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A subalgebra of a semisimple Lie algebra is wide if every simple module of the semisimple Lie algebra remains indecomposable when restricted to the subalgebra. A subalgebra is narrow if the restrictions of all non-trivial simple modules to the subalgebra have proper decompositions. A semisimple Lie algebra is regular extreme if any regular subalgebra of the semisimple Lie algebra is either narrow or wide. We determine necessary and sufficient conditions for a simple module of a semisimple Lie algebra to remain indecomposable when restricted to a regular subalgebra. As a natural consequence, we establish necessary and sufficient conditions for regular subalgebras to be wide, a result which has already been established by Panyushev for essentially all regular solvable subalgebras [10]. Next, we show that establishing whether or not a regular subalgebra of a simple Lie algebra is wide does not require consideration of all simple modules. It is necessary and sufficient to only consider the adjoint representation. Then, we show that all simple Lie algebras are regular extreme. Finally, we show that no non-simple, semisimple Lie algebra is regular extreme. [ABSTRACT FROM AUTHOR]

As rice is one of the most crucial staple food sources worldwide, enhancing rice yield is paramount for ensuring global food security. Fulvic acid (FA), serving as a plant growth promoter and organic fertilizer, holds significant practical importance in studying its impact on rice root growth for improving rice yield and quality. This study investigated the effects of different concentrations of FA on the growth of rice seedlings. The results indicated that 0.05 g/L FA could promote the growth of rice seedlings, while 0.5 g/L FA inhibited root growth, reduced cell activity and enzyme activity in the root tips, and accumulated reactive oxygen species in root cells. To further elucidate the molecular mechanisms underlying these effects, we performed transcriptomic analysis and found that auxin (Aux) may be involved in the growth process mediated by FA. Furthermore, transcriptome heatmap analysis revealed a significant upregulation of the Aux/indoleacetic acid (Aux/IAA) gene family after FA treatment, suggesting that this gene family plays a crucial role in the impact of FA on root growth. Additionally, by detecting endogenous Aux content and adding exogenous Aux inhibitors, we confirmed the involvement of FA in rice seedling root growth as well as in the synthesis and transduction pathway of Aux. [ABSTRACT FROM AUTHOR]

Surface runoff in mining areas transports dissolved and suspended particles into water bodies, known as mine spoil rainwater, contributing to increases in turbidity. The aim of this study was to evaluate the effectiveness of horizontal flow wetlands, free water surface (FWS), and subsurface flow (HSSF) in reducing turbidity >1500 NTU from a synthetic mine spoil rainwater. Macrophytes, support media, hydraulic retention time (HRT), and hydraulic loading rate (HLR) were analyzed. The HSSF T. domingensis in gravel #1 achieved a 99% reduction for 4-day HRT, with residual turbidity of 7 ± 3 NTU for 27.43 L m−2 d−1 HLR. The FWS P. stratiotes achieved a 99% reduction for 6-day HRT, with residual turbidity of 11 ± 5 NTU for 36.53 L m−2 d−1 HLR. P. stratiotes free root structures promoted interception of suspended colloidal particles, resulting in a better performance. The dense root structure of T. domingensis spreading through the pores of the substrate provided better efficiency than N. humboldtiana. However, N. humboldtiana proved to be promising as a native species. The use of small granulometry alkaline support media (9 to 19 mm) was highlighted. Therefore, this research proves the efficiency of constructed wetlands in reducing high turbidity and provides optimized parameters for this technology application. [ABSTRACT FROM AUTHOR]

Typically, root system architecture (RSA) is not visible, and realistically, high-throughput methods for RSA trait phenotyping should capture key features of developing root systems in solid substrates in 3D. In a published 2-D study using thin rhizoboxes, vermiculite as a growing medium, and photography for imaging, triplicates of 137 soybean cultivars were phenotyped for their RSA. In the transition to 3-D work using X-ray computed tomography (CT) scanning and mineral soil, two research questions are addressed: (1) how different is the soybean RSA characterization between the two phenotyping systems; and (2) is a direct comparison of the results reliable? Prior to a full-scale study in 3D, we grew, in pots filled with sand, triplicates of the Casino and OAC Woodstock cultivars that had shown the most contrasting RSAs in the 2-D study, and CT scanned them at the V1 vegetative stage of development of the shoots. Differences between soybean cultivars in RSA traits, such as total root length and fractal dimension (FD), observed in 2D, can change in 3D. In particular, in 2D, the mean FD values are 1.48 ± 0.16 (OAC Woodstock) vs. 1.31 ± 0.16 (Casino), whereas in 3D, they are 1.52 ± 0.14 (OAC Woodstock) vs. 1.24 ± 0.13 (Casino), indicating variations in RSA complexity. [ABSTRACT FROM AUTHOR]

Given a root system Φ of type A n , B n , C n , or D n in Euclidean space E, let W be the associated Weyl group. For a point p ∈ E not orthogonal to any of the roots in Φ , we consider the W-permutohedron P W , which is the convex hull of the W-orbit of p. The representation of W on the rational cohomology ring H ∗ (X Φ) of the toric variety X Φ associated to (the normal fan to) P W has been studied by various authors. Let { s 1 , ... , s n } be a complete set of simple reflections in W. For K ⊆ [ n ] , let W K be the standard parabolic subgroup of W generated by { s k : k ∈ K } . We show that the fixed subring H ∗ (X Φ) W K is isomorphic to the cohomology ring of the toric variety X Φ (K) associated to a polytope obtained by intersecting P W with half-spaces bounded by reflecting hyperplanes for the given generators of W K . We also obtain explicit formulas for h-vectors of these polytopes. By a result of Balibanu–Crooks, the cohomology rings H ∗ (X Φ (K)) are isomorphic with cohomology rings of certain regular Hessenberg varieties. [ABSTRACT FROM AUTHOR]

