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.

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.

Preliminary notions

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Root System ◽  
Maximal Torus ◽  
Reductive Group ◽  
Root Systems ◽  
Positive System ◽  
Reductive Groups ◽  
Dynkin Diagrams ◽  
Standard Construction ◽  
Minimal Type ◽  
Split Maximal Torus

This chapter considers some preliminary notions, starting with standard pseudo-reductive groups, Levi subgroups, and root systems. It reviews the “standard construction” of pseudo-reductive k-groups and shows that any connected reductive group equipped with a chosen split maximal torus is generated by that maximal torus and its root groups for the simple positive and negative roots relative to a choice of positive system of roots in the root system. It also discusses the basic exotic construction, noting that the only nontrivial multiplicities that occur for the edges of Dynkin diagrams of reduced irreducible root systems are 2 and 3. Finally, it explains the minimal type pseudo-reductive k-group G, along with quotient homomorphism between pseudo-reductive groups.

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.

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.

Kinematic calibration of serial robot using dual quaternions

2019 ◽  
Vol 46 (2) ◽  
pp. 247-258 ◽  
Author(s):  
Guozhi Li ◽  
Fuhai Zhang ◽  
Yili Fu ◽  
Shuguo Wang
Keyword(s):  
Kinematic Analysis ◽  
Quaternion Algebra ◽  
Error Model ◽  
Kinematic Calibration ◽  
Dual Quaternion ◽  
Robot Calibration ◽  
Geometrical Meaning ◽  
Content Type ◽  
Dual Quaternions ◽  
Serial Robot

Purpose The purpose of this paper is to propose an error model for serial robot kinematic calibration based on dual quaternions. Design/methodology/approach The dual quaternions are the combination of dual-number theory and quaternion algebra, which means that they can represent spatial transformation. The dual quaternions can represent the screw displacement in a compact and efficient way, so that they are used for the kinematic analysis of serial robot. The error model proposed in this paper is derived from the forward kinematic equations via using dual quaternion algebra. The full pose measurements are considered to apply the error model to the serial robot by using Leica Geosystems Absolute Tracker (AT960) and tracker machine control (T-MAC) probe. Findings Two kinematic-parameter identification algorithms are derived from the proposed error model based on dual quaternions, and they can be used for serial robot calibration. The error model uses Denavit–Hartenberg (DH) notation in the kinematic analysis, so that it gives the intuitive geometrical meaning of the kinematic parameters. The absolute tracker system can measure the position and orientation of the end-effector (EE) simultaneously via using T-MAC. Originality/value The error model formulated by dual quaternion algebra contains all the basic geometrical parameters of serial robot during the kinematic calibration process. The vector of dual quaternion error can be used as an indicator to represent the trend of error change of robot’s EE between the nominal value and the actual value. The accuracy of the EE is improved after nearly 20 measurements in the experiment conduct on robot SDA5F. The simulation and experiment verify the effectiveness of the error model and the calibration algorithms.

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.

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