Root Space Research Articles

On the full Kostant–Toda hierarchy and its ℓ-banded reductions for the Lie algebras of type A,B and G

Abstract This paper is concerned with the solutions of the full Kostant–Toda (f-KT) hierarchy in the Hessenberg form and their reductions to the ℓ -banded Kostant–Toda ( ℓ -KT) hierarchies for 1 ⩽ ℓ ⩽ n , where the case with ℓ = 1 is the classical tridiagonal Toda hierarchy and the case with ℓ = n is the f-KT hierarchy. Based on the Hessenberg form of the f-KT hierarchy, we study the f-KT hierarchy and the corresponding ℓ -KT hierarchies on simple Lie algebras of type A , B and G as the root space reductions with proper choices of Chevalley systems. Explicit formulas of the polynomial solutions for the τ-functions are also given in terms of the Schur functions and Schur’s Q-functions.

Accurate Measurement of Tower Grounding Resistance for Single-Tower and Multi-Tower Parallel Scenarios Based on the Clamp Meter Method: For the Sustainable Operation of Towers

Transmission tower grounding safety is a critical element that significantly impacts the reliability and sustainability of power grids. Tower grounding resistance is a significant grounding characteristic parameter. In order to accurately measure the grounding resistance of towers using the clamp meter method in both single-tower and multi-tower parallel scenarios, this paper establishes theoretical calculation models for measuring tower grounding resistance using the clamp meter method. Considering the interaction between the artificial grounding device and the tower foundation, this paper models and simulates transmission towers and lightning shield wires, analyzing the influencing factors on the grounding resistance measurement results using the clamp meter method in both single- and multi-tower parallel scenarios. The results show that for single towers, the clamp meter measurement results increase with decreasing foundation root spacing and increasing length of the artificial device’s extension lines, but the changes are relatively small. In multi-tower grounding scenarios, due to the shunting effect of the parallel branches formed by the towers and lightning shield wires, the value is smaller than in single-tower grounding scenarios. Changing the number of parallel towers and the type of lightning shield wire produces little effect on the measurement results. However, changes in soil resistivity have the most significant impact. Therefore, correction formulas for the impact of soil resistivity on clamp meter measurements are proposed and then verified by applying them to field tests.

Negatively Charged Carbon Dots Employed Symplastic and Apoplastic Pathways to Enable Better Plant Delivery than Positively Charged Carbon Dots.

Efficient delivery of nanoparticles (NPs) to plants is important for agricultural application. However, to date, we still lack knowledge about how NPs' charge matters for its translocation pathway, i.e., symplastic and apoplastic pathways, in plants. In this study, we synthesized and used negatively charged citrate sourced carbon dots (C-CDs, -37.97 ± 1.89 mV), Cy5 coated C-CDs (Cy5-C-CDs, -41.90 ± 2.55 mV), positively charged PEI coated carbon dots (P-CDs, +43.03 ± 1.71 mV), and Cy5 coated P-CDs (Cy5-P-CDs, +48.80 ± 1.21 mV) to investigate the role of surface charges and coatings on the employed translocation pathways (symplastic and apoplastic pathways) of charged NPs in plants. Our results showed that, different from the higher fluorescence intensity of P-CDs and Cy5-P-CDs in extracellular than intracellular space, the fluorescence intensity of C-CDs and Cy5-C-CDs was similar between intracellular and extracellular space in cucumber and cotton roots. It suggests that the negatively charged CDs were translocated via both symplastic and apoplastic pathways, but the positively charged CDs were mainly translocated via the apoplastic pathway. Furthermore, our results showed that root applied negatively charged C-CDs demonstrated higher leaf fluorescence than did positively charged P-CDs in both cucumber (8.09 ± 0.99 vs 3.75 ± 0.23) and cotton (7.27 ± 1.06 vs 3.23 ± 0.22), indicating that negatively charged CDs have a higher translocation efficiency from root to leaf than do positively charged CDs. It should be noted that CDs do not affect root cell activities, ROS level, and photosynthetic performance in cucumber and cotton, showing its good biocompatibility. Overall, this study not only figured out that root applied negatively charged CDs employed both symplastic and apoplastic pathways to do the transportation in roots compared with mainly the employment of apoplastic pathway for positively charge CDs, but also found that negatively charge CDs could be more efficiently translocated from root to leaf than positively charged CDs, indicating that imparting negative charge to NPs, at least CDs, matters for its efficient delivery in crops.