We introduce a notion of representation for a class of generalised quivers known as Coxeter quivers. These representations are built using fusion categories associated to U q (s l 2) at roots of unity and we show that many of the classical results on representations of quivers can be generalised to this setting. Namely, we prove a generalised Gabriel's theorem for Coxeter quivers that encompasses all Coxeter–Dynkin diagrams—including the non-crystallographic types H and I. Moreover, a similar relation between reflection functors and Coxeter theory is used to show that the indecomposable representations correspond bijectively to the (extended) positive roots of Coxeter root systems over fusion rings. [ABSTRACT FROM AUTHOR]

Modifying root systems by crop breeding has been attracting increasing attention as a potentially effective strategy to enhance the sustainability of agriculture by increasing soil organic matter (SOM) stocks and soil quality, whilst maintaining or even improving yields. We used the new soil‐crop model USSF (Uppsala model of Soil Structure and Function) to investigate the potential of this management strategy using winter wheat as a model crop. USSF combines a simple (generic) crop growth model with physics‐based descriptions of soil water flow, water uptake and transpiration by plants. It also includes a model of the interactions between soil structure dynamics and organic matter turnover that considers the effects of physical protection and microbial priming on the decomposition of SOM. The model was first calibrated against field data on soil water contents and both above‐ground and root biomass of winter wheat measured during one growing season in a clay soil in Uppsala, Sweden using the GLUE method to identify five 'acceptable' parameter sets. We created four model crops (ideotypes) by modifying root‐related parameters to mimic winter wheat phenotypes with improved root traits. Long‐term (30‐year) simulations of a conventionally tilled monoculture of winter wheat were then performed to evaluate the potential effects of cultivating these ideotypes on the soil water balance, soil organic matter stocks and grain yields. Our results showed that ideotypes with deeper root systems or root systems that are more effective for water uptake increased grain yields by 3% and SOM stocks in the soil profile by ca. 0.4%–0.5% in a 30‐year perspective (as an average of the five parameter sets). An ideotype in which below‐ground allocation of dry matter was increased at the expense of stem growth gave even larger increases in SOM stocks (ca. 1.4%). An ideotype combining all three modifications (deeper and more effective root systems and greater root production) showed even more promising results: compared with the baseline scenario, surface runoff decreased while yields were predicted to increase by ca. 7% and SOM stocks in the soil profile by ca. 2%, which is roughly equivalent to ca. 20% of the 4‐per‐mille target (https://4p1000.org/). [ABSTRACT FROM AUTHOR]

We study algebras of meromorphic modular forms whose poles lie on Heegner divisors for orthogonal and unitary groups associated with root lattices. We give a uniform construction of 147 hyperplane arrangements on type IV symmetric domains for which the algebras of modular forms with constrained poles are free and therefore the Looijenga compactifications of the arrangement complements are weighted projective spaces. We also construct eight free algebras of modular forms on complex balls with poles on hyperplane arrangements. The most striking example is the discriminant kernel of the 2U ⊕D11 lattice, which admits a free algebra on 14 meromorphic generators. Along the way, we determine minimal systems of generators for non-free algebras of orthogonal modular forms for 26 reducible root lattices and prove the modularity of formal Fourier-Jacobi series associated with them. By exploiting an identity between weight 1 singular additive and multiplicative lifts on 2U ⊕ D11, we prove that the additive lift of any (possibly weak) theta block of positive weight and q-order 1 is a Borcherds product. The special case of holomorphic theta blocks of one elliptic variable is the theta block conjecture of Gritsenko, Poor and Yuen. [ABSTRACT FROM AUTHOR]

There has been recent interest in novel Clifford geometric invariants of linear transformations. This motivates the investigation of such invariants for a certain type of geometric transformation of interest in the context of root systems, reflection groups, Lie groups and Lie algebras: the Coxeter transformations. We perform exhaustive calculations of all Coxeter transformations for A 8 , D 8 and E 8 for a choice of basis of simple roots and compute their invariants, using high-performance computing. This computational algebra paradigm generates a dataset that can then be mined using techniques from data science such as supervised and unsupervised machine learning. In this paper we focus on neural network classification and principal component analysis. Since the output—the invariants—is fully determined by the choice of simple roots and the permutation order of the corresponding reflections in the Coxeter element, we expect huge degeneracy in the mapping. This provides the perfect setup for machine learning, and indeed we see that the datasets can be machine learned to very high accuracy. This paper is a pump-priming study in experimental mathematics using Clifford algebras, showing that such Clifford algebraic datasets are amenable to machine learning, and shedding light on relationships between these novel and other well-known geometric invariants and also giving rise to analytic results. [ABSTRACT FROM AUTHOR]

