paranaíba were classified in the most productive group under both water regimes. The most productive genotypes under water deficit showed higher root density in the deeper soil layers. The most productive genotypes in the two water regimes were also those that showed the highest transpiration. RESUMO A identificação de genótipos mais tolerantes à deficiência hídrica e mais eficientes no uso de nutrientes pouco disponíveis no solo é uma importante estratégia de baixo custo para promover a agricultura sustentável em regiões marginalizadas. Neste sentido, foi conduzido um estudo na Plataforma SITIS de Fenotipagem da Embrapa Arroz e Feijão, com o objetivo de avaliar a tolerância à deficiência hídrica de genótipos de arroz de terras altas tolerantes à deficiência de fósforo. O delineamento experimental foi blocos ao acaso, com parcelas subsubdivididas, com duas repetições. Nas parcelas foram estabelecidos dois regimes hídricos (com e sem deficiência hídrica), nas subparcelas dois teores de fósforo no solo (25 e 200 mg dm-3) e nas subsubparcelas 48 genótipos de arroz de terras altas. A deficiência hídrica teve mais impacto na produtividade de grãos do que o suprimento de fósforo. Os genótipos AB 062037, AB 062041, AB 062138, Arroz Mato Grosso, BRA 02601, BRA 052045, CNA 4098, CNA 6187, Guapa, Guaporé e Rio Paranaíba foram classificados no grupo mais produtivo sob ambos regimes hídricos. Os genótipos mais produtivos sob deficiência hídrica apresentaram maior densidade do sistema radicular nas camadas mais profundas do solo. Os genótipos mais produtivos nos dois regimes hídricos foram também aqueles que apresentaram maior transpiração.

Cluster structures have been established on numerous algebraic varieties. These lectures focus on the Grassmannian variety and explain the cluster structures on it. The tools include dimer models on surfaces, associated algebras, and the study of associated module categories.

We consider a certain linear combination of zeta-functions of root systems for a root system. Showing two different expressions of this linear combination, we find that a certain signed sum of zeta-functions of root systems is equal to a sum involving Bernoulli functions of root systems. This identity gives a non-trivial functional relation among zeta-functions of root systems, if the signed sum does not identically vanish. This is a generalization of the authors' previous result (Proc. London Math. Soc. 100 (2010), 303–347). We present several explicit examples of such functional relations. We give a criterion of the non-vanishing of the signed sum, in terms of Poincaré polynomials of associated Weyl groups. Moreover we prove a certain converse theorem.

International audience; Sparse interpolation refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. For multivariate functions, the case of the monomial basis is well studied, as is now the basis of exponential functions. Beyond the multivariate Chebyshev polynomial obtained as tensor products of univariate Chebyshev polynomials, the theory of root systems allows to define a variety of generalized multivariate Chebyshev polynomials that have connections to topics such as Fourier analysis and representations of Lie algebras. We present a deterministic algorithm to recover a function that is the linear combination of at most r such polynomials from the knowledge of r and an explicitly bounded number of evaluations of this function.

Coastal areas are at increasing risk from flooding and erosion due to coastal development and climate change. Seagrass meadows, like other coastal ecosystems, can stabilise sediments, indirectly reducing coastal erosion. There is evidence that seagrass can reduce both the erosion of seabeds and lateral erosion of seagrass cliffs. However, these previous studies have only considered seagrass species whose below-ground biomass form dense root mats. This study aimed to investigate whether Enhalus acoroides, a species which does not form root mats, reduces lateral erosion rates of seagrass cliffs. To do this, samples of E. acoroides collected in sandy sediments within Xincun Bay China were transplanted into a wave flume. Lateral erosion rates were measured under exposure to waves for five cliff heights over 1 h. There was an interaction between cliff height and time on the rate of erosion, with higher erosion rates earlier in the experiment for larger cliff heights. The cores with the E. acoroides persisted longer than unvegetated cores, regardless of cliff height, but this result was not assessed statistically due to constraints in the analyses. The below-ground biomass of E. acoroides samples in this study was higher than values previously reported in the literature. Overall, these results suggest that non-dense root-mat forming seagrass species may not provide equivalent sediment stabilisation as dense root-mat-forming species, likely due to the reduced binding structure of the roots in species such as Enhalus.

In most ecosystems, plant roots from different species decompose in mixtures and in the presence of living roots; however much root decomposition research has focused on how roots of individual species or artificial mixtures decompose in the absence of living plants. We thus examined two poorly studied components of root litter decomposition: 1) whether decomposition of root mixtures can be predicted from the sum of the decomposition rates of each component species and 2) how living plants influence rates of root decomposition. Decomposition rates of roots from three perennial herbaceous Mediterranean species grown in monocultures and in two- and three-species mixtures were determined after a one-year incubation period under their living community and in non-vegetated soil (bare soil). Soil respiration in the presence of glucose (substrate induced respiration, SIR) was measured in each plant community and in bare soil. Decomposition rates of root mixtures cannot be predicted from decomposition rates of the component species, both additive and non-additive effects were observed; the presence of low quality roots of Carex humilis in mixtures strongly negatively influenced root decomposition. The presence of living plants stimulated root decomposition in monocultures and two-species communities, likely through an enhanced microbial activity (SIR) under plant communities. This study highlights that root decomposition cannot be predicted from decomposition rates of the component species and is more influenced by endogenous factors or root litter functional composition than by plant community composition.

Made available in DSpace on 2018-12-11T17:30:25Z (GMT). No. of bitstreams: 0 Previous issue date: 2017-05-01 Agence Nationale de la Recherche Centre de Coopération Internationale en Recherche Agronomique pour le Développement Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Universidade de São Paulo Background and aims: Comparing root functioning under contrasting rainfall regimes can help assessing the capacity of plant species to cope with more intense and frequent drought predicted under climate change context. While the awareness of the need to study the whole root system is growing, most of the studies of root functioning through rhizosphere analyses have been restricted to the topsoil. Our study aimed to assess whether the depth in the soil and the rainfall amount affect root functioning, and notably the fate of nutrients within the rhizosphere. Methods: We compared pH and nutrient availability within the rhizosphere and bulk soil along a 4-m deep soil profile in a 5-year-old eucalypt (Eucalyptus grandis) plantation under undisturbed and reduced rainfall treatments. Results: The exchangeable K concentration and the pH of the bulk soil were not influenced by the reduced rainfall treatment. By contrast, the H3O+ concentration in the rhizosphere was significantly greater than that of the bulk soil, only in the reduced rainfall plot. The concentrations of exchangeable K in the rhizosphere were significantly larger than those of the bulk soil in both treatments but this difference was higher in the reduced rainfall plot, notably below the depth of 2 m. Both exchangeable K and H3O+ concentration significantly increased within the rhizosphere in the reduced rainfall treatment at soil depth down to 4 m. Conclusions: The amount of K brought to the roots by mass flow was estimated and could not explain the observed increase in exchangeable K concentration within the rhizosphere. A more likely explanation was root-induced weathering of K-bearing minerals, partly related to enhanced rhizosphere acidification. Our results demonstrate that root functioning can be considerably altered as a response to drought down to large depths. CIRAD UMR Eco&Sols, 2 Place Viala INRA UMR Eco&Sols, 2 Place Viala FCA UNESP LCF ESALQ CENA ESALQ FCA UNESP Agence Nationale de la Recherche: ANR-13-AGRO-0005 FAPESP: 2013/25998-4