A hyperbolic Kac-Moody Calogero model

A new kind of quantum Calogero model is proposed, based on a hyperbolic Kac-Moody algebra. We formulate nonrelativistic quantum mechanics on the Minkowskian root space of the simplest rank-3 hyperbolic Lie algebra AE3 with an inverse-square potential given by its real roots and reduce it to the unit future hyperboloid. By stereographic projection this defines a quantum mechanics on the Poincaré disk with a unique potential. Since the Weyl group of AE3 is a ℤ2 extension of the modular group PSL(2,ℤ), the model is naturally formulated on the complex upper half plane, and its potential is a real modular function. We present and illustrate the relevant features of AE3, give some approximations to the potential and rewrite it as an (almost everywhere convergent) Poincaré series. The standard Dunkl operators are constructed and investigated on Minkowski space and on the hyperboloid. In the former case we find that their commutativity is obstructed by rank-2 subgroups of hyperbolic type (the simplest one given by the Fibonacci sequence), casting doubt on the integrability of the model. An appendix with Don Zagier investigates the computability of the potential. We foresee applications to cosmological billards and to quantum chaos.

Soil depth constraints in the revegetation of reclaimed coal mine land

Abstract. Hartati W, Syahrinudin, Sudarmadji T. 2024. Soil depth constraints in the revegetation of reclaimed coal mine land. Biodiversitas 25: 1838-1845. Plants require adequate root space, nutrients, water, and air for optimal growth. This study analyzed the relationship between plant growth (i.e., diameter) and soil properties (i.e., thickness, organic matter content, texture) on revegetated post-coal mining land in East Kalimantan, Indonesia. Field research was conducted at revegetated area primarily planted with four species (Hibiscus tiliaceus, Enterolobium cyclocarpum, Samanea saman and Acacia sp.) with age of 8 years. Data was collected from 30 plots of 0.05 ha. Plant diameter were recorded and soil samples were taken from 0-10, 10-30, and >30 cm depths using soil auger at three points per plot. Organic matter content was determined by the Walkley-Black method, and texture by the Pipette method. Multiple regression analysis was used to used to investigate the relationship between plant diameter and soil properties. The study found that different species showed varying correlations between soil properties and growth. Acacia sp. and S. saman exhibited strong correlations (r=0.78), while E. cyclocarpum had a low correlation (r=0.35), and H. tiliaceus a moderate one (r=0.49). Soil thickness had a stronger influence on H. tiliaceus (?=0.05) and S. saman (?=0.01) than other tested properties. For some species, diameter growth was primarily determined by soil thickness in the revegetated area. The study highlights the importance of considering species-specific soil requirements for successful revegetation and plant growth on post-coal mining sites.

Linkage of graphs with flows

We prove several linkage properties of graphs with flows, generalizing some results on linkage of graphs. This translates in properties of connectedness through codimension one of certain posets. For example, the poset of flows and the posets of odd and even tropical spin curves. These posets are, respectively, the posets underlying the moduli space of roots of divisors on tropical curves and the moduli spaces of odd and even tropical spin curves.

Chaos in Stochastic 2d Galerkin-Navier–Stokes

We prove that all Galerkin truncations of the 2d stochastic Navier–Stokes equations in vorticity form on any rectangular torus subjected to hypoelliptic, additive stochastic forcing are chaotic at sufficiently small viscosity, provided the frequency truncation satisfies N≥392\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 392$$\\end{document}. By “chaotic” we mean having a strictly positive Lyapunov exponent, i.e. almost-sure asymptotic exponential growth of the derivative with respect to generic initial conditions. A sufficient condition for such results was derived in previous joint work with Alex Blumenthal which reduces the question to the non-degeneracy of a matrix Lie algebra implying Hörmander’s condition for the Markov process lifted to the sphere bundle (projective hypoellipticity). The purpose of this work is to reformulate this condition to be more amenable for Galerkin truncations of PDEs and then to verify this condition using (a) a reduction to genericity properties of a diagonal sub-algebra inspired by the root space decomposition of semi-simple Lie algebras and (b) computational algebraic geometry executed by Maple in exact rational arithmetic. Note that even though we use a computer assisted proof, the result is valid for all aspect ratios and all sufficiently high dimensional truncations; in fact, certain steps simplify in the formal infinite dimensional limit.

Infinite affine, hyperbolic and Lorentzian Weyl groups with their associated Calogero models

We propose generalizations of Calogero models that exhibit invariance with respect to the infinite Weyl groups of affine, hyperbolic, and Lorentzian types. Our approach involves deriving closed analytic formulas for the action of the associated Coxeter elements of infinite order acting on arbitrary roots within their respective root spaces. These formulas are then utilized in formulating the new type of Calogero models.