Trees sequester carbon dioxide from the atmosphere through photosynthesis, storing it in branches, stems, and roots, where the belowground carbon fraction, approximately 1/4 of the total amount, exhibits significant interspecies root biomass variability. Estimating the amount of carbon stored in tree roots of different species is key to understanding an important aspect of climate change and exploring how natural forests, urban tree planting policies, and reforestation projects might help to address it. In this context, one of the most prominent Non-Destructive Testing methods capable of estimating the diameter and length of roots at different depths is ground penetrating radar (GPR). It has been widely used for geological, archaeological, and geotechnical studies due to its accuracy in locating buried material in different contexts, although standards for the correct management of datasets related to belowground root systems are still been developed. This paper reports a GPR signal processing flow to estimate the root diameter of three species of tropical forest trees, and to demonstrate the method's viability, a dataset was collected in a study area with a 900 MHz shielded antenna. A multistage data processing flow is then presented, including raw data, file format conversion, zero-time adjustment, background removal, signal gain, Stolt FK migration, and time-to-depth conversion with hyperbolic adjustment velocity. The resulting data were converted from true amplitude data to a trace envelope. High amplitudes on the envelope section, with lateral continuity in parallel sections, were interpreted as roots. However, the interpretation of outcomes encounters notable complexities, primarily attributable to the intricate nature of subsurface root architectures, the soil matrix characterized by significant clay content, and the co-occurrence of buried materials proximate to the arboreal subjects. Consequently, amplitudes discerned within ground penetrating radar (GPR) 2D sections necessitate cautious interpretation, as they are not immediately indicative of subsurface roots. To overcome this difficulty, this study used direct measurements of the roots in the field, to confirm the GPR data. Despite these complexities, the study demonstrates GPR's efficacy, particularly in the uppermost soil layer-a pivotal carbon reservoir with a 96% correlation (R²) between GPR-derived coarse-root diameter estimates and field measurements. [ABSTRACT FROM AUTHOR]

Let G be a non-compact classical semisimple Lie group and let G / V be the adjoint orbit with respect to a fixed element in G. These manifolds can be equipped with an almost-Kähler structure and we provide explicit formulae for the existence of special almost-complex structures on G / V purely in terms of the combinatorics of the associated Vogan diagram. The formulae are given separately for Lie groups whose Lie algebras are of type A ℓ , B ℓ , C ℓ , D ℓ , where ℓ denotes the rank of the Lie algebra. [ABSTRACT FROM AUTHOR]

Surface runoff in mining areas transports dissolved and suspended particles into water bodies, known as mine spoil rainwater, contributing to increases in turbidity. The aim of this study was to evaluate the effectiveness of horizontal flow wetlands, free water surface (FWS), and subsurface flow (HSSF) in reducing turbidity >1500 NTU from a synthetic mine spoil rainwater. Macrophytes, support media, hydraulic retention time (HRT), and hydraulic loading rate (HLR) were analyzed. The HSSF T. domingensis in gravel #1 achieved a 99% reduction for 4-day HRT, with residual turbidity of 7 ± 3 NTU for 27.43 L m−2 d−1 HLR. The FWS P. stratiotes achieved a 99% reduction for 6-day HRT, with residual turbidity of 11 ± 5 NTU for 36.53 L m−2 d−1 HLR. P. stratiotes free root structures promoted interception of suspended colloidal particles, resulting in a better performance. The dense root structure of T. domingensis spreading through the pores of the substrate provided better efficiency than N. humboldtiana. However, N. humboldtiana proved to be promising as a native species. The use of small granulometry alkaline support media (9 to 19 mm) was highlighted. Therefore, this research proves the efficiency of constructed wetlands in reducing high turbidity and provides optimized parameters for this technology application.

Typically, root system architecture (RSA) is not visible, and realistically, high-throughput methods for RSA trait phenotyping should capture key features of developing root systems in solid substrates in 3D. In a published 2-D study using thin rhizoboxes, vermiculite as a growing medium, and photography for imaging, triplicates of 137 soybean cultivars were phenotyped for their RSA. In the transition to 3-D work using X-ray computed tomography (CT) scanning and mineral soil, two research questions are addressed: (1) how different is the soybean RSA characterization between the two phenotyping systems; and (2) is a direct comparison of the results reliable? Prior to a full-scale study in 3D, we grew, in pots filled with sand, triplicates of the Casino and OAC Woodstock cultivars that had shown the most contrasting RSAs in the 2-D study, and CT scanned them at the V1 vegetative stage of development of the shoots. Differences between soybean cultivars in RSA traits, such as total root length and fractal dimension (FD), observed in 2D, can change in 3D. In particular, in 2D, the mean FD values are 1.48 ± 0.16 (OAC Woodstock) vs. 1.31 ± 0.16 (Casino), whereas in 3D, they are 1.52 ± 0.14 (OAC Woodstock) vs. 1.24 ± 0.13 (Casino), indicating variations in RSA complexity.

We construct the basic representation of the double affine Hecke algebra at critical level q = 1 associated to an irreducible reduced affine root system R with a reduced gradient root system. For R of untwisted type such a representation was studied by Oblomkov [A. Oblomkov, Double affine Hecke algebras and Calogero–Moser spaces, Represent. Theory 8(2004) 243–266] and further detailed by Gehles [K. E. Gehles, Properties of Cherednik algebras and graded Hecke algebras, PhD thesis, University of Glasgow (2006)] in the presence of minuscule weights. [ABSTRACT FROM AUTHOR]

