Description of reflection-generated polytopes using decorated Coxeter diagrams

1995 ◽  
Vol 73 (9-10) ◽  
pp. 566-584 ◽  
Author(s):  
B. Champagne ◽  
M. Kjiri ◽  
J. Patera ◽  
R. T. Sharp
Keyword(s):  
Symmetry Group ◽  
Weyl Group ◽  
Lie Group ◽  
Single Point ◽  
Explicit Description ◽  
New Method ◽  
Maximal Dimension ◽  
Group D

A new method of explicit description of n-dimensional polytopes generated by reflections of a single point (D-polytopes) or a face of maximal dimension (V-polytopes) is used to provide a comprehensive uniform description of all such polytopes in dimensions three and four. In addition a class of six-dimensional polytopes is also described; its symmetry group is the Weyl group of the simple Lie group D6.

scholarly journals Construction and Characterization of Representations of SU(7) for GUT Model Builders

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1044
Author(s):  
Daniel Jones ◽  
Jeffery A. Secrest
Keyword(s):  
Gauge Symmetry ◽  
Symmetry Group ◽  
Lie Group ◽  
Cartan Subalgebra ◽  
Natural Extension ◽  
Grand Unification ◽  
Raising And Lowering Operators ◽  
Higher Dimensional ◽  
Lowering Operators

The natural extension to the SU(5) Georgi-Glashow grand unification model is to enlarge the gauge symmetry group. In this work, the SU(7) symmetry group is examined. The Cartan subalgebra is determined along with their commutation relations. The associated roots and weights of the SU(7) algebra are derived and discussed. The raising and lowering operators are explicitly constructed and presented. Higher dimensional representations are developed by graphical as well as tensorial methods. Applications of the SU(7) Lie group to supersymmetric grand unification as well as applications are discussed.

scholarly journals Three-dimensional analysis of upper airway morphology in skeletal Class III patients with and without mandibular asymmetry

2017 ◽  
Vol 87 (4) ◽  
pp. 526-533 ◽  
Author(s):  
Xi Wen ◽  
Xiyu Wang ◽  
Shuqi Qin ◽  
Lorenzo Franchi ◽  
Yan Gu
Keyword(s):  
Symmetry Group ◽  
Three Dimensional ◽  
Upper Airway ◽  
Class Iii ◽  
Cross Sectional ◽  
Skeletal Class ◽  
Mandibular Asymmetry ◽  
Subgroup Ii ◽  
Group D ◽  
Inferior Limit

ABSTRACT Objective: To compare the three-dimensional (3D) morphology of the upper airway in skeletal Class III patients with and without mandibular asymmetry and to investigate the possible underlying correlations between the morphology of the upper airway and mandibular deviation. Materials and Methods: Cone-beam computed tomography images of 54 subjects with skeletal Class III malocclusion (ANB angle ≤ 0.4°, Wits ≤ −5.5°) were taken and 3D upper airway models were reconstructed using Dolphin 3D software. According to the distance (d) from symphysis menti to the sagittal plane, all subjects were divided into a symmetry group (d ≤ 2 mm) and an asymmetry group (d ≥ 4 mm). Based on the severity of mandibular deviation, the asymmetry group was divided into subgroup I (4 mm ≤d <10 mm) and subgroup II (d ≥ 10 mm). Cross-sectional linear distances, areas, and volumetric variables of the upper airway were measured in the 3D airway model. Results: Width of the inferior limit of the glossopharynx (P3W), cross-sectional area of the anterior limit of the nasal airway (P5S), and height of the glossopharynx (GPH) in the asymmetry group were significantly larger than in the symmetry group. As for subjects with severe mandibular deviation in subgroup II (d ≥ 10 mm), volume of the glossopharynx (GPV), total volume of the pharynx (TPV), length of the inferior limit of the velopharynx (P2L), and ratio of length to width of the inferior limit of the velopharynx (P2L/P2W) showed significantly negative correlations with mandibular deviation (r > 0.7, P < .05). Conclusions: In Class III subjects with severe mandibular asymmetry, the pharyngeal airway showed a tendency toward constriction and presented a more elliptical shape as mandibular deviation became more severe (P < .01).

Exotic Holonomies ${\rm E}^{(a)}_7$

1997 ◽  
Vol 08 (05) ◽  
pp. 583-594 ◽  
Author(s):  
Quo-Shin Chi ◽  
Sergey Merkulov ◽  
Lorenz Schwachhöfer
Keyword(s):  
Symmetry Group ◽  
Lie Groups ◽  
Lie Group ◽  
Moduli Spaces ◽  
Local Symmetry ◽  
Positive Dimension ◽  
Affine Connections ◽  
Finite Dimensional ◽  
Torsion Free

It is proved that the Lie groups [Formula: see text] and [Formula: see text] represented in ℝ56 and the Lie group [Formula: see text] represented in ℝ112 occur as holonomies of torsion-free affine connections. It is also shown that the moduli spaces of torsion-free affine connections with these holonomies are finite dimensional, and that every such connection has a local symmetry group of positive dimension.

scholarly journals Invariant Poisson–Nijenhuis structures on Lie groups and classification

2018 ◽  
Vol 15 (04) ◽  
pp. 1850059 ◽  
Author(s):  
Zohreh Ravanpak ◽  
Adel Rezaei-Aghdam ◽  
Ghorbanali Haghighatdoost
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Symmetry Group ◽  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Text Structure ◽  
Baxter Equation ◽  
Text Structures ◽  
Group A