Dissimilarity in root traits and spatial distribution promotes the productivity of Piper nigrum L. and tree species in mixture systems

Background and aimsClarifying why species mixtures produce greater biomass compared to monocultures is important for understanding the effect of biodiversity in natural and agricultural systems. At present, it remains unclear whether differential root traits and spatial distribution between the liana and tree species can promote productivity of mixtures. MethodsA 4-month glasshouse experiment and a 3-year field trial were conducted to examine the influence of root traits and the vertical and horizontal root distribution on productivity of independently supported liana Piper nigrum (Pn) and five other species. In the glasshouse experiment, Pn was grown in monoculture or mixed with one of five species: Coffea arabica, Areca catechu, Artocarpus heterophyllus, Camellia oleifera, and Coffea canephora. In the field trial, the Pn/Ac system and each monoculture were tested. ResultsCompared with monoculture, all Pn and mixture systems showed a positive net effect and a high complementarity effect on shoot biomass. The variation in complementarity effect was mainly associated with dissimilarity in the multidimensional root space trait and a few specific individual traits (root phosphatase activity and mycorrhizal colonization) between Pn and the neighbors. In the field, Pn in the mixture increased proportion of root length in the 20–40 cm soil layer and allocated less root length in the space shared between Pn and Ac, thus achieving spatial separation of roots and promoting the final yield. ConclusionThese findings suggest that the dissimilarity in root traits and spatial distribution may be an important mechanism for promoting the productivity of mixtures of liana and tree species.

Diazotrophic nitrogen fixation through aerial roots occurs in Avicennia marina: implications for adaptation of mangrove plant growth to low-nitrogen tidal flats.

Nitrogen limitation of primary production is common in coastal ecosystems. Mangrove trees maintain high levels of nitrogen fixation around their roots. The interior aerial space of mangrove roots, in which atmospheric gas is supplied through lenticels, could be efficient sites for nitrogen fixation. We measured tidal variations of partial pressure of N2 in root aerenchyma and conducted field experiments using 15 N2 as a tracer to track N2 movement through aerial roots of Avicennia marina. We used the acetylene reduction assay to identify the root parts harboring diazotrophs. The nitrogenase activity and estimated nitrogen fixation through aerenchyma were higher in pneumatophores and absorbing roots than in cable roots. Positive correlations between root nitrogen contents and turnover rates of root nitrogen derived from N2 through aerenchyma suggested that the internal supply of N2 to diazotrophs could be the main source for nitrogen assimilation by A. marina roots. Our results confirmed that N2 is supplied to diazotrophs through aerial roots and that nitrogen fixation occurs in A. marina roots. The aerial root structures, which occur across families of mangrove plants, could be an adaptation to survival in not only low-oxygen environments but also tidal flats with little plant-available nitrogen.

Character variation of root space microbial community composition in the response of drought-tolerant spring wheat to drought stress.

Drought is the most prevalent environmental stress in crop production, posing a significant danger to food security. Microorganisms in the crop root zone affect crop growth and development, enhance effective nutrient use, and resist adversity hazards. To analyze the changes and functional differences of root space microbial (endosphere-rhizosphere-bulk soil) communities in spring wheat under drought stress. In this study, the root, rhizosphere, and bulk soil of the drought-tolerant group (DTG, three varieties) and drought-sensitive group (DSG, three varieties) were collected. The control (CK, 25-28%), moderate drought (MD, 15-18%), and severe drought (SD, 9-12%) were analyzed by high-throughput sequencing and bioinformatics. The results showed significant differences in the diversity of Bacteria and Fungi in the root space of spring wheat under drought stress (P < 0.05), with the drought-tolerant group exhibiting higher microbial diversity. The microbial community change in spring wheat root space was mainly determined by the niche differentiation of endosphere, rhizosphere, and bulk soil and declined from endosphere to bulk soil due to drought. The antagonism between microbial and root-space species increased, and the community's complexity and stability deteriorated. Enriching drought-resistant preference groups like Actinobaciota, Variovorax, Streptomyces, and Conocybe altered the structure and function of the microbial community in the root space of spring wheat. Spring wheat's root space Bacteria and Fungi have different strategies to respond to drought.