On RN$\mathbb {R}^N$ equipped with a root system R, multiplicity function k≥0$k \ge 0$, and the associated measure dw(x)=∏α∈R|⟨x,α⟩|k(α)dx$dw(\mathbf {x})=\prod _{\alpha \in R}|\langle \mathbf {x},\alpha \rangle |^{k(\alpha)}\,d\mathbf {x}$, we consider a (nonradial) kernel K(x)${K}(\mathbf {x})$, which has properties similar to those from the classical theory of singular integrals and the Dunkl convolution operator Tf=f∗K$\mathbf {T}f=f*K$ associated with K. Assuming that b belongs to the BMO space on the space of homogeneous type X=(RN,∥·∥,dw)$X=(\mathbb {R}^N,\Vert \cdot \Vert ,dw)$, we prove that the commutator [b,T]f(x)=b(x)Tf(x)−T(bf)(x)$[b,\mathbf {T}]f(\mathbf {x})=b(\mathbf {x})\mathbf {T}f(\mathbf {x})-\mathbf {T}(bf)(\mathbf {x})$ is a bounded operator on Lp(dw)$L^p(dw)$ for all 1

Given any smooth germ of a 3-fold flopping contraction, we first give a combinatorial characterisation of which Gopakumar--Vafa (GV) invariants are non-zero, by prescribing multiplicities to the walls in the movable cone. On the Gromov--Witten (GW) side, this allows us to describe, and even draw, the critical locus of the associated quantum potential. We prove that the critical locus is the infinite hyperplane arrangement of Iyama and the second author and, moreover, that the quantum potential can be reconstructed from a finite fundamental domain. We then iterate, obtaining a combinatorial description of the matrix that controls the transformation of the non-zero GV invariants under a flop. There are three main ingredients and applications: (1) a construction of flops from simultaneous resolution via cosets, which describes how the dual graph changes; (2) a closed formula, which describes the change in dimension of the contraction algebra under flop; and (3) a direct and explicit isomorphism between quantum cohomologies of different crepant resolutions, giving a Coxeter-style, visual proof of the Crepant Transformation Conjecture for isolated cDV singularities. [ABSTRACT FROM AUTHOR]

Trees sequester carbon dioxide from the atmosphere through photosynthesis, storing it in branches, stems, and roots, where the belowground carbon fraction, approximately ¼ of the total amount, exhibits significant interspecies root biomass variability. Estimating the amount of carbon stored in tree roots of different species is key to understanding an important aspect of climate change and exploring how natural forests, urban tree planting policies, and reforestation projects might help to address it. In this context, one of the most prominent Non-Destructive Testing methods capable of estimating the diameter and length of roots at different depths is ground penetrating radar (GPR). It has been widely used for geological, archaeological, and geotechnical studies due to its accuracy in locating buried material in different contexts, although standards for the correct management of datasets related to belowground root systems are still been developed. This paper reports a GPR signal processing flow to estimate the root diameter of three species of tropical forest trees, and to demonstrate the method’s viability, a dataset was collected in a study area with a 900 MHz shielded antenna. A multi-stage data processing flow is then presented, including raw data, file format conversion, zero-time adjustment, background removal, signal gain, Stolt FK migration, and time-to-depth conversion with hyperbolic adjustment velocity. The resulting data were converted from true amplitude data to a trace envelope. High amplitudes on the envelope section, with lateral continuity in parallel sections, were interpreted as roots. However, the interpretation of outcomes encounters notable complexities, primarily attributable to the intricate nature of subsurface root architectures, the soil matrix characterized by significant clay content, and the co-occurrence of buried materials proximate to the arboreal subjects. Consequently, amplitudes discerned within ground penetrating radar (GPR) 2D sections necessitate cautious interpretation, as they are not immediately indicative of subsurface roots. To overcome this difficulty, this study used direct measurements of the roots in the field, to confirm the GPR data. Despite these complexities, the study demonstrates GPR’s efficacy, particularly in the uppermost soil layer-a pivotal carbon reservoir with a 96% correlation (R2) between GPR-derived coarse-root diameter estimates and field measurements.

We propose a refined version of the existing conjectural asymptotic formula for the moments of the family of quadratic Dirichlet L -functions over rational function fields. Our prediction is motivated by two natural conjectures that provide sufficient information to determine the analytic properties (meromorphic continuation, location of poles, and the residue at each pole) of a certain generating function of moments of quadratic L -functions. The number field analogue of our asymptotic formula can be obtained by a similar procedure, the only difference being the contributions coming from the archimedean and even places, which require a separate analysis. To avoid this additional technical issue, we present, for simplicity, the asymptotic formula only in the rational function field setting. This has also the advantage of being much easier to test. [ABSTRACT FROM AUTHOR]

Abstract Background Various growth systems are available for studying plant root growth and plant–microbe interactions including hydroponics and aeroponics. Although some of these systems work well with Arabidopsis thaliana and smaller cereal model plants, they may not scale up as well for use with hundreds of plants at a time from a larger plant species. The aim of this study is to present step-by-step instructions for fabricating an aeroponic system, also called a “caisson,” that has been in use in several legume research labs studying the development of symbiotic nitrogen fixing nodules, but for which detailed directions are not currently available. The aeroponic system is reusable and is adaptable for many other types of investigations besides root nodulation. Results An aeroponic system that is affordable and reusable was adapted from a design invented by French engineer René Odorico. It consists of two main components: a modified trash can with a lid of holes and a commercially available industrial humidifier that is waterproofed with silicon sealant. The humidifier generates a mist in which plant roots grow, suspended from holes in trash can lid. Results from use of the aeroponic system have been available in the scientific community for decades; it has a record as a workhorse in the lab. Conclusions Aeroponic systems present a convenient way for researchers to grow plants for studying root systems and plant–microbe interactions in root systems. They are particularly attractive for phenotyping roots and following the progress of nodule development in legumes. Advantages include the ability to precisely control the growth medium in which the plants grow and easy observations of roots during growth. In this system, mechanical shear potentially killing microbes found in some other types of aeroponic devices is not an issue. Disadvantages of aeroponic systems include the likelihood of altered root physiology compared to root growth on soil and other solid substrates and the need to have separate aeroponic systems for comparing plant responses to different microbial strains.