Background and Aims Root anchorage function is crucial for tree survival as most trees are exposed to recurrent wind throughout their lifespan. Trees exhibit a large variability of root system architecture (RSA) due genetic and environmental factors. This study aims to understand the links between RSA and tree stability. Methods A 3D biomechanical model was used to simulate tree overturning. To capture the variability of sinker RSA, fourteen virtual root patterns were created from an ensemble average of measured Pinus pinaster root systems. Root virtual patterns and tree-pulling simulations were verified against experimental data. Results The model predicts realistic tree anchorage strength, root stress, and failure patterns. Only a few root components contribute significantly to anchorage strength. The taproot contributes the most to anchorage rigidity, representing 61 % of the anchorage strength. The windward roots failure drives ultimate anchorage failure, representing 25 % of the anchorage strength. Simulations show that root secondary thickening induces higher anchorage rigidity and increases anchorage strength by 58 %. Conclusions This innovative approach appears promising for describing tree stability and its acclimation to external constraints.

A practical and cost-effective destructive sampling method for estimating above and belowground biomass of Corymbia citriodora subspecies variegata grown in plantations is described. The methodology includes details on selecting sample trees, weighing tree components in the field, excavating root systems and obtaining fresh weights and laboratory analyses of components to determine oven dry weights. The development of these sampling procedures is a basic step towards successful, consistent collection of biomass data in 18-20 years old plantation forests. This methodology was developed for eucalypt hardwood timber plantations in Queensland, Australia. However, these procedures can be applied to plantations elsewhere as well as to trees in native forest environments with minor modifications. The methodology developed for field sampling of the tree components and the derivation of allometric relationships for predicting individual tree biomass (above and belowground) highlighted the following:•Accurate quantification of above and belowground biomass of eucalypts.•Description of measured variables for developing allometric relationships.•Integration of field and laboratory measurements to streamline data collection., Graphical abstract Image, graphical abstract

Background Tree allometric equations are critical tools for determining tree volume, biomass and carbon stocks. However, there is a lack of species-specific biomass equations for juvenile trees of many of New Zealand’s indigenous species. The aim of this study was to provide allometric equations for total above- and below-ground biomass and total root biomass and length for eight common evergreen conifer and broadleaved species. Methods In a plot-based field trial, growth metrics of conifers Prumnopitys taxifolia (matai), Agathis australis (kauri), Prumnopitys ferruginea (miro), Podocarpus totara (totara), Dacrycarpus dacrydioides (kahikatea) and Dacrydium cupressinum (rimu) and broadleaved species Alectryon excelsus (titoki) and Vitex lucens (puriri) were measured annually. These species were selected based on their potential role as a long-term solution for mitigating erosion in areas of marginal land proposed for new afforestation/reforestation and as an important carbon (C) sink. Results Root collar diameter (RCD) provided the best fit for tree height, total above-ground biomass (AGB) and total below-ground biomass (BGB), and all regressions were highly significant (P = 0.001). Most species showed significant increases in annual growth and, by year 5, the BGB ranged between 21 and 42% of total biomass and decreased with increasing plant age. Of the conifers, Podocarpus totara had the greatest mean maximum root spread (2.2 m) exceeded only by the broadleaved Vitex lucens (2.5 m). For all species, and in each year of the trial, 100% of the BGB remained confined to within 0.5 m of the ground surface. With the exception of Vitex lucens and Podocarpus totara, > 90% of the total root length remained within a 0.5-m radius of the root bole. The species-specific mean tree biomass of 5-year-old plants ranged from 0.32 to 4.28 kg plant−1. A mixed-species forest established at 1000 stems per hectare (spha), consisting of 200 of each of the best performed of the trialled species, would amass ~ 2.3 t ha−1 of biomass and a forest carbon stock of 3.8 t CO2 ha−1 within 5 years. Conclusions Inter-species differences in the allocation of BGB and AGB appeared to be age dependent. The root-growth metrics of these common indigenous forest species, as candidates for erosion control, have improved our understanding of their potential usefulness for stabilising marginal land. Whole-plant biomass of juvenile trees will greatly improve the accuracy of current estimates of forest carbon stocks for proposed new areas of indigenous afforestation/reforestation.

International audience; This paper shows that for any Lie group G whose Lie algebra L is the split real form of a complex simple Lie algebra, and for any arbitrary root ź, there exists a Cartan decomposition of L, related to ź, which characterizes some controllability properties by using the adjoint orbits of sl(2, ź). For a class of invariant control systems evolving on G, it is proved that the necessary full rank condition for controllability is also sufficient.

