Sorghum is an important staple crop in Sub-Saharan Africa. In the Sudano-Sahelian zone of West Africa, sorghum is mainly intercropped with cowpea, but these intercropping systems are facing low-productivity problems. The overall aim of this research was to identify sorghum varieties with different agro-morphological and physiological traits that could improve the performance of the intercropping systems. We followed a two-step methodology comprising (i) identification of varieties and plant traits of interest in intercropping systems, using participatory methods, and (ii) agro-morpho-physiological characterization of 50 sorghum varieties, to examine the range of variation in traits of interest. The results show that landraces are the varieties most widely used by farmers, and that 82.5% of farmers consider the variety type they choose for intercropping to be important. Farmers mentioned plant height, number of leaves and stem diameter as important traits to consider. Analysis of variance showed significant differences between varieties for half of the 24 agro-morpho-physiological traits studied. Hierarchical clustering identified three main groups of varieties, distinguished by morphological traits such as stem diameter, total number and size of leaves (group 1), root traits (depth, growth angle, dry matter) and relative chlorophyll content (groups 2 and 3). Based on this classification, we recommend several varieties from each of the three groups, exhibiting contrasting traits, for an assessment of their performances in intercropping systems.

We derive two-sided bounds for the Newton and Poisson kernels of the W-invariant Dunkl Laplacian in the geometric complex case when the multiplicity $k(\alpha )=1$ i.e., for flat complex symmetric spaces. For the invariant Dunkl–Poisson kernel $P^{W}(x,y)$ , the estimates are

$$ \begin{align*} P^{W}(x,y)\asymp \frac{P^{\mathbf{R}^{d}}(x,y)}{\prod_{\alpha> 0 \ }|x-\sigma_{\alpha} y|^{2k(\alpha)}}, \end{align*} $$

$\alpha $

$\mathbf {R}^{d}$

$\sigma _{\alpha }$

$P^{\mathbf {R}^{d}}$

${\mathbf {R}^{d}}$

$d\ge 3$

The same estimates are derived in the rank one direct product case $\mathbb {Z}_{2}^{N}$ and conjectured for general W-invariant Dunkl processes.

As an application, we get a two-sided bound for the Poisson and Newton kernels of the classical Dyson Brownian motion and of the Brownian motions in any Weyl chamber.

Given a root system, the Weyl chambers in the co-weight lattice give rise to a real toric variety, called the real toric variety associated with the Weyl chambers. We compute the integral cohomology groups of real toric varieties associated with the Weyl chambers of type Cn and Dn, completing the computation for all classical types.

We define a natural lattice structure on all subsets of a finite root system that extends the weak order on the elements of the corresponding Coxeter group. For crystallographic root systems, we show that the subposet of this lattice induced by antisymmetric closed subsets of roots is again a lattice. We then study further subposets of this lattice that naturally correspond to the elements, the intervals, and the faces of the permutahedron and the generalized associahedra of the corresponding Weyl group. These results extend to arbitrary finite crystallographic root systems the recent results of G. Chatel, V. Pilaud, and V. Pons on the weak order on posets and its induced subposets.

This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of $W$ to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general).

Experiments are conducted to evaluate the tolerance of three deciduous species: black locust, honeylocust, apple, and three coniferous species: Douglas fir, ponderosa pine, and Japanese black pine, in the seedling stage to napropamide. Deciduous species were more susceptible to napropamide than coniferous species. Domestic apple had the slowest seed germination and root development and was more susceptible. The deciduous species had more secondary roots in the shallow soil layer which contained most of the herbicide than the coniferous species. The roots of all woody species tested in vitro were inhibited significantly by contact with the herbicide, but only shoot growth of domestic apple and black locust was inhibited. Injury to woody nursery crop seedlings may be avoided by delaying herbicide application.

We examined effects of supplementary nitrogen on young Canada thistle stands in Alberta, Canada. Urea applied at 70 or 100 kg N ha–1 increased shoot population densities, mostly through increased root growth in the top 20 cm of soil rather than through released root bud dormancy. Nitrogen fertilization, therefore, may increase the severity of a Canada thistle infestation. In 1-yr-old stands, the number of emerged and unemerged root buds/m of root was higher near the soil surface in fertilized plots than in unfertilized plots, and no underground shoots were detected at the time of sampling. In 2-yr-old stands, more unemerged root buds/m of root were found at depths below 20 cm in fertilized plots than in unfertilized plots, and twice as many underground shoots/m of root occurred between 20 and 40 cm in fertilized plots as in unfertilized plots. However, no effect of nitrogen on the number of emerged root buds was detected. Nitrogen treatment increased shoot production only for those replanted root fragments from deeper than 60 cm in 1-yr-old stands and deeper than 160 cm in 2-yr-old stands.

Field research was conducted to evaluate the performance of chlorsulfuron {2-chloro-N-[[4-methoxy-6-methyl-1,3,5-triazin-2-yl)amino] carbonyl] benzenesulfonamide} applied to established Canada thistle plants [Cirsium arvense L. scop. # CIRAR] in the spring or fall to young vegetative shoots, and in June to plants in the bud growth stage. Canada thistle shoot densities measured in the spring following chlorsulfuron applications the previous year were lower when chlorsulfuron was applied in the bud and fall growth stage as compared to applications made in the spring to young vegetative shoots. Suppression of Canada thistle shoots from chlorsulfuron application during the bud and fall growth stage increased with rate, and was greatest at 269 g ai/ha. Chlorsulfuron applied at 269 g/ha at the fall growth stage only partially controlled the root system of Canada thistle.