Any symmetric closed subset of a finite crystallographic root system must be a closed subroot system. This is not, in general, true for real affine root systems. In this paper, we determine when this is true and also give a very explicit description of symmetric closed subsets of real affine root systems. At the end, using our results, we study the correspondence between symmetric closed subsets of real affine root systems and the regular subalgebras generated by them. [ABSTRACT FROM AUTHOR]

Summary: The propagation of xylem embolism throughout the root systems of drought‐affected plants remains largely unknown, despite this process being comparatively well characterized in aboveground tissues.We used optical and X‐ray imaging to capture xylem embolism propagation across the intact root systems of bread wheat (Triticum aestivum L. 'Krichauff') plants subjected to drying. Patterns in vulnerability to xylem cavitation were examined to investigate whether vulnerability may vary based on root size and placement across the entire root system.Individual plants exhibited similar mean whole root system vulnerabilities to xylem cavitation but showed enormous 6 MPa variation within their component roots (c. 50 roots per plant). Xylem cavitation typically initiated in the smallest, peripheral parts of the root system and moved inwards and upwards towards the root collar last, although this trend was highly variable.This pattern of xylem embolism spread likely results in the sacrifice of replaceable small roots while preserving function in larger, more costly central roots. A distinct pattern of embolism‐spread belowground has implications for how we understand the impact of drought in the root system as a critical interface between plant and soil. [ABSTRACT FROM AUTHOR]

Purpose: Incorporating biosolids into the soil improves plant yield compared with surface application, but it can result in the increased uptake of trace elements. However, there is a lack of knowledge about how different types of biosolids applications affect soil quality. We aimed to determine the effect of the type and rate of biosolids application on soil quality and the mobility of contaminants. Methods: Soil quality was determined by soil fertility (inorganic N, exchangeable P, Mg, Ca, K), exchangeable trace and non-essential elements (Al, Mn, Zn, Cu and Cd) and biological activity (dehydrogenase activity). We measured the properties of soil pore water, bulk soil and rhizosphere in a pot and a rhizobox experiment, with increasing concentration of biosolids (equiv. 16 t ha− 1, 48 t ha− 1 and 145 t ha− 1 dry weight), applied on the surface, incorporated to 25 cm, or incorporated into a patch. Results and discussion: The incorporation of biosolids into the soil increased the exchangeable Zn, Cu, Cr, Ni and Cd, compared with surface application. The surface application of biosolids increased the inorganic N in the soil compared with biosolids incorporation (680 mg kg− 1 vs. 380 mg kg− 1), and decreased soil pH by 1.1 units. This aligned with solubilisation of Al (43 mg kg− 1 vs. 6 mg kg− 1) and Mn (43 mg kg− 1 vs. 33 mg kg− 1) and explains the decreased microbial activity in the soil compared with the unamended soil. Incorporating biosolids in the soil increased the biological activity, likely due to biosolids-borne microbes. The root systems significantly increased microbial activity, pH, and the concentration of NH4+, NO3−, and exchangeable P, S, Mg, Na, Zn, Cu and Ni, and significantly decreased exchangeable concentration of Mn and Fe. [ABSTRACT FROM AUTHOR]

On R N equipped with a root system R and a multiplicity function k > 0 , we study the generalized (Dunkl) translations τ x g (- y) of not necessarily radial kernels g. Under certain regularity assumptions on g, we derive bounds for τ x g (- y) by means the Euclidean distance ‖ x - y ‖ and the distance d (x , y) = min σ ∈ G ‖ x - σ (y) ‖ , where G is the reflection group associated with R. Moreover, we prove that τ does not preserve positivity, that is, there is a non-negative Schwartz class function φ , such that τ x φ (- y) < 0 for some points x , y ∈ R N . [ABSTRACT FROM AUTHOR]

We study the ring of Weyl invariant E 8 weak Jacobi forms. Wang recently proved that the ring is not a polynomial algebra. We consider a polynomial algebra which contains the ring as a subset and clarify the necessary and sufficient condition for an element of the polynomial algebra to be a Weyl invariant E 8 weak Jacobi form. This serves as a new algorithm for constructing all the Jacobi forms of given weight and index. The algorithm is purely algebraic and does not require Fourier expansion. Using this algorithm we determine the generators of the free module of Weyl invariant E 8 weak Jacobi forms of given index m for m ≤ 20. We also determine the lowest weight generators of the free module of index m for m ≤ 28. Our results support the lower bound conjecture of Sun and Wang and prove explicitly that there exist generators of the ring of Weyl invariant E 8 weak Jacobi forms of weight − 4 m and index m for all 12 ≤ m ≤ 28. [ABSTRACT FROM AUTHOR]