Orientador: Rafael Vasconcelos Ribeiro Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Biologia Resumo: As plantas cítricas estão sujeitas à variação sazonal da disponibilidade de água em várias áreas de cultivo e uma estratégia eficaz para amenizar os impactos da escassez de água na citricultura é a utilização de porta-enxertos tolerantes à seca. É importante entender quais características fazem alguns porta-enxertos mais tolerantes que outros para que espécies alternativas com características fenotípicas e fisiológicas semelhantes sejam selecionadas. A redistribuição hidráulica é o movimento passivo de água nas raízes seguindo um gradiente de potencial da água, sendo a água absorvida por raízes em solo úmido e liberada em tecidos em contato com solo seco. Outra estratégia fisiológica é o ajuste osmótico, caracterizado pelo acúmulo ativo de solutos na célula em resposta à redução da disponibilidade de água no ambiente. A redução do potencial osmótico facilita a extração da água em solos secos e mantém o turgor das células. Nesse contexto, os objetivos deste trabalho foram avaliar a redistribuição hidráulica em duas espécies de porta-enxerto, bem como verificar se o ajuste osmótico está associado a este fenômeno. Laranjeiras Valência foram enxertadas com limoeiro Cravo e citrumeleiro Swingle de tal forma que as mudas formadas possuíssem dois sistemas radiculares de porta-enxertos diferentes ou de um mesmo porta-enxerto para simular uma situação em que apenas parte das raízes tinha disponibilidade de água enquanto a outra parte estava sob déficit hídrico. Os experimentos permitiram constatar que as duas espécies realizavam redistribuição de água e que o Cravo possui maior capacidade de enviar água para o lado seco da raiz. A maior habilidade de redistribuir água do limoeiro Cravo em relação ao citrumeleiro Swingle foi associada à maior absorção de água pelas raízes úmidas ocasionada pela diminuição do potencial osmótico. As copas de laranjeira Valência enxertadas sobre limoeiro Cravo também apresentaram sensibilidade estomática à variação da disponibilidade de água, sugerindo que as raízes sob seca sinalizaram para que o fechamento estomático ocorresse mesmo havendo suprimento de água pela parte úmida e a folha estando hidratada. As trocas gasosas da copa enxertada sobre o citrumeleiro Swingle não foram significativamente afetadas pela redução da disponibilidade hídrica, permanecendo com os estômatos abertos durante o período experimental. Quando comparados os porta-enxertos, as modificações fisiológicas e mecanismos descritos levaram à maior resistência do limoeiro Cravo à seca Abstract: Citrus plants are subject to a seasonal variation of water availability in many growing areas, an effective strategy to mitigate the impacts of citrus water scarcity is the use of drought tolerant rootstocks. It is important to understand which characteristics make some rootstocks more tolerant than others so that alternative rootstocks with similar phenotypic and physiological characteristics are selected. Hydraulic redistribution is a passive movement of water in roots following a gradient of water potential, in moist soil sites water is uptaken by roots and then released in tissues in contact with drier soil. Osmotic adjustment is another physiological strategy, characterized by an active cell accumulation of solutes in response to a reduction of water availability in the environment. A reduction in osmotic potential facilitates the extraction of water in dry soils and maintains the cells turgor. In this context, the aims of this work were to evaluate the hydraulic redistribution in two rootstock species, as well as to verify if the osmotic adjustment is associated with this phenomenon. Seedlings of Valencia sweet orange were inarched with Rangpur lime and Swingle citrumelo rootstocks, each seedling had two root systems to simulate a situation where only part of the roots had water availability while the other part was under water limiting conditions. The experiments allowed us to verify that both species redistributed water and that Rangpur lime has greater capacity to send water to the roots under drought. The greater ability to redistribute water of Rangpur lime in comparison to Swingle citrumelo was associated to a greater absorption of water by the moist roots caused by a decrease in osmotic potential. Valencia sweet orange canopies grafted on Rangpur lime also showed a stomatal sensitivity to the variation of water availability, suggesting that roots under drought signaled for stomatal closure even though water was supplied by the moist part and leaves were hydrated. The gas exchanges of the canopy grafted on Swingle citrumelo were not significantly affected by the reduction of water availability, keeping the stomata open during the experimental period. When comparing rootstocks, physiological modifications and mechanisms described led to a greater drought tolerance of Rangpur lime Mestrado Biologia Vegetal Mestra em Biologia Vegetal FAPESP 2016/02199-7 CNPQ 131126/2016-4

Orientador: Marcelo Menossi Teixeira Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Biologia Resumo: A cana-de-açúcar é uma cultura de importância econômica mundial, atuando como matéria-prima principalmente na produção de açúcar e etanol. A cultura de cana é afetada por vários tipos de estresses bióticos e abióticos, sendo a seca um dos principais fatores que reduzem a sua produtividade. A seca atua sobre as variedades vegetais de diversas formas, provocando alterações na parte aérea e no sistema radicular. As raízes desempenham um papel importante na manutenção das funções da planta em condições de estresse, de modo que suas propriedades exercem grande influência na resposta à escassez hídrica. Entretanto, o conhecimento acerca das alterações da expressão gênica em raízes de cana sujeitas à seca ainda é muito restrito, embora seja uma etapa importante para a identificação de genes com potencial para a produção de plantas mais tolerantes ao deficit hídrico. Deste modo, este projeto objetivou a identificação de genes modulados por seca em raízes de cana para estudos funcionais empregando plantas transgênicas de Arabidosis thaliana e cana-de-açúcar. Utilizando dados de RNA-Seq de dois cultivares de cana (RB92579 e RB72454) foram selecionados 10 genes potencialmente relevantes nas respostas radiculares frente à condições de seca. Testes de estresse severo e moderado utilizando plantas transgênicas de Arabidopsis e cana-de-açúcar superexpressando alguns desses genes apontaram o envolvimento no mecanismo de tolerância à seca Abstract: Sugarcane is a crop of global economic importance, acting as raw material especially in the production of sugar and ethanol. Sugarcane is affected by various types of biotic and abiotic stresses, and drought is one of the main factors reducing sugarcane productivity. Drought acts on the plant varieties of several ways, causing alterations in shoots and roots. Roots play an important role in the maintenance of plant functions under stress conditions, and their properties have great influence on responses to water scarcity. However, the knowledge on gene expression changes in plant roots subject to drought is still very restricted, although it is an important step for the identification of genes with potential for the production of plants more tolerant to water deficit. Therefore, this project aims to identify genes modulated by drought in sugarcane roots for functional studies using Arabidopsis thaliana and sugarcane transgenic plants. Using RNA-Seq data from two sugarcane cultivars (RB92579 and RB72454) we selected 10 potentially relevant genes in root responses in drought conditions. Severe and moderate stress tests using Arabidopsis and sugarcane transgenic plants overexpressing some of these genes indicated the involvement in drought tolerance mechanisms Doutorado Genética Vegetal e Melhoramento Doutora em Genética e Biologia Molecular FAPESP 2014/26521-0

Carbon isotope composition in respired CO2 and organic matter of individual organs were measured on peanut seedlings during early ontogeny in order to compare fractionation during heterotrophic growth and transition to autotrophy in a species with lipid seed reserves with earlier results obtained on beans. Despite a high lipid content in peanut seeds (48 %) compared with bean seeds (1.5 %), the isotope composition of leaf- and root-respired CO2 as well as its changes during ontogeny were similar to already published data on bean seedlings: leaf-respired CO2 became C-13-enriched reaching -21.5 parts per thousand, while root-respired CO2 became C-13-depleted reaching around -31 parts per thousand at the four-leaf stage. The opposite respiratory fractionation in leaves vs. roots already reported for C-3 herbs was thus confirmed for peanuts. However, contrarily to beans, the peanut cotyledon-respired CO2 was markedly C-13-enriched, and its C-13-depletion was noted from the two-leaf stage onwards only. Carbohydrate amounts being very low in peanut seeds, this cannot be attributed solely to their use as respiratory substrate. The potential role of isotope fractionation during glyoxylate cycle and/or gluconeogenesis on the C-13-enriched cotyledon-respired CO2 is discussed.

We classify all non-abelian groups G such that there exists a pair (V,W) of absolutely simple Yetter-Drinfeld modules over G such that the Nichols algebra of the direct sum of V and W is finite-dimensional under two assumptions: the square of the braiding between V and W is not the identity, and G is generated by the support of V and W. As a corollary, we prove that the dimensions of such V and W are at most six. As a tool we use the Weyl groupoid of (V,W)., Comment: 21 pages. Final version. Accepted for publication in Journal of Group Theory

28 pags.- 4 Tabls.- 4 Figs. The definitive version is available at: https://www.sciencedirect.com/science/journal/03418162, The accurate measurement of soil infiltration rates is a challenging task in hydrological studies, as they are governed by complex interactions between the plant and soil components. To explore the effects of plant roots and soil moisture (SM) on infiltration process, an automatic measurement system based on a point source device was used to determine infiltration rates at different cultivated grasslands (leguminous, gramineous and mixed) in an arid region of China. The results showed that SM and plant roots significantly affected the infiltration process at different stages in the three grasslands. The soil infiltration rates were negatively correlated to SM, but a positive correlation was observed with the below-ground biomass (BGB). SM was the main factor influencing the infiltration rates in the initial stage of infiltration, whereas both SM and BGB determined the steady-state infiltration rates. The influence of the combined effects of SM and plant roots decreased over infiltration time. The decreased rate of determining coefficient showed that SM was more influent than plant roots. These results may contribute to interpret the observed higher soil infiltration capacity in the leguminous grasslands than in the gramineous and the mixed grasslands, and were conducive to better understand the effects of plant and soil properties on infiltration processes in arid areas., This research was funded by the Projects of the National Natural Science Foundation of China (NSFC 41722107), the Light of West China Program of the Chinese Academy of Sciences (XAB2015A04; XAB2018B09), the Fundamental Research Funds for the Central Universities (2452018086; 2452019187), and the Youth Talent Plan Foundation of the Northwest A&F University (2452018025).