Coupling resonances and spectral properties of the product of resolvent and perturbation

Given a self-adjoint operator H0 and a relatively compact self-adjoint perturbation V, we study in some detail the spectral properties of the product (H0−z)−1V, z∈C. The eigenvalues of this product are related simply to those of the corresponding product in the situation when the operator H0 is perturbed to H0+sV for s∈C. For some numbers rz, the eigenvalues of (H0+sV−z)−1V, s∈C, are (s−rz)−1. We study the root spaces of the eigenvalues (s−rz)−1 and complex-analytic properties of the functions rz such as branching points. In particular, for a generic case, we give a variety of necessary and sufficient conditions for branching. The functions rz, called coupling resonances, are important in the spectral analysis of H0+rV for any real number r∈R. For instance, they afford a description of the spectral shift function (SSF) of the pair H0 and V, as well as the absolutely continuous and singular parts of the SSF. A thorough study of real-valued coupling resonances rλ for real λ outside of the essential spectrum was carried out in a recent work by the first author. Here we extend this study to the complex domain, motivated by the fact, which is well known in the case of a rank-one perturbation, that the behaviour of coupling resonances rz near the essential spectrum provides valuable information about the latter.

Trees in planters – Growth, structure and ecosystem services of Platanus x hispanica and Tilia cordata and their reaction to soil drought

Urban environments are often characterized by extensive paved surfaces, exacerbating the urban heat island effect. At the same time, limited root space due to underground infrastructure poses a challenge for planting new trees in these areas. Trees in planters have emerged as popular design elements, offering innovative and sustainable greening solutions, particularly in urban environments with limited rooting space. However, growing conditions in planters may strongly impact tree growth and the provision of environmental ecosystem services (ES). In this 3-year study, we analyzed tree growth and ecosystem services (cooling by shading, CO2-fixation) of London plane (Platanus x hispanicaMünchh.) and small-leaved lime (Tilia cordataMill.) in four planting treatments: in-ground (G), planters in the ground (PG), non-insulated plastic planters (P), and insulated planters (PI). We also recorded soil temperature throughout the experiment and implemented soil drought conditions by reducing soil irrigation for half of the trees after one year. Our findings revealed higher thermal fluctuations in soil temperature within non-insulated plastic planters (P), reaching a maximum of 45 °C, surpassing the critical temperature threshold for plant growth (>38 °C). In contrast, insulated planters (PI) effectively mitigated soil temperatures, staying below 33.8 °C. When planted in the ground (G), P. x hispanica exhibited a significantly higher stem diameter increment (52–66%) compared to other planting treatments, aligning with the provision of ecosystem services. However, T. cordata trees showed a more moderate response to planting treatments in terms of growth and ecosystem service provision. Furthermore, the implementation of soil drought conditions resulted in a reduction of up to 34% in stem diameter increment for P. x hispanica and up to 25% for T. cordata. Our results underscore the necessity of tree species-specific knowledge about growth responses to different planting treatments for effective urban planning perspectives, as the provision of ecosystem services may be influenced differently.

On moduli spaces of roots in algebraic and tropical geometry

In this paper we construct a tropical moduli space parametrizing roots of divisors on tropical curves. We study the relation between this space and the skeleton of Jarvis moduli space of nets of limit roots on stable curves. We show that the combinatorics of the moduli space of tropical roots is governed by the poset of flows, a poset parametrizing certain flows on graphs.

Dynamic changes of soil microorganisms in rotation farmland at the western foot of the Greater Khingan range.

Crop rotation and other tillage systems can affect soil microbial communities and functions. Few studies have reported the response of soil spatial microbial communities to rotation under drought stress. Therefore, the purpose of our study was to explore the dynamic changes of the soil space microbial community under different drought stress-rotation patterns. In this study, two water treatments were set up, control W1 (mass water content 25%-28%), and drought W2 (mass water content 9%-12%). Four crop rotation patterns were set in each water content, spring wheat continuous (R1), spring wheat-potato (R2), spring wheat-potato-rape (R3) and spring wheat-rape (R4), for a total of eight treatments (W1R1, W1R2, W1R3, W1R4, W2R1, W2R2, W2R3, W2R4). Endosphere, rhizosphere and bulk soil of spring wheat in each treatment were collected, and root space microbial community data were generated. The soil microbial community changed under different treatments and their relationship with soil factors were analyzed using a co-occurrence network, mantel test, and other methods. The results revealed that the alpha diversity of microorganisms in the rhizosphere and bulk soil did not differ significantly, but it was significantly greater than in the endosphere. The bacteria community structure was more stable, fungi alpha-diversity significant changes (p < 0.05), that were more sensitive to the response of various treatments than bacteria. The co-occurrence network between fungal species was stable under rotation patterns (R2, R3, R4), while the community stability was poor under continuous cropping pattern (R1), and interactions were strengthened. Soil organic matter (SOM), microbial biomass carbon (MBC), and pH value were the most important factors dominating the bacteria community structural changed in the endosphere, rhizosphere, and bulk soil. The dominant factor that affected the fungal community structural changed in the endosphere, rhizosphere, and bulk soil was SOM. Therefore, we conclude that soil microbial community changes under the drought stress-rotation patterns are mainly influenced by soil SOM and microbial biomass content.