Background: Various growth systems are available for studying plant root growth and plant–microbe interactions including hydroponics and aeroponics. Although some of these systems work well with Arabidopsis thaliana and smaller cereal model plants, they may not scale up as well for use with hundreds of plants at a time from a larger plant species. The aim of this study is to present step-by-step instructions for fabricating an aeroponic system, also called a "caisson," that has been in use in several legume research labs studying the development of symbiotic nitrogen fixing nodules, but for which detailed directions are not currently available. The aeroponic system is reusable and is adaptable for many other types of investigations besides root nodulation. Results: An aeroponic system that is affordable and reusable was adapted from a design invented by French engineer René Odorico. It consists of two main components: a modified trash can with a lid of holes and a commercially available industrial humidifier that is waterproofed with silicon sealant. The humidifier generates a mist in which plant roots grow, suspended from holes in trash can lid. Results from use of the aeroponic system have been available in the scientific community for decades; it has a record as a workhorse in the lab. Conclusions: Aeroponic systems present a convenient way for researchers to grow plants for studying root systems and plant–microbe interactions in root systems. They are particularly attractive for phenotyping roots and following the progress of nodule development in legumes. Advantages include the ability to precisely control the growth medium in which the plants grow and easy observations of roots during growth. In this system, mechanical shear potentially killing microbes found in some other types of aeroponic devices is not an issue. Disadvantages of aeroponic systems include the likelihood of altered root physiology compared to root growth on soil and other solid substrates and the need to have separate aeroponic systems for comparing plant responses to different microbial strains. [ABSTRACT FROM AUTHOR]

In this note, we give a classification of the maximal order Abelian subgroups of finite irreducible Coxeter groups. We also prove a Weyl group analog of Cartan's theorem that all maximal tori in a connected compact Lie group are conjugate. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Root metrics and plant height for 256 excavated saplings and small trees of 27 species, including sown plants, were used to describe belowground structure and assess factors that influence shoot growth in a tropical dry forest (TDF) in Zambia. Models were developed to (i) estimate taproot depth from incomplete excavations and (ii) coarse lateral root biomass from proximal diameter data. The majority of the species studied are slow-growing and had a median height of <200 cm at the age of 16 years. Root development advanced sequentially from taproot elongation to thickening to coarse lateral root development. Shrubs in shallow soil had short taproots with a lower wood density. Plant age explained <10% of the variance in shoot height. Root variables explained the majority of the variance in shoot height. More research is needed to improve our knowledge about how belowground structures influence shoot growth and tree recruitment in TDFs of southern Africa. [ABSTRACT FROM AUTHOR]

Summary: Although the above and belowground sizes and shapes of plants strongly influence plant competition, community structure, and plant–environment interactions, plant sizes and shapes remain poorly characterized across climate regimes. We investigated relationships among shoot and root system size and climate.We assembled and analyzed, to our knowledge, the largest global database describing the maximum rooting depth, lateral spread, and shoot size of terrestrial plants – more than doubling the Root Systems of Individual Plants database to 5647 observations.Water availability and growth form greatly influence shoot size, and rooting depth is primarily influenced by temperature seasonality. Shoot size is the strongest predictor of lateral spread, with root system diameter being two times wider than shoot width on average for woody plants.Shoot size covaries strongly with rooting system size; however, the geometries of plants differ considerably across climates, with woody plants in more arid climates having shorter shoots, but deeper, narrower root systems. Additionally, estimates of the depth and lateral spread of plant root systems are likely underestimated at the global scale. See also the Commentary on this article by Kattge, 235: 821–823. [ABSTRACT FROM AUTHOR]

This review focuses on the systematization of works in the field of adaptive reactions of woody plants to technogenesis at the main hierarchical structural and functional levels of living organization, and in the field of adaptive strategies. The influence of various types of industrial pollution on micromorphology, micromorphology, the physiological reactions of needles and leaves, radial increment of the trunk, root systems, and the vital state of forest stands, as well as issues of determining adaptive reactions and adaptive strategies of woody species to technogenesis are considered. The relative independence of adaptive reactions within organs is shown, and specific and non-specific reactions are identified. The causes of these polyvariant reactions that underlie adaptive potential, the principles of multiple provision of biologically necessary functions for the preservation of homeostasis, and species tolerance to technogenesis are discussed. The issues of biosystem resistance to technogenesis based on the adaptive reactions taking place at all levels of living matter organization (from cytogenetic to ecosystem) are considered. Industrial pollution, as a new historical factor for plants, has necessitated the development discussion of the adaptive strategies of species to technogenesis, based on adaptive potential, variability, sustainability and environmental plasticity of species. [ABSTRACT FROM AUTHOR]

Let 𝔤 be a semisimple Lie algebra, j a Cartan subalgebra of 𝔤, j*, the dual of j, jv the bidual of j and B(., .) the restriction to j of the Killing form of 𝔤. In this work, we will construct a chain of reproducing kernel Cartan subalgebras ordered by inclusion.

High-throughput, field-based characterization of root systems for hundreds of genotypes in thousands of plots is necessary for breeding and identifying loci underlying variation in root traits and their plasticity. We designed a large-scale sampling of root pulling force, the vertical force required to extract the root system from the soil, in a maize diversity panel under differing irrigation levels for two growing seasons. We then characterized the root system architecture of the extracted root crowns. We found consistent patterns of phenotypic plasticity for root pulling force for a subset of genotypes under differential irrigation, suggesting that root plasticity is predictable. Using genome-wide association analysis, we identified 54 SNPs as statistically significant for six independent root pulling force measurements across two irrigation levels and four developmental timepoints. For every significant GWAS SNP for any trait in any treatment and timepoint we conducted post hoc tests for genotype-by-environment interaction, using a mixed model ANOVA. We found that 8 of the 54 SNPs showed significant GxE. Candidate genes underlying variation in root pulling force included those involved in nutrient transport. Although they are often treated separately, variation in the ability of plant roots to sense and respond to variation in environmental resources including water and nutrients may be linked by the genes and pathways underlying this variation. While functional validation of the identified genes is needed, our results expand the current knowledge of root phenotypic plasticity at the whole plant and gene levels, and further elucidate the complex genetic architecture of maize root systems. [ABSTRACT FROM AUTHOR]