Moringa oleifera Lam. (Moringa), Jatropha curcas L. (Jatropha), and Ricinus communis L. (Ricinus) are oily species known by their capability to grow in tropical and subtropical lands. However, there are no studies comparing their growth and recovery capabilities after root pruning and transplant. The purpose of this research was to compare and analyze propagation, growth, and recovery performance of these species after root pruning and transplant. We sowed 100 seeds per species and monitored their survival and growth during a 63-day period, after this, we uprooted the plants and pruned their roots 4.0 cm from their base and transplanted them. We monitored their recovery over 83 days, and then uprooted plants and measured above- and belowground data, digitized their roots in three dimensions, and calculated biomass fractions. With this information, we established allometric equations to estimate biomass fractions and root distribution models. Results indicated that Ricinus had the highest propagation capabilities. Jatropha and Ricinus had similar recovery after root pruning and transplant. Moringa had the lowest propagation and recovery from transplant. Concerning belowground data, root pruning increased root density more than three times in Moringa, four times in Ricinus, and six times in Jatropha. Nevertheless, the three species maintained natural root trays. Ricinus had the longest and thinnest roots and the highest number of branches, followed by Jatropha, and finally Moringa, with the smallest quantity and the shortest and thickest roots. We concluded that the three species recovered well from root pruning and transplant, with improved root structure upon applying these practices.

If x and y are roots in the root system with respect to the standard (Tits) geometric realization of a Coxeter group W, we say that x dominates y if for all w ∈ W , wy is a negative root whenever wx is a negative root. We call a positive root elementary if it does not dominate any positive root other than itself. The set of all elementary roots is denoted by E . It has been proved by B. Brink and R.B. Howlett [B. Brink, R.B. Howlett, A finiteness property and an automatic structure of Coxeter groups, Math. Ann. 296 (1993) 179–190] that E is finite if (and only if) W is a finite-rank Coxeter group. Amongst other things, this finiteness property enabled Brink and Howlett to establish the automaticity of all finite-rank Coxeter groups. Later Brink has also given a complete description of the set E for arbitrary finite-rank Coxeter groups [B. Brink, The set of dominance-minimal roots, J. Algebra 206 (1998) 371–412]. However the set of non-elementary positive roots has received little attention in the literature. In this paper we answer a collection of questions concerning the dominance behavior between such non-elementary positive roots. In particular, we show that for any finite-rank Coxeter group and for any non-negative integer n, the set of roots each dominating precisely n other positive roots is finite. We give upper and lower bounds for the sizes of all such sets as well as an inductive algorithm for their computation.

For certain Lie algebras g, we can use a Z/5Z-grading and define a quartic form and a skew-symmetric bilinear form on the degree 1 component, g_1, thereby constructing a Freudenthal triple system. The structure of the Freudenthal triple system is examined using root system methods available in the Lie algebra context. In the cases g = E_8 (where g_1 is the minuscule representation of E_7) and g = D_4, we determine the groups stabilizing the quartic form and both the quartic and bilinear forms., Comment: 28 pages, no figures

We exploit the structure of the critical orbital sets of symmetry classes of tensors associated to sign uniform partitions and we establish new connections between symmetry classes of tensors, matchings on bipartite graphs and coding theory. In particular, we prove that the orthogonal dimension of the critical orbital sets associated to single hook partitions λ = ( w , 1 n - w ) equals the value of the coding theoretic function A ( n , 4 , w ) . When w = 2 we reobtain this number as the independence number of the Dynkin diagram A n - 1 .

In this paper we show that for a simply-laced root system a choice of a Coxeter element C gives rise to a natural construction of the Dynkin diagram, in which vertices of the diagram correspond to C-orbits in R; moreover, it gives an identification of R with a certain subset Iˆ of I×Z2h, where h is the Coxeter number. The set Iˆ has a natural quiver structure; we call it the periodic Auslander–Reiten quiver. This gives a combinatorial construction of the root system associated with the Dynkin diagram I: roots are vertices of Iˆ, and the root lattice and the inner product admit an explicit description in terms of Iˆ. Finally, we relate this construction to the theory of quiver representations.

The plethysms of the Weyl characters associated to a classical Lie group by the symmetric functions stabilize in large rank. In the case of a power sum plethysm, we prove that the coefficients of the decomposition of this stabilized form on the basis of Weyl characters are branching coefficients which can be determined by a simple algorithm. This generalizes in particular some classical results by Littlewood on the power sum plethysms of Schur functions. We also establish explicit formulas for the outer multiplicities appearing in the decomposition of the tensor square of any irreducible finite-dimensional module into its symmetric and antisymmetric parts. These multiplicities can notably be expressed in terms of the Littlewood–Richardson coefficients.

To be able to use oxygen control in the root environment as cultivation tool, it is necessary to obtain representative profiles of the oxygen content within substrates. It was the aim of the current project to study whether it is possible to obtain oxygen profiles within substrates using a limited amount of oxygen measurements in the root environment, which are subsequently coupled to a model. In this model, correlations between oxygen and some other growth conditions should be described. Next, it was studied whether this technology can be used to develop a measurement tool for practical use. The following results were obtained: A multi-oxygen sensor was constructed for determination of large numbers of oxygen measurements, which can be simultaneously measured at different locations in growing media (stone wool). Data has been generated, both from measurements at the laboratory and in the green house during growth of tomatoes. Besides oxygen, several other parameters were measured, which may be related to oxygen contents, such as water content, light, temperature, EC, RV. Data sets were analyzed, and a concept model was developed, that describes relationships between the oxygen and other parameters. Our results indicate that it is possible to develop an oxygen measuring strategy, which can be used to determine a representative pattern of oxygen levels in growing media. Also, it seems likely that a sensor can be constructed (i.e., the "software oxygen sensor"), which contains besides sensor hardware, also a (software) model, which describes relationships between different growth conditions.