We prove simplicity for incomplete rank 2 Kac—Moody groups over algebraic closures of finite fields with trivial commutation relations between root groups corresponding to prenilpotent pairs. We don't use the (yet unknown) simplicity of the corresponding finitely generated groups (i.e., when the ground field is finite). Nevertheless we use the fact that the latter groups are just infinite (modulo center).

Let $W$ be an infinite Coxeter group. We initiate the study of the set $E$ of limit points of “normalized” roots (representing the directions of the roots) of $\text{W}$. We show that $E$ is contained in the isotropic cone $Q$ of the bilinear form $B$ associated with a geometric representation, and we illustrate this property with numerous examples and pictures in rank 3 and 4. We also define a natural geometric action of $W$ on $E$, and then we exhibit a countable subset of $E$, formed by limit points for the dihedral reflection subgroups of $W$. We explain how this subset is built fromthe intersection with $Q$ of the lines passing through two positive roots, and finally we establish that it is dense in $E$.

We show that every orbital measure, ${\mu }_{x} $, on a compact exceptional Lie group or algebra has the property that for every positive integer either ${ \mu }_{x}^{k} \in {L}^{2} $ and the support of ${ \mu }_{x}^{k} $ has non-empty interior, or ${ \mu }_{x}^{k} $ is singular to Haar measure and the support of ${ \mu }_{x}^{k} $ has Haar measure zero. We also determine the index $k$ where the change occurs; it depends on properties of the set of annihilating roots of $x$. This result was previously established for the classical Lie groups and algebras. To prove this dichotomy result we combinatorially characterize the subroot systems that are kernels of certain homomorphisms.

Let Un(q) denote the upper triangular group of degree n over the finite field with q elements. It is known that irreducible constituents of supercharacters partition the set of all irreducible characters Irr(Un(q)). In this paper we present a correspondence between supercharacters and pattern subgroups of the form Uk(q) ∩ wUk(q), where w is a monomial matrix in GLk(q) for some k < n.

This note proves Cellini’s conjecture that, in a Coxeter system (W,S) with reflections T, the T-increasing paths in W are self-avoiding. Here, a T-increasing path is a sequence v,t1v,…,tn⋯t1v in W with ti∈T and t1≺⋯≺tn in a reflection order ⪯ of T.

We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these formulas involve a family of polynomials associated with a particular choice of ‘principal’ coefficients. We show that the exchange graph of a cluster algebra with principal coefficients covers the exchange graph of any cluster algebra with the same exchange matrix. We investigate two families of parameterizations of cluster monomials by lattice points, determined, respectively, by the denominators of their Laurent expansions and by certain multi-gradings in cluster algebras with principal coefficients. The properties of these parameterizations, some proven and some conjectural, suggest links to duality conjectures of Fock and Goncharov. The coefficient dynamics leads to a natural generalization of Zamolodchikov's $Y$-systems. We establish a Laurent phenomenon for such $Y$-systems, previously known in finite type only, and sharpen the periodicity result from an earlier paper. For cluster algebras of finite type, we identify a canonical ‘universal’ choice of coefficients such that an arbitrary cluster algebra can be obtained from the universal one (of the same type) by an appropriate specialization of coefficients.

Let V be a finite-dimensional real vector space on which a root system Σ is given. Consider a meromorphic function φ on V$\Bbb C$=V+iV, the singular locus of which is a locally finite union of hyperplanes of the form {λ ∈ V$\Bbb C$[mid ]〈 λ, α 〉 = s}, α ∈ Σ, s ∈ $\Bbb R$. Assume φ is of suitable decay in the imaginary directions, so that integrals of the form ∫η +iV φ λ, dλ make sense for generic η ∈ V. A residue calculus is developed that allows shifting η. This residue calculus can be used to obtain Plancherel and Paley–Wiener theorems on semisimple symmetric spaces.

The effects of potassium deficiency (KD) and all-macronutrient deficiency (MD) on elongation of tap and lateral roots were studied on maritime pine seedlings (Pinus pinaster Ait.) in hydroponic culture.

Tap root elongation was unaffected by either of the two deficiencies. By marked contrast, lateral root elongation was strongly reduced. The analyses of cell turgor pressure and relative elemental growth rate (REGR) profile in the growing zone allowed us to determine the effects of the nutrient stresses on cell-wall properties. For both deficiency treatments, elongation rate, REGR profile (measured only for control and KD) and turgor pressure in the fastest growing cells were unaffected in the tap root, suggesting that KD and MD did not modify cell-wall properties in the growing zone. In lateral roots, KD shortened the growing zone and significantly reduced REGR. However, turgor pressure remained unaffected in this region. The absence of turgor pressure change suggests that KD reduced elongation of lateral roots by tightening cell walls. In mature cells of the two types of roots, turgor and osmotic pressures tended to be reduced by the nutrient deficiencies, indicating that these parameters were better maintained in the growing cells.

Cell turgor and osmotic pressures of control plants were 0·1 MPa lower at 30 mm (mature cells) than at 2–4 mm (expanding cells) from the meristem. Moreover, these parameters were 0·1 MPa lower in expanding cells of lateral roots than in those of tap the root. Turgor and osmotic pressures were not homogeneous throughout the root system and were affected differently by the nutrient deficiencies depending on the location in the root system.