The role and features of the European spruce root systems structure in the conditions of recreational load in the tourist routes area of the Carpathian National Nature Park (NNP) are described. The aim of the study is to establish the ratio of the volume and surface area that interacts with the soil for conductive and physiologically active roots as a prerequisite for anti-erosion resistance of spruce phytocenoses under recreational load conditions. To determine the silvicultural and biometric indicators of tree species at the forest stand 13 circular and semicircular temporary trial areas were laid down according to the method which had become well established and commonly usedin forest measurement and forest management. The studied plantations are mostly pure in composition, less often mixed with the predominance of European spruce and 1-2 units’ participation in the forest composition by beech (Fagus sylvatica L). or single specimens of the genus Salix L. The age structure is dominated by medieval and pre-mature stands. The root systems are the main anti-erosion element of forest phytocenosis on sloping lands, where the leading (skeletal) roots provide consolidation of trees in a vertical position, and physiologically active - provides soil consolidation, participates in the formation of waterproof aggregates, improving soil permeability and activation of soil-forming processes. An important indicator of the roots involvement in the landscapes erosion resistance increasing is their volume and surface area, which interacts with the soil solution. The main role in the absorption of nutrients by the underground part of the tree is played by physiologically active roots. There is a natural competition between the root systems in both pure and mixed stands. A denser interweaving of root systems is observed on nutrient-rich soils. However, the high density of root systems is due to the water limit on limited access to nutrients. The plasticity of the physiological root, its biomass and the density of distribution indicate the level of ability to compete for habitat. 136 samples of root-containing soil layer at a depth of 0-10 cm, 10-20, 20-30 and 30-40 cm were taken from the sample areas of stands growing in clean and with a small participation of medieval beech forest and ripening stands, from which the roots of European spruce were isolated from division into conductive and physiologically active roots. Also volumes and surface areas of these roots have been calculated. Statistical correlations between altitude and the volume of the fraction of physiologically active and conductive roots were established, the correlation coefficients of which are 0.78 and 0.47, respectively. The distribution of the root systems of European spruce depending on the depth of the parent rock showed that in the upper mineral layers of the soil by volume the fraction of the leading root predominates. In some cases, its volume increases with depth, which affects the density of the conductive root. The surface area of the roots is dominated by physiologically active roots at a depth of 0-20 cm, while at a depth of 20-30 there is a predominance of the leading root. In plantations with stronger soil profiles (over 30 cm) in the upper layers (0-20 cm) and at a depth of 30-40 cm the fraction of the leading root predominates, but at a depth of 20-30 cm this tendency changes in favor of physiologically active roots. The main part of the roots is located in the upper mineral layer of the soil at a depth of up to 10 cm on the objects of nature reserves, which are more sensitive to recreational loads in thе mountain conditions. In this soil horizon the share of conducting and physiological roots of the total volume of each fraction is 67.5 and 75.3%, and in places with rock outcrops - 72.7 and 87.5%, respectively.

High-throughput, field-based characterization of root systems for hundreds of genotypes in thousands of plots is necessary for breeding and identifying loci underlying variation in root traits and their plasticity. We designed a large-scale sampling of root pulling force, the vertical force required to extract the root system from the soil, in a maize diversity panel under differing irrigation levels for two growing seasons. We then characterized the root system architecture of the extracted root crowns. We found consistent patterns of phenotypic plasticity for root pulling force for a subset of genotypes under differential irrigation, suggesting that root plasticity is predictable. Using genome-wide association analysis, we identified 54 SNPs as statistically significant for six independent root pulling force measurements across two irrigation levels and four developmental timepoints. For every significant GWAS SNP for any trait in any treatment and timepoint we conducted post hoc tests for genotype-by-environment interaction, using a mixed model ANOVA. We found that 8 of the 54 SNPs showed significant GxE. Candidate genes underlying variation in root pulling force included those involved in nutrient transport. Although they are often treated separately, variation in the ability of plant roots to sense and respond to variation in environmental resources including water and nutrients may be linked by the genes and pathways underlying this variation. While functional validation of the identified genes is needed, our results expand the current knowledge of root phenotypic plasticity at the whole plant and gene levels, and further elucidate the complex genetic architecture of maize root systems.

We develop an algorithm for computing the cohomology of complements of toric arrangements. In the case a finite group Γ is acting on the arrangement, the algorithm determines the cohomology groups as representations of 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. [ABSTRACT FROM AUTHOR]