Highlight Root hairs anchored maize roots to the sides of pore walls, helping root tips to penetrate soil at intermediate soil bulk densities, shown by comparing wild-type seedlings with hairless mutants., The physical role of root hairs in anchoring the root tip during soil penetration was examined. Experiments using a hairless maize mutant (Zea mays: rth3–3) and its wild-type counterpart measured the anchorage force between the primary root of maize and the soil to determine whether root hairs enabled seedling roots in artificial biopores to penetrate sandy loam soil (dry bulk density 1.0–1.5g cm−3). Time-lapse imaging was used to analyse root and seedling displacements in soil adjacent to a transparent Perspex interface. Peak anchorage forces were up to five times greater (2.5N cf. 0.5N) for wild-type roots than for hairless mutants in 1.2g cm−3 soil. Root hair anchorage enabled better soil penetration for 1.0 or 1.2g cm−3 soil, but there was no significant advantage of root hairs in the densest soil (1.5g cm−3). The anchorage force was insufficient to allow root penetration of the denser soil, probably because of less root hair penetration into pore walls and, consequently, poorer adhesion between the root hairs and the pore walls. Hairless seedlings took 33h to anchor themselves compared with 16h for wild-type roots in 1.2g cm−3 soil. Caryopses were often pushed several millimetres out of the soil before the roots became anchored and hairless roots often never became anchored securely.The physical role of root hairs in anchoring the root tip may be important in loose seed beds above more compact soil layers and may also assist root tips to emerge from biopores and penetrate the bulk soil.

© 2015 Elsevier B.V. As significant indicators of deep-time palaeoclimate, a number of new palaeontological, pedological and geochemical characteristics are provided for the Chemnitz Fossil Lagerstätte to depict more precisely its environmental conditions. For the first time, several lines of evidence indicate that this fossil forest, instantaneously preserved by volcanic deposits, once received an annual precipitation of around 800-1100 mm, but grew on a nearly unweathered palaeosol. Although the composition of this rich and diverse T0 assemblage suggests a hygrophilous, dense and multi-aged vegetation dominated by conservative lineages, the habitat was affected by environmental disturbances and pronounced seasonality. Repeated changes in local moisture availability are suggested by geochemical proxies, the co-occurrence to intergrowth of calcic and ferric glaebules in the palaeosol and developmental traits of perennial vegetational elements. Specific substrate adaptation is reflected by different root systems and cyclic growth interruptions recorded in the stems, branches and roots of long-lived woody plants. Many differentially adapted terrestrial animals complete the more comprehensive reconstruction of a late Sakmarian ecosystem and its climatic and preservational controls. Albeit spatially confined, this diverse in-situ record may contribute to understand wetland-dryland dynamics of sub-tropical Northern Hemisphere Pangaea.

Orientador: Rafael Silva Oliveira Tese (doutorado) Universidade Estadual de Campinas, Instituto de Biologia Resumo: As mudanças climáticas influenciam a distribuição e o volume de chuvas ao longo do ano, o que pode provocar a intensificação de secas extremas nas regioes tropicais. As espécies vegetais existentes em áreas montanhosas estão entre as mais vulneráveis aos efeitos das mudanças climáticas, com possibilidade de redução na sua distribuição, migração para regiões mais altas, ou mesmo desaparecimento. Neste estudo, avaliamos a diversidade de respostas ecofisiológicas de cinco espécies de plantas de campos de altitude submetidas a um evento extremo de seca. Testamos a hipótese de que a profundidade do sistema radicular e o grau de iso ou anisohidria são bons preditores da fotossíntese, do status hídrico, da dinâmica de crescimento e fenológica de plantas expostas a condições contrastantes de disponibilidade de água. Para avaliar essas hipóteses, simulamos condições de seca em plantas de campos de altitude mediante experimentos de exclusão de água da chuva em campos de altitude na Serra da Mantiqueira, Mata Atlântica, Brasil. Ao longo de 22 meses, monitoramos variáveis micrometeorológicas e dinâmica de água no solo e avaliamos os efeitos da exclusão de água no potencial hídrico, trocas gasosas, dinâmica de crescimento e fenológica das plantas. As espécies responderam de maneiras distintas às mudanças na disponibilidade de água, formando um continuum de variação quanto à regulação do potencial hídrico, da fotossíntese, e da dinâmica de crescimento e fenológica. Encontramos que a profundidade do sistema radicular e as estratégias hidráulicas iso e anisohídricas foram bons preditores das mudanças temporais da fotossíntese e status hídrico das plantas, pois as espécies de raízes mais profundas isohídricas mantiveram-se mais hidratadas, com maiores taxas fotossintéticas e abertura estomática durante a seca do que as plantas de raízes superficiais isohídricas e anisohídricas. Também observamos que a profundidade do sistema radicular e as estratégias hidráulicas iso e anisohídricas foram bons preditores de respostas à seca quanto ao crescimento e a fenologia, sendo que espécies isohídricas de raízes superficiais e profundas foram as primeiras a apresentar redução na dinâmica de crescimento e reprodução, enquanto as espécies anisohídricas reduziram a produção de folhas de forma mais acentuada. A espécie anisohídrica e de raiz superficial, Myrsine ferruginea, foi a única a apresentar mortalidade significativa (30% dos indivíduos) nas parcelas de exclusão de chuva. Concluímos que plantas com sistema radicular superficial e anisohídricas foram mais sensíveis que as espécies isohídricas de raízes superficiais e profundas, pois são mais propensas a sofrerem mortalidade por falhas hidraúlicas. A profundidade do sistema radicular e o grau de regulação estomática representam bons preditores do grau de resistência das espécies de campo de altitude a secas extremas Abstract: Intensification of extreme droughts are predicted for tropical regions in the coming decades. Plant species in tropical mountains are amongst the most vulnerable to climate change because they can have their distribution reduced, be displaced to higher altitudes or even go extinct. In this study, we evaluated the diversity of ecophysiological responses in five plant species from "campos de altitude" (montane grasslands) subjected to an extreme drought event. We tested the hypothesis that rooting depth and stomata regulation capacity (the iso or anisohydric hydraulic strategies) are good predictors of ecophysiological, phenological and growth performance of plants when subjected to lower water availability. In order to test this hypothesis, we established a rainfall exclusion experiment to simulated drier conditions for plants in campos de altitude in Serra da Mantiqueira, Mata Atlântica, Brazil. We evaluated the effects of water exclusion on plant water potential, gas exchange, phenology, and growth dynamics. We found a continuum of drought responses in the campos de altitude community, Rooting depth and stomatal regulation capacity were good predictors of photosynthesis and water status of plants, as isohydric species with deeper roots maintained higher water potential and photosynthetic rates throughout the drought. Isohydric species with both shallow and deep roots were the first ones to reduce their growth and reproduction. The anisohydric and shallow rooted species, M. ferruginea, was the only species that presented significant morality (30%) in the rainfall exclusion plots. We concluded that anisohydric and shallow rooted plants are more sensitive and more likely to die of hydraulic failure than isohydric species with shallow and deep roots. Rooting depth and stomatal regulation capacity are good traits to predict the sensitivity of campos de altitude species to drought Doutorado Biologia Vegetal Doutora em Biologia Vegetal CAPES 3354-13-0 FAPESP 10/17204-0 CNPQ 160710/2011-1