We study right-invariant (respectively, left-invariant) Poisson–Nijenhuis structures ([Formula: see text]-[Formula: see text]) on a Lie group [Formula: see text] and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra [Formula: see text]. We show that [Formula: see text]-[Formula: see text] structures can be used to find compatible solutions of the classical Yang–Baxter equation (CYBE). Conversely, two compatible [Formula: see text]-matrices from which one is invertible determine an [Formula: see text]-[Formula: see text] structure. We classify, up to a natural equivalence, all [Formula: see text]-matrices and all [Formula: see text]-[Formula: see text] structures with invertible [Formula: see text] on four-dimensional symplectic real Lie algebras. The result is applied to show that a number of dynamical systems which can be constructed by [Formula: see text]-matrices on a phase space whose symmetry group is Lie group a [Formula: see text], can be specifically determined.

On the construction of supergrassmannians as homogeneous superspaces

2020 ◽  
pp. 2150053
Author(s):  
Mohammad Mohammadi ◽  
Saad Varsaie
Keyword(s):  
Lie Group ◽  
Explicit Description ◽  
Functor Of Points

In this paper, we give an explicit description of the action of the super Lie group [Formula: see text] on supergrassmannian [Formula: see text] in the functor of points language. In particular, we give a concrete proof of the transitively of this action, and the gluing of the local charts of the supergrassmannian.

scholarly journals On the construction of $ \mathbb Z^n_2- $grassmannians as homogeneous $ \mathbb Z^n_2- $spaces

2021 ◽  
Vol 30 (1) ◽  
pp. 221-241
Author(s):  
Mohammad Mohammadi ◽  
◽  
Saad Varsaie
Keyword(s):  
Lie Group ◽  
Explicit Description ◽  
Functor Of Points

<abstract><p>In this paper, we construct the $ \mathbb Z^n_2- $grassmannians by gluing of the $ \mathbb Z^n_2- $domains and give an explicit description of the action of the $ \mathbb Z^n_2- $Lie group $ GL(\overrightarrow{\textbf{m}}) $ on the $ \mathbb Z^n_2- $grassmannian $ G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}}) $ in the functor of points language. In particular, we give a concrete proof of the transitively of this action, and the gluing of the local charts of the $ \mathbb Z^n_2- $grassmannian.</p></abstract>

scholarly journals Reconstructing the wave speed and the source

2021 ◽  
Author(s):  
Amin Boumenir ◽  
Vu Kim Tuan
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Wave Speed ◽  
Simple Procedure ◽  
Single Point ◽  
Spectral Estimation ◽  
New Method ◽  
Estimation Techniques ◽  
Nodal Lines

We are concerned with the inverse problem of recovering the unknown wave speed and also the source in a multidimensional wave equation. We show that the wave speed coefficient can be reconstructed from the observations of the solution taken at a single point. For the source, we may need a sequence of observation points due to the presence of multiple spectrum and nodal lines. This new method, based on spectral estimation techniques, leads to a simple procedure that delivers both uniqueness and reconstruction of the coefficients at the same time.

Decomposition Strategies for the Synthesis of Single Input-Single Output and Dual Input-Single Output Compliant Mechanisms

2007 ◽  
Author(s):  
Charles Kim
Keyword(s):  
Compliant Mechanisms ◽  
Compliant Mechanism ◽  
Single Point ◽  
Building Blocks ◽  
New Method ◽  
Single Input Single Output ◽  
Single Output ◽  
Alternative Method ◽  
Decomposition Strategies ◽  
Single Input

In this paper a new method for the synthesis of compliant mechanism topologies is presented which involves the decomposition of motion requirements into more easily solved sub-problems. The decomposition strategies are presented and demonstrated for both single input-single output (SISO) and dual input-single output (DISO) planar compliant mechanisms. The methodology makes use of the single point synthesis (SPS) which effectively generates topologies which satisfy motion requirements at one point by assembling compliant building blocks. The SPS utilizes compliance and stiffness ellipsoids to characterize building blocks and to combine them in an intelligent manner. Both the SISO and DISO problems are decomposed into sub-problems which may be addressed by the SPS. The decomposition strategies are demonstrated with illustrative example problems. This paper presents an alternative method for the synthesis of compliant mechanisms which augments designer insight.

A Novel Blind Source Separation Method Based on the Extended Natural Gradient

2012 ◽  
Vol 433-440 ◽  
pp. 3570-3576
Author(s):  
Yu Feng Xue ◽  
Yu Jia Wang ◽  
Qiu Dong Sun
Keyword(s):  
Iterative Algorithm ◽  
Blind Source Separation ◽  
Lie Group ◽  
Mathematical Proof ◽  
Source Separation ◽  
Separation Method ◽  
New Method ◽  
Invariance Property ◽  
Natural Gradient ◽  
Group Invariance

In this paper, a new method is introduced to derive the extended natural gradient, which was proposed by Lewicki and Sejnowski in [1]. However, they made their derivation under many approximations, and the proof is also very complicated. To give a more rigors mathematical proof for this gradient, the Lie group invariance property is introduced which makes the proof much easier and straightforward. In addition, an iterative algorithm through Newton's method is also given to estimate the sources efficiently. The results of the experiments confirm the efficiency of the proposed method.

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