scholarly journals Clifford Spinors and Root System Induction: $$H_4$$ and the Grand Antiprism

2021 ◽  
Vol 31 (3) ◽  
Author(s):  
Pierre-Philippe Dechant
Keyword(s):  
Root System ◽  
Root Systems ◽  
Invariant Sets ◽  
Simple Construction ◽  
Reflection Groups ◽  
Computational Work ◽  
Algebraic Framework ◽  
Underlying Geometry ◽  
Coxeter Plane ◽  
Insight Into

AbstractRecent work has shown that every 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of $$H_3\rightarrow H_4$$ H 3 → H 4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan–Dieudonné theorem, giving a simple construction of the $${\mathrm {Pin}}$$ Pin and $${\mathrm {Spin}}$$ Spin covers. Using this connection with $$H_3$$ H 3 via the induction theorem sheds light on geometric aspects of the $$H_4$$ H 4 root system (the 600-cell) as well as other related polytopes and their symmetries, such as the famous Grand Antiprism and the snub 24-cell. The uniform construction of root systems from 3D and the uniform procedure of splitting root systems with respect to subrootsystems into separate invariant sets allows further systematic insight into the underlying geometry. All calculations are performed in the even subalgebra of $${\mathrm {Cl}}(3)$$ Cl ( 3 ) , including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes, and are shared as supplementary computational work sheets. This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework.

scholarly journals Twisted quadratic foldings of root systems

2021 ◽  
Vol 33 (1) ◽  
pp. 65-84
Author(s):  
M. Lanini ◽  
K. Zainoulline
Keyword(s):  
Root System ◽  
Equivariant Cohomology ◽  
Quaternion Algebra ◽  
Root Systems ◽  
Group Cohomology ◽  
Characteristic Classes ◽  
Reflection Groups ◽  
Content Type ◽  
Dynkin Diagrams ◽  
Projective Homogeneous Varieties

The present paper is devoted to twisted foldings of root systems that generalize the involutive foldings corresponding to automorphisms of Dynkin diagrams. A motivating example is Lusztig’s projection of the root system of type E 8 E_8 onto the subring of icosians of the quaternion algebra, which gives the root system of type H 4 H_4 . By using moment graph techniques for any such folding, a map at the equivariant cohomology level is constructed. It is shown that this map commutes with characteristic classes and Borel maps. Restrictions of this map to the usual cohomology of projective homogeneous varieties, to group cohomology and to their virtual analogues for finite reflection groups are also introduced and studied.

On Closed Subsets of Root Systems

1994 ◽  
Vol 37 (3) ◽  
pp. 338-345 ◽  
Author(s):  
D. Ž. Doković ◽  
P. Check ◽  
J.-Y. Hée
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Irreducible Component ◽  
Product Space ◽  
Root Systems ◽  
Inner Product Space ◽  
Inner Product ◽  
Closed Sets ◽  
Finite Dimensional ◽  
Real Inner Product Space

AbstractLet R be a root system (in the sense of Bourbaki) in a finite dimensional real inner product space V. A subset P ⊂ R is closed if α, β ∊ P and α + β ∊ R imply that α + β ∊ P. In this paper we shall classify, up to conjugacy by the Weyl group W of R, all closed sets P ⊂ R such that R\P is also closed. We also show that if θ:R —> R′ is a bijection between two root systems such that both θ and θ-1 preserve closed sets, and if R has at most one irreducible component of type A1, then θ is an isomorphism of root systems.

Extension and Longitudinal Growth During the Development of Red Pine Root Systems

1975 ◽  
Vol 5 (1) ◽  
pp. 109-121 ◽  
Author(s):  
D. C. F. Fayle
Keyword(s):  
Root System ◽  
Root Systems ◽  
Red Pine ◽  
Longitudinal Growth ◽  
Rooting Depth ◽  
Soil Conditions ◽  
Moisture Availability ◽  
Below Ground

Extension of the root system and stem during the first 30 years of growth of plantation-grown red pine (Pinusresinosa Ait.) on four sites was deduced by root and stem analyses. Maximum rooting depth was reached in the first decade and maximum horizontal extension of roots was virtually complete between years 15 and 20. The main horizontal roots of red pine seldom exceed 11 m in length. Elongation of vertical and horizontal roots was examined in relation to moisture availability and some physical soil conditions. The changing relations within the tree in lineal dimensions and annual elongation of the roots and stem are illustrated. The development of intertree competition above and below ground is considered.

Effects of Temperature on Root Morphology and Ectendomycorrhizal Development in Pinusresinosa Ait.