Engel’s Theorem in Mathlib

We discuss the theory of Lie algebras in Lean’s Mathlib library. Using nilpotency as the theme, we outline a computer formalisation of Engel’s theorem and an application to root space theory. We emphasise that all arguments work with coefficients in any commutative ring.

Root vertical spatial stress: A method for enhancing rhizosphere effect of plants in subsurface flow constructed wetland

The depth of the substrate of subsurface flow (SSF) constructed wetlands (CWs) is closely related to their cost and operation stability. To explore the physiological regulation mechanism of wetland plants and pollutant removal potential of SSF CWs under “vertical spatial stress of roots” (by greatly reducing the depth of the substrate in SSF CWs to limit the vertical growth space of roots, VSSR), the physiological response and wetland purification effect of a 0.1 m Canna indica L. CW under VSSR were studied compared with conventional SSF CWs (0.6 m, 1.2 m). The results demonstrated that VSSR significantly enhanced the dissolved oxygen (DO) concentration (p < 0.05) within the SSF CWs, with the DO in 0.1 m CW remaining stable at over 3 mg/L. Under the same hydraulic retention time (HRT), VSSR significantly improved the removal effect of pollutants (p < 0.05). The removal rates of COD, NH4+-N, and total phosphorus (TP) remained above 87%, and the mean removal rates of total nitrogen (TN) reached 91.71%. VSSR promoted the morphological adaptation mechanisms of plants, such as significantly increased root-shoot ratio (p < 0.05), changed biomass allocation. Plants could maintain the stability of the photosynthetic mechanism by changing the distribution of light energy. The results of microbial community function prediction demonstrated that aerobic denitrification was the main mechanism of N transformation in the 0.1 m CW under VSSR. VSSR could induce the high root activity of plants, augment the concentration of root exudates, enhance the redox environment of the plant rhizosphere, further foster the enrichment of aerobic denitrifying bacteria, and strengthen the absorption efficiency of wetland plants and substrate, thus achieving an efficient pollutant removal capacity. Studies showed that VSSR was an effective means to enhance the rhizosphere effect of plants and pollutant removal in SSF CWs.

"Endodontic Retreatment in the Cure of Apical Periodontitis; Lateral Canal Involvement: Report of a Case with 18 Years of Evolution"

The objectives of endodontics are the prevention and cure of periapical disease (Apical Periodontitis); which is achieved by cleaning and disinfection of the root canal system and subsequent obturation of root spaces by filling with inert materials, avoiding reinfection and persistent inflammation [1].

Logarithmic moduli of roots of line bundles on curves

We use the theory of logarithmic line bundles to construct compactifications of spaces of roots of a line bundle on a family of curves, generalising work of a number of authors. This runs via a study of the torsion in the tropical and logarithmic jacobians (recently constructed by Molcho and Wise). Our moduli space carries a ‘double ramification cycle’ measuring the locus where the given root is isomorphic to the trivial bundle, and we give a tautological formula for this class in the language of piecewise polynomial functions (as recently developed by Molcho–Pandharipande–Schmitt and Holmes–Schwarz).

Geomorphodynamics, evolution, and ecology of vertical roots.

The roots of some coastal and wetland trees grow peculiar vertical protrusions, the function of which remains unclear. Here, using computational simulations based on first-principles fluid and sedimentation dynamics, we argue that the protrusions work together to create an elevated patch of sediment downstream of the tree, thereby creating its own fertile flood-protected breeding grounds for the seedlings. In our simulations, we vary the vertical root diameter, root spacing and total root area and show that there is an optimal vertical root spacing that depends on root thickness. Next, we quantify and discuss the cooperative effects between adjacent vertical root patches. Lastly, by varying vertical root spacing of a patch of trees, we estimate a maximal vegetation density for which vertical-root production has a beneficial geomorphological response. Our hypothesis suggests that vertical roots, such as the 'knee roots' of baldcypress trees, have an important role in shaping riparian geomorphology and community structure.