Although most of the symmetry groups or ''invariance groups'' associated with two term transformations between (basic) hypergeometric series have been studied and identified, this is not the case for the most general transformation formulae in the theory of basic hypergeometric series, namely Bailey's transformations for @f910-series. First, we show that the invariance group for both Bailey's two term transformations for terminating @f910-series and Bailey's four term transformations for non-terminating @f910-series (rewritten as a two term transformation of a so-called @F-series) is isomorphic to the Weyl group of type E"6. We continue our recent research concerning the group structure underlying three term transformations [S. Lievens, J. Van der Jeugt, Invariance groups of three term transformations for basic hypergeometric series, J. Comput. Appl. Math. 197 (2006) 1-14] and demonstrate that the group associated with a three term transformation between these @F-series, each admitting Bailey's two term transformation, is the Weyl group of type E"7. We do this by giving a description of the root system of type E"7 that allows to find a transformation between equivalent three term identities in an easy way. A computation shows that there are five, essentially different, three term transformations between these @F-series; we give an explicit form of each of these five transformations in an elegant way. To our knowledge only one of these transformations has appeared in the literature.

We complete the classification of the irreducible 2F-modules for quasi-simple groups by dealing with the two open cases for sporadic groups from [R.M. Guralnick, G. Malle, Classification of 2F-modules, I, J. Algebra 257 (2002) 348–372]. The techniques developed involve abelian sets of roots within root systems, and enable us to determine the 2-ranks of the Monster, the Baby Monster and the double cover of the latter.

Dynkin has shown how subsystems of real root systems may be constructed. As the concept of subsystems of complex root systems is not as well developed as in the real case, in this paper we give an algorithm to classify the proper subsystems of complex proper root systems. Furthermore, as an application of this algorithm, we determine the proper subsystems of imprimitive complex proper root systems. These proper subsystems are useful in giving combinatorial constructions of irreducible representations of properly generated finite complex reflection groups.

Let @[email protected]?Z^n denote one of the classical irreducible root systems A"n"-"1, B"n, C"n and D"n, and write @F^(^+^) for the configuration consisting of all positive roots of @F together with the origin of R^n. Gelfand, Graev and Postnikov, in: V.I. Arnold, I.M. Gelfand, M. Smirnov, V.S. Retakh (Eds.), Arnold-Gelfand, Mathematics Seminars, Geometry and Singularity Theory, Birkhauser, Boston, 1997, pp. 205-221 showed that by constructing an explicit unimodular triangulation, the normalized volume of the convex hull of A"n"-"1^(^+^) is equal to the Catalan number. On the other hand, Fong (Triangulations and Combinatorial Properties of Convex Polytopes, Dissertation, MIT Press, Cambridge, MA, 2000) computed the normalized volume of the convex hull of each of the configurations B"n^(^+^), C"n^(^+^) and D"n^(^+^). Moreover, the normalized volume of the convex hull of the subconfiguration of A"n"-"1^(^+^) arising from a complete bipartite graph was computed by Ohsugi and Hibi (Illinois J. Math. 44 (2000) 391) and Fong. The purpose of the present paper is, via the theory of Grobner bases of toric ideals and triangulations, to compute the normalized volume of the convex hull of each of the subconfigurations of B"n^(^+^), C"n^(^+^) and D"n^(^+^) arising from a complete bipartite graph.

We study multiple zeta values (MZVs) from the viewpoint of zeta-functions associated with the root systems which we have studied in our previous papers. In fact, the $r$-ple zeta-functions of Euler-Zagier type can be regarded as the zeta-function associated with a certain sub-root system of type $C_r$. Hence, by the action of the Weyl group, we can find new aspects of MZVs which imply that the well-known formula for MZVs given by Hoffman and Zagier coincides with Witten's volume formula associated with the above sub-root system of type $C_r$. Also, from this observation, we can prove some new formulas which especially include the parity results of double and triple zeta values. As another important application, we give certain refinement of restricted sum formulas, which gives restricted sum formulas among MZVs of an arbitrary depth $r$ which were previously known only in the cases of depth $2,3,4$. Furthermore, considering a sub-root system of type $B_r$ analogously, we can give relevant analogues of the Hoffman-Zagier formula, parity results and restricted sum formulas., Comment: 27 pages

International audience; Stand water balance was calculated in a floodplain forest in Southern Moravia. A model was applied to actual and theoretical scenarios of climate and soil water supply. Rooted and root-free soil volumes were considered separately because root development was poor for these trees. Input data, i.e., measured flows within the system including sap flow rate, characterize both the period of regular natural floods and period when floods were interrupted for over 20 years because of canalization of rivers in the region. Under non-limiting underground water supply, net precipitation supplied only about 50 and 25% of water for actual evapotranspiration, $E_{\rm T}$, under mild and dry weather, respectively, and the other 30 and 60% came from underground sources. The model also characterizes the theoretical situation of no underground water supply, when $E_{\rm T}$ may decrease significantly. An important limit for water supply to the trees may be the maximum hydraulic conductivity $K$, allowing horizontal transport in heavy soils, because a small decrease in soil water content (2 to 4% $_{{\rm vol}}$) causes a large drop in $K. K$ may become supply-limiting before soil water potential becomes a limiting factor. Trees with smaller or damaged root systems or lower root/shoot ratio were especially threatened by drought even on relatively moist heavy soils.; Bilan hydrique d'une forêt dans une plaine alluviale du Sud de la Moravie en conditions naturelles ou sous des régimes hydriques modifiés : ses conséquences écologiques. Le bilan hydrique a été calculé dans des peuplements situés dans une plaine alluviale du Sud de la Moravie. Un modèle a été appliqué à des scénarios actuels ou théoriques de climat et d'alimentation en eau. Les volumes de sol, avec et sans racines, ont été pris en compte séparément du fait que le développement racinaire de ces arbres était faible. Les valeurs entrées, par exemple les flux mesurés dans le système incluant le flux de sève, caractérisent l'ensemble des périodes, celles des inondations naturelles régulières et celles durant lesquelles les inondations furent interrompues pendant plus de 20 ans du fait de la canalisation des rivières de cette région. Sous conditions non limitatives d'alimentation en eau du sol, les précipitations nettes fournissent seulement environ 50 et 25 % de l'eau pour l'évaporation réelle, $E_{\rm T}$, sous un temps moyen et sec, respectivement, et les autres 30 et 60 % ont pour origine l'eau du sol. Le modèle caractérise aussi les situations théoriques d'une absence d'alimentation en eau provenant du sol, lorsque $E_{\rm T}$ peut décroître significativement. Un facteur limitant important pour l'alimentation en eau des arbres peut être le maximum de la conductivité hydraulique $K$, permettant un transport latéral dans les sols lourds. À cause de la faible diminution de la teneur en eau (2 à 4 % $_{{\rm vol}}$) cela provoque une forte chute dans la valeur de $K. K$ peut devenir facteur limitant de l'alimentation avant que le potentiel hydrique ne devienne le facteur limitant. Les arbres ayant un système racinaire réduit ou endommagé ou bien avec un rapport racine/pousse faible, étaient spécialement menacés par la sécheresse, même sur les sols lourds relativement humides.