1975 ◽  
Vol 5 (2) ◽  
pp. 171-175 ◽  
Author(s):  
Hugh E. Wilcox ◽  
Ruth Ganmore-Neumann
Keyword(s):  
Root Growth ◽  
Root System ◽  
Root Morphology ◽  
Root Systems ◽  
Second Order ◽  
Root Temperature ◽  
First Order ◽  
Effects Of Temperature

Seedlings of Pinusresinosa were grown at root temperatures of 16, 21 and 27 °C, both aseptically and after inoculation with the ectendomycorrhizal fungus BDG-58. Growth after 3 months was significantly influenced by the presence of the fungus at all 3 temperatures. The influence of the fungus on root growth was obscured by the effects of root temperature on morphology. The root system at 16 and at 21 °C possessed many first-order laterals with numerous, well developed second-order branches, but those at 27 °C had only a few, relatively long, unbranched first-order laterals. Although the root systems of infected seedlings were larger, the fungus increased root growth in the same pattern as determined by the temperature.

scholarly journals Contemporary Concepts of Root System Architecture of Urban Trees

2010 ◽  
Vol 36 (4) ◽  
pp. 149-159
Author(s):  
Susan Day ◽  
P. Eric Wiseman ◽  
Sarah Dickinson ◽  
J. Roger Harris
Keyword(s):  
Built Environment ◽  
Root Growth ◽  
Root System ◽  
Root Architecture ◽  
Root Systems ◽  
Growth Patterns ◽  
Urban Trees ◽  
Rooting Depth ◽  
Similar Species ◽  
Urban Settings

Knowledge of the extent and distribution of tree root systems is essential for managing trees in the built environment. Despite recent advances in root detection tools, published research on tree root architecture in urban settings has been limited and only partially synthesized. Root growth patterns of urban trees may differ considerably from similar species in forested or agricultural environments. This paper reviews literature documenting tree root growth in urban settings as well as literature addressing root architecture in nonurban settings that may contribute to present understanding of tree roots in built environments. Although tree species may have the genetic potential for generating deep root systems (>2 m), rooting depth in urban situations is frequently restricted by impenetrable or inhospitable soil layers or by underground infrastructure. Lateral root extent is likewise subject to restriction by dense soils under hardscape or by absence of irrigation in dry areas. By combining results of numerous studies, the authors of this paper estimated the radius of an unrestricted root system initially increases at a rate of approximately 38 to 1, compared to trunk diameter; however, this ratio likely considerably declines as trees mature. Roots are often irregularly distributed around the tree and may be influenced by cardinal direction, terrain, tree lean, or obstacles in the built environment. Buttress roots, tap roots, and other root types are also discussed.

scholarly journals Lorentzian Toda field theories

2021 ◽  
pp. 2150017
Author(s):  
Andreas Fring ◽  
Samuel Whittington
Keyword(s):  
Lie Algebras ◽  
Mass Spectra ◽  
Energy Momentum Tensor ◽  
Root Systems ◽  
Momentum Tensor ◽  
Field Theories ◽  
Simple Lie Algebras ◽  
Different Types ◽  
Algebraic Framework ◽  
Complex Valued

We propose several different types of construction principles for new classes of Toda field theories based on root systems defined on Lorentzian lattices. In analogy to conformal and affine Toda theories based on root systems of semi-simple Lie algebras, also their Lorentzian extensions come about in conformal and massive variants. We carry out the Painlevé integrability test for the proposed theories, finding in general only one integer valued resonance corresponding to the energy-momentum tensor. Thus most of the Lorentzian Toda field theories are not integrable, as the remaining resonances, that grade the spins of the W-algebras in the semi-simple cases, are either non-integer or complex valued. We analyze in detail the classical mass spectra of several massive variants. Lorentzian Toda field theories may be viewed as perturbed versions of integrable theories equipped with an algebraic framework.

A biochemical study of BAS 517 using excised corn and soybean root systems

Weed Science ◽  
1999 ◽  
Vol 47 (1) ◽  
pp. 28-36
Author(s):  
Hwei-Yiing Li ◽  
Chester L. Foy
Keyword(s):  
Root System ◽  
Biochemical Study ◽  
Lipid Fraction ◽  
Root Systems ◽  
Water Soluble ◽  
Root Tips ◽  
Soybean Root ◽  
Corn Roots ◽  
Tracer Techniques ◽  
High Degree

The mode of action of BAS 517 in a susceptible plant species, corn, was investigated using an excised root system and14C-tracer techniques. The root system of a tolerant species, soybean, was used for comparison. When UL-14C- glucose was used as a precursor,14C incorporation into lipids was reduced in BAS 517-treated corn roots, although14C incorporation from UL-14C-glucose into lipids was relatively low. Inhibition of14C incorporation into water-soluble compounds was not definite because of a high degree of variability. Using14C-acetate as a precursor, 49, 43, and 34% of the recovered radioactivity was found in the lipid fractions of root tips treated with 0, 1.0, and 10 μM BAS 517, respectively. In nontreated soybean root tips, 47% of the recovered radioactivity was found in the lipid fraction compared to 49% in root tips treated with 10 μM BAS 517. Further analysis of lipids showed that BAS 517 inhibited the incorporation of14C from14C-acetate into phosphatidylethanolamine, a phospholipid, whereas the labeling of sterols in treated corn roots was not adversely affected. Acetyl CoA carboxylase extracted from root systems of corn and soybean showed different sensitivity to BAS 517, suggesting its role as the herbicide target site and as a basis for the selectivity.