Let $U_q'({\mathfrak {g}})$ be a quantum affine algebra with an indeterminate $q$ , and let $\mathscr {C}_{\mathfrak {g}}$ be the category of finite-dimensional integrable $U_q'({\mathfrak {g}})$ -modules. We write $\mathscr {C}_{\mathfrak {g}}^0$ for the monoidal subcategory of $\mathscr {C}_{\mathfrak {g}}$ introduced by Hernandez and Leclerc. In this paper, we associate a simply laced finite-type root system to each quantum affine algebra $U_q'({\mathfrak {g}})$ in a natural way and show that the block decompositions of $\mathscr {C}_{\mathfrak {g}}$ and $\mathscr {C}_{\mathfrak {g}}^0$ are parameterized by the lattices associated with the root system. We first define a certain abelian group $\mathcal {W}$ (respectively $\mathcal {W} _0$) arising from simple modules of $\mathscr {C}_{\mathfrak {g}}$ (respectively $\mathscr {C}_{\mathfrak {g}}^0$) by using the invariant $\Lambda ^\infty$ introduced in previous work by the authors. The groups $\mathcal {W}$ and $\mathcal {W} _0$ have subsets $\Delta$ and $\Delta _0$ determined by the fundamental representations in $\mathscr {C}_{\mathfrak {g}}$ and $\mathscr {C}_{\mathfrak {g}}^0$ , respectively. We prove that the pair $(\mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} _0, \Delta _0)$ is an irreducible simply laced root system of finite type and that the pair $(\mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} , \Delta)$ is isomorphic to the direct sum of infinite copies of $(\mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} _0, \Delta _0)$ as a root system. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We give a full classification of vertex-transitive zonotopes. We prove that a vertex-transitive zonotope is a Γ -permutahedron for some finite reflection group Γ ⊂ O (R d) . The same holds true for zonotopes in which all vertices are on a common sphere, and all edges are of the same length. The classification of these then follows from the classification of finite reflection groups. We prove that root systems can be characterized as those centrally symmetric sets of vectors, for which all intersections with half-spaces, that contain exactly half the vectors, are congruent. We provide a further sufficient condition for a centrally symmetric set to be a root system. [ABSTRACT FROM AUTHOR]

We call a finite, spanning set of a semi-simple real Lie algebra a distinguished set if it satisfies the following property: The Lie bracket of any two elements out of the set is, up to some constant, another element in the set; conversely, for any element in the set, there are two elements out of the set whose Lie bracket is, up to some constant, the given element. We show that every semi-simple real Lie algebra has a distinguished set. [ABSTRACT FROM AUTHOR]

We construct Frobenius structures on the C × -bundle of the complement of a toric arrangement associated with a root system, by making use of a one-parameter family of torsion free and flat connections on it. This gives rise to a class of Frobenius manifolds parameterizing Frobenius algebras in terms of root systems in a trigonometric setting. We then determine their potential functions, which amounts to giving the trigonometric solutions of WDVV equations. [ABSTRACT FROM AUTHOR]

We consider some new limits for the elliptic hypergeometric integrals on root systems. After the degeneration of elliptic beta integrals of type I and type II for root systems A n and C n to the hyperbolic hypergeometric integrals, we apply the limit ω 1 → − ω 2 for their quasiperiods (corresponding to b → i in the two-dimensional conformal field theory) and obtain complex beta integrals in the Mellin–Barnes representation admitting exact evaluation. Considering type I elliptic hypergeometric integrals of a higher order obeying nontrivial symmetry transformations, we derive their descendants to the level of complex hypergeometric functions and prove the Derkachov–Manashov conjectures for functions emerging in the theory of non-compact spin chains. We describe also symmetry transformations for a type II complex hypergeometric function on the C n -root system related to the recently derived generalized complex Selberg integral. For some hyperbolic beta integrals we consider a special limit ω 1 → ω 2 (or b → 1) and obtain new hypergeometric identities for sums of integrals of rational functions. [ABSTRACT FROM AUTHOR]

Belowground biomass represents a significant carbon reserve with a crucial role in terms of ecosystem functioning, but very little is known about root systems across tropical dry forests. Here we investigate patterns of root biomass, nutrient concentration/stocks, and the effect of aboveground biomass, chronic anthropogenic disturbance, precipitation, and soil depth as their potential drivers in a Caatinga dry forest, northeastern Brazil. Belowground biomass was obtained by a destructive method from July 2020 to March 2021 across forest stands in three 0.49-m2 and 1-m deep trenches per stand. Root biomass ranged from 3.23 Mg ha−1 to 29.6 Mg ha−1, accounting for less than 1/3 of total forest biomass. Belowground biomass greatly varied across forest stands mainly represented by large roots concentrated in the superficial soil layer. In general, nutrient concentrations and stocks did not vary significantly across forest stands regarding both root size class and soil depth, although fine roots support a higher concentration of P than large-sized roots. Finally, chronic disturbance, aboveground biomass, precipitation, and soil depth were important predictors affecting root biomass and nutrients. Our results suggest that a combination of natural and human-related drivers modulate root biomass and nutrients in Caatinga dry forests immersed in human-modified landscapes. • Belowground biomass is a key component of the Caatinga forest biomass. • Root biomass concentrates in the soil superficial layer and it is highly sensitive to precipitation. • Nutrient stocks are negatively affected by chronic anthropogenic disturbance. • Both natural and human-related drivers modulate root biomass and nutrients. [ABSTRACT FROM AUTHOR]

We report a generic smartphone app for quantitative annotation of complex images. The app is simple enough to be used by children, and annotation tasks are distributed across app users, contributing to efficient annotation. We demonstrate its flexibility and speed by annotating >30,000 images, including features of rice root growth and structure, stem cell aggregate morphology, and complex worm (Caenorhabditis elegans) postures, for which we show that the speed of annotation is >130-fold faster than state-of-the-art techniques with similar accuracy.

A Lie algebra L is said to be (Θ n , sl n) -graded if it contains a simple subalgebra g isomorphic to s l n such that the g -module L decomposes into copies of the adjoint module, the trivial module, the natural module V , its symmetric and exterior squares S 2 V and ∧ 2 V and their duals. We describe the multiplicative structures and the coordinate algebras of (Θ n , sl n) -graded Lie algebras for n ≥ 5 , classify these Lie algebras and determine their central extensions. [ABSTRACT FROM AUTHOR]