O objetivo deste trabalho foi avaliar o efeito da compactação do solo na anatomia da raiz e no desenvolvimento de duas cultivares de soja (Glycine max (L.) Merrill), IAC-8 e IAC-14, em vasos. O solo utilizado foi um Latossolo Roxo, tirado da camada superficial (0-200 mm). Os níveis de compactação consistiram de 0,95 (controle), 1,20, 1,35 e 1,50 kg L-1, e foram obtidos pela compactação de determinada massa de solo por prensa hidráulica, em cilindros de ferro. A compactação do solo não afetou a estrutura anatômica da raiz, em nenhuma das duas cultivares. Na cultivar IAC-8, o número de folhas, a altura das plantas e o peso da matéria seca da raiz e do caule diminuíram com o aumento da densidade do solo, e o comprimento da raiz aumentou. Na cultivar IAC-14, o número de folhas, a área foliar, o número de vagens, a altura das plantas, o peso da matéria seca da raiz, do caule e das folhas foram diminuindo à medida que aumentava a densidade do solo, embora nas densidades intermediárias (1,20 e 1,35) se tenha notado uma tendência de aumento naquelas variáveis. A produção de soja não foi afetada pela compactação do solo, com exceção do número de vagens da cultivar IAC-14, que apresentou diferenças em razão do tratamento. Os resultados sugerem que as duas cultivares podem ser bem adaptadas a solos compactados, nos níveis analisados. The present study aimed at evaluating the effect of soil compaction on the root anatomy and plant development of two soybean (Glycine max (L.) Merrill) cultivars, IAC-8 and IAC-14, in pots. A clayey soil collected, from a superficial (0-200 mm) layer was used. The compaction levels consisted of 0.95 (control), 1.20, 1.35 and 1.50 kg L-1, which were obtained by compaction of some soil mass with a hydraulic press inside of iron cylinders. The soil compaction did not affect the anatomic structure of the roots, in any of the two cultivars. In the IAC-8, the leaf number, plant height and the dry matter of roots and stem decreased with the increase of soil density, and the root length was increased. In the IAC-14, the leaf number, foliar area, pod number, plant height, and weight of the dry matter of roots, stem and leaves decreased as the soil density increased, except for the intermediate densities (1.20 and 1.35), in which a trend to increase was observed in these variables. The soybean production was not affected by soil compaction, except for pod number in IAC-14, which showed differences as a function of the treatment. The results suggest that both cultivars can be well adapted to compacted soils in the analysed levels of compaction.

Real physical systems with reflective and rotational symmetries such as viruses, fullerenes and quasicrystals have recently been modeled successfully in terms of three-dimensional (affine) Coxeter groups. Motivated by this progress, we explore here the benefits of performing the relevant computations in a Geometric Algebra framework, which is particularly suited to describing reflections. Starting from the Coxeter generators of the reflections, we describe how the relevant chiral (rotational), full (Coxeter) and binary polyhedral groups can be easily generated and treated in a unified way in a versor formalism. In particular, this yields a simple construction of the binary polyhedral groups as discrete spinor groups. These in turn are known to generate Lie and Coxeter groups in dimension four, notably the exceptional groups D_4, F_4 and H_4. A Clifford algebra approach thus reveals an unexpected connection between Coxeter groups of ranks 3 and 4. We discuss how to extend these considerations and computations to the Conformal Geometric Algebra setup, in particular for the non-crystallographic groups, and construct root systems and quasicrystalline point arrays. We finally show how a Clifford versor framework sheds light on the geometry of the Coxeter element and the Coxeter plane for the examples of the two-dimensional non-crystallographic Coxeter groups I_2(n) and the three-dimensional groups A_3, B_3, as well as the icosahedral group H_3., Comment: Best Paper Award at AGACSE 2012, La Rochelle, France; 18 pages, 4 figures, 2 tables; accepted in Advances in Applied Clifford Algebras

Many combinatorial and topological invariants of a hyperplane arrangement can be computed in terms of its Tutte polynomial. Similarly, many invariants of a hypertoric arrangement can be computed in terms of its arithmetic Tutte polynomial. We compute the arithmetic Tutte polynomials of the classical root systems of types A,B,C, and D, with respect to their integer, root, and weight lattices. We do it in two ways: by introducing a finite field method for arithmetic Tutte polynomials, and by enumerating signed graphs with respect to six parameters., Comment: 39 pages, 3 figures

We study a family of bivariate orthogonal polynomials associated to the deltoid curve. These polynomials arise when classifying bivariate diffusion operators that have discrete spectral decomposition given by orthogonal polynomials with respect to some compactly-supported probability measure on the interior of the deltoid curve., Comment: 33

Agradecimentos: This study was supported by grants from São Paulo Research Foundation (FAPESP), grant #2011/16174-2, and FAPESP/Microsoft Research Connections - BIOEN, grant #2010/52139-4. RCMP and RVR acknowledge the fellowships granted by the National Council for Scientific and Technological Development (CNPq, Brazil) Abstract: Despite of the agronomic importance for water management, few studies of sugarcane roots have been performed under field conditions during the crop cycle. The aim of this study was to determine the root distribution in the soil profile of four sugarcane cultivars. The field experiment was carried out in Campinas, SP, Brazil, and the cultivars present two different leaf architectures: IACSP95-5000 and IACSP94-2094 with arched leaves; IACSP94-2101 and SP79-1011 with upright leaves. The fertigation was performed by subsurface drip system and the soil moisture was monitored by capacitance probes. Three access acrylic tubes with 1.05 m-length were used for each cultivar. The images were obtained using the CID Root Scanner CI-600™ in five dates, 38, 58, 123, 185 and 205 days after harvest (DAH) during the third cycle (2nd cane ratoon) in four depths, up to 0.8 m. The data were analyzed by the software CID CI-690™. The results showed root distribution patterns slightly different among the cultivars. The root distribution pattern of the cultivars presented accurate adjust with exponential fit, except for IACSP94-2101 at 58 DAH. It was observed differences between root distribution among cultivars and sampling dates, especially up to 0.4 m depth. In all four cultivars, at 205 DAH, more than 50% of root distribution was observed up to 0.2 m depth and more than 80% up to 0.4 m depth CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO - CNPQ FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESP Fechado

The paper comprises a graphical calculus which is designed to deal with the Coxeter-Dynkin series of type E and some generalizations. Temperley-Lieb algebras of type E are defined as quotients of Hecke algebras and the module structure of the algebra associated to E 6 is determined. The graphical calculus is a refinement of the calculus for the ordinary Temperley-Lieb algebra: a planar strip is decomposed by the arcs of a diagram into domains and the domains are used to incorporate additional information into the figure.

Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4, as well as those of E_8, have been used extensively in the literature. The present paper analyses such Coxeter groups in the Clifford Geometric Algebra framework, which affords a simple way of performing reflections and rotations whilst exposing more clearly the underlying geometry. The Clifford approach shows that the quaternionic representations in fact have very simple geometric interpretations. The representations of the groups A_1 x A_1 x A_1, A_3, B_3 and H_3 of rank 3 in terms of pure quaternions are shown to be simply the Hodge dualised root vectors, which determine the reflection planes of the Coxeter groups. Two successive reflections result in a rotation, described by the geometric product of the two reflection vectors, giving a Clifford spinor. The spinors for the rank-3 groups A_1 x A_1 x A_1, A_3, B_3 and H_3 yield a new simple construction of binary polyhedral groups. These in turn generate the groups A_1 x A_1 x A_1 x A_1, D_4, F_4 and H_4 of rank 4, and their widely used quaternionic representations are shown to be spinors in disguise. Therefore, the Clifford geometric product in fact induces the rank-4 groups from the rank-3 groups. In particular, the groups D_4, F_4 and H_4 are exceptional structures, which our study sheds new light on., Comment: 20 pages, 1 figure; to appear in Advances in Applied Clifford Algebras

The effects of photosynthetic photon flux density (PPFD) and soil temperature on root system elongation rate have been analysed by using an architectural framework. Root elongation rate was analysed by considering three terms, (i) the branch appearance rate, (ii) the individual elongation rates of the taproot and branches and (iii) the proportion of branches which stop elongating. Large ranges ofPPFD and soil temperature were obtained in a series of field and growth chamber experiments. In the field, the growth of root systems experiencing day-to-day natural fluctuation of PPFD and temperature was followed, and some of the plants under study were shaded. In the growth chamber, plants experienced contrasting and constant PPFDs and root temperatures. The direct effect of apex temperature on individual root elongation rate was surprisingly low in the range 13–25°C, except for the first days after germination. Root elongation rate was essentially related to intercepted PPFD and to distance to the source, both in the field and in the growth chamber. Branch appearance rate substantially varied among days and environmental conditions. It was essentially linked to taproot elongation rate, as the profile of branch density along the taproot was quite stable. The length of the taproot segment carrying newly appeared branches on a given day was equal to taproot elongation on this day, plus a 'buffering term' which transiently increased if taproot elongation rate slowed down. The proportion of branches which stopped elongating a short distance from the taproot ranged from 50–80% and was, therefore, a major architectural variable, although it is not taken into account in current architectural models. A set of equations accounting for the variabilities in elongation rate, branch appearance rate and proportion of branches which stop elongating, as a function of intercepted PPFD and apex temperature is proposed. These equations apply for both field and growth chamber experiments. EEA Balcarce Fil: Aguirrezábal, Luis Adolfo Nazareno. Universidad Nacional de Mar del Plata. Facultad de Ciencias Agrarias. Cátedra de Fisiología Vegetal; Argentina. Instituto Nacional de Tecnología Agropecuaria (INTA). Estación Experimental Agropecuaria Balcarce; Argentina Fil: Tardieu, F. INRA. Laboratory d’Ecophysiologie des Plantes sous Stress Environnementaux; Francia

Sustainable intensification is seen as the main route for meeting the world’s increasing demands for food and fibre. As demands mount for greater efficiency in the use of resources to achieve this goal, so the focus on roots and rootstocks and their role in acquiring water and nutrients, and overcoming pests and pathogens, is increasing. The purpose of this review is to explore some of the ways in which understanding root systems and their interactions with soils could contribute to the development of more sustainable systems of intensive production. Physical interactions with soil particles limit root growth if soils are dense, but root–soil contact is essential for optimal growth and uptake of water and nutrients. X-ray microtomography demonstrated that maize roots elongated more rapidly with increasing root–soil contact, as long as mechanical impedance was not limiting root elongation, while lupin was less sensitive to changes in root–soil contact. In addition to selecting for root architecture and rhizosphere properties, the growth of many plants in cultivated systems is profoundly affected by selection of an appropriate rootstock. Several mechanisms for scion control by rootstocks have been suggested, but the causal signals are still uncertain and may differ between crop species. Linkage map locations for quantitative trait loci for disease resistance and other traits of interest in rootstock breeding are becoming available. Designing root systems and rootstocks for specific environments is becoming a feasible target.

We construct a large family of evidently integrable Hamiltonian systems which are generalizations of the KM system. The Hamiltonian vector field is homogeneous cubic but in a number of cases a simple change of variables transforms such a system to a quadratic Lotka-Volterra system. We present in detail all such systems in dimensions 4 and 5 and we also give some examples from higher dimensions. This construction generalizes easily to each complex simple Lie algebra., Comment: 26 pages

We construct a divided difference operator using GKM theory. This generalizes the classical divided difference operator for the cohomology of the complete flag variety. This construction proves a special case of a recent conjecture of Shareshian and Wachs. Our methods are entirely combinatorial and algebraic, and rely heavily on the combinatorics of root systems and Bruhat order.

This is a continuation of our previous result, in which properties of multiple zeta-functions associated with simple Lie algebras of Ar type have been studied. In the present paper we consider more general situation, and discuss the Lie theoretic background structure of our theory. We show a recursive structure in the family of zeta-functions of sets of roots, which can be explained by the order relation among roots. We also point out that the recursive structure can be described in terms of Dynkin diagrams. Then we prove several analytic properties of zeta-functions associated with simple Lie algebras of Br, Cr, and Dr types.

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra and $M$ be a $\mathfrak{g}$-module. The Fernando-Kac subalgebra of $\mathfrak{g}$ associated to $M$ is the subset $\mathfrak{g}[M]\subset\mathfrak{g}$ of all elements $g\in\mathfrak{g}$ which act locally finitely on $M$. A subalgebra $\mathfrak{l}\subset\mathfrak{g}$ for which there exists an irreducible module $M$ with $\mathfrak{g}[M]=\mathfrak{l}$ is called a Fernando-Kac subalgebra of $\mathfrak{g}$. A Fernando-Kac subalgebra of $\mathfrak{g}$ is of finite type if in addition $M$ can be chosen to have finite Jordan-H\"older $\mathfrak{l}$-multiplicities. Under the assumption that $\mathfrak{g}$ is simple, I. Penkov has conjectured an explicit combinatorial criterion describing all Fernando-Kac subalgebras of finite type which contain a Cartan subalgebra. In the present paper we prove this conjecture for $\mathfrak{g}\neq E_8$.

In this paper we study a random walk on an affine building of type Ãr, whose radial part, when suitably normalized, converges toward the Brownian motion of the Weyl chamber. This gives a new discrete approximation of this process, alternative to the one of Biane (Probab. Theory Related Fields 89 (1991) 117–129). This extends also the link at the probabilistic level between Riemannian symmetric spaces of the noncompact type and their discrete counterpart, which had been previously discovered by Bougerol and Jeulin in rank one (C. R. Acad. Sci. Paris Sér. I Math. 333 (2001) 785–790). The main ingredients of the proof are a combinatorial formula on the building and the estimate of the transition density proved in Anker et al. (2006).