scholarly journals MARSHAL, a novel tool for virtual phenotyping of maize root system hydraulic architectures

2019 ◽  
Vol 2 (1) ◽  
Author(s):  
Félicien Meunier ◽  
Adrien Heymans ◽  
Xavier Draye ◽  
Valentin Couvreur ◽  
Mathieu Javaux ◽  
...  
Keyword(s):  
Root System ◽  
Web Application ◽  
Root Traits ◽  
Root Systems ◽  
R Package ◽  
Maize Root ◽  
Model Parameters ◽  
System Modelling ◽  
Architecture Model ◽  
System Models

Abstract Functional-structural root system models combine functional and structural root traits to represent the growth and development of root systems. In general, they are characterized by a large number of growth, architectural and functional root parameters, generating contrasted root systems evolving in a highly non-linear environment (soil, atmosphere), which makes the link between local traits and functioning unclear. On the other end of the root system modelling continuum, macroscopic root system models associate to each root system a set of plant-scale, easily interpretable parameters. However, as of today, it is unclear how these macroscopic parameters relate to root-scale traits and whether the upscaling of local root traits is compatible with macroscopic parameter measurements. The aim of this study was to bridge the gap between these two modelling approaches. We describe here the MAize Root System Hydraulic Architecture soLver (MARSHAL), a new efficient and user-friendly computational tool that couples a root architecture model (CRootBox) with fast and accurate algorithms of water flow through hydraulic architectures and plant-scale parameter calculations. To illustrate the tool’s potential, we generated contrasted maize hydraulic architectures that we compared with root system architectural and hydraulic observations. Observed variability of these traits was well captured by model ensemble runs. We also analysed the multivariate sensitivity of mature root system conductance, mean depth of uptake, root system volume and convex hull to the input parameters to highlight the key model parameters to vary for virtual breeding. It is available as an R package, an RMarkdown pipeline and a web application.

scholarly journals Comparison of Root System Morphology of Cucurbit Rootstocks for Use in Watermelon Grafting

2018 ◽  
Vol 28 (5) ◽  
pp. 629-636 ◽  
Author(s):  
Matthew B. Bertucci ◽  
David H. Suchoff ◽  
Katherine M. Jennings ◽  
David W. Monks ◽  
Christopher C. Gunter ◽  
...  
Keyword(s):  
Root System ◽  
Surface Area ◽  
Root Length ◽  
Dry Weight ◽  
Root Systems ◽  
Root Surface ◽  
Root Diameter ◽  
Root Surface Area ◽  
Diameter Class ◽  
Main Effect

Grafting of watermelon (Citrullus lanatus) is an established production practice that provides resistance to soilborne diseases or tolerance to abiotic stresses. Watermelon may be grafted on several cucurbit species (interspecific grafting); however, little research exists to describe root systems of these diverse rootstocks. A greenhouse study was conducted to compare root system morphology of nine commercially available cucurbit rootstocks, representing four species: pumpkin (Cucurbita maxima), squash (Cucurbita pepo), bottle gourd (Lagenaria siceraria), and an interspecific hybrid squash (C. maxima × C. moschata). Rootstocks were grafted with a triploid watermelon scion (‘Exclamation’), and root systems were compared with nongrafted (NG) and self-grafted (SG) ‘Exclamation’. Plants were harvested destructively at 1, 2, and 3 weeks after transplant (WAT), and data were collected on scion dry weight, total root length (TRL), average root diameter, root surface area, root:shoot dry-weight ratio, root diameter class proportions, and specific root length. For all response variables, the main effect of rootstock and rootstock species was significant (P < 0.05). The main effect of harvest was significant (P < 0.05) for all response variables, with the exception of TRL proportion in diameter class 2. ‘Ferro’ rootstock produced the largest TRL and root surface area, with observed values 122% and 120% greater than the smallest root system (‘Exclamation’ SG), respectively. Among rootstock species, pumpkin produced the largest TRL and root surface area, with observed values 100% and 82% greater than those of watermelon, respectively. These results demonstrate that substantial differences exist during the initial 3 WAT in root system morphology of rootstocks and rootstock species available for watermelon grafting and that morphologic differences of root systems can be characterized using image analysis.

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