scholarly journals CLASSIFICATION OF SYMMETRY GROUPS FOR PLANAR -BODY CHOREOGRAPHIES

2013 ◽  
Vol 1 ◽  
Author(s):  
JAMES MONTALDI ◽  
KATRINA STECKLES
Keyword(s):  
Braid Group ◽  
Connected Components ◽  
Symmetry Groups ◽  
Exceptional Groups ◽  
Strong Force ◽  
Symmetry Properties ◽  
Foundational Work ◽  
Force Potential ◽  
Planar Body

AbstractSince the foundational work of Chenciner and Montgomery in 2000 there has been a great deal of interest in choreographic solutions of the $n$-body problem: periodic motions where the $n$ bodies all follow one another at regular intervals along a closed path. The principal approach combines variational methods with symmetry properties. In this paper, we give a systematic treatment of the symmetry aspect. In the first part, we classify all possible symmetry groups of planar $n$-body collision-free choreographies. These symmetry groups fall into two infinite families and, if $n$ is odd, three exceptional groups. In the second part, we develop the equivariant fundamental group and use it to determine the topology of the space of loops with a given symmetry, which we show is related to certain cosets of the pure braid group in the full braid group, and to centralizers of elements of the corresponding coset. In particular, we refine the symmetry classification by classifying the connected components of the set of loops with any given symmetry. This leads to the existence of many new choreographies in $n$-body systems governed by a strong force potential.

scholarly journals Finite simple exceptional groups of Lie type in which all subgroups of odd index are pronormal

2020 ◽  
Vol 23 (6) ◽  
pp. 999-1016
Author(s):  
Anatoly S. Kondrat’ev ◽  
Natalia V. Maslova ◽  
Danila O. Revin
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups ◽  
Exceptional Groups ◽  
Odd Index ◽  
Groups Of Lie Type

AbstractA subgroup H of a group G is said to be pronormal in G if H and {H^{g}} are conjugate in {\langle H,H^{g}\rangle} for every {g\in G}. In this paper, we determine the finite simple groups of type {E_{6}(q)} and {{}^{2}E_{6}(q)} in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal.

scholarly journals Performance of Various Deep-Learning Networks in the Seed Classification Problem

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1892
Author(s):  
Recep Eryigit ◽  
Bulent Tugrul
Keyword(s):  
Performance Metrics ◽  
Classification Problem ◽  
Grass Species ◽  
Test Accuracy ◽  
Learning Networks ◽  
Symmetry Properties ◽  
Depth Study ◽  
Image Set ◽  
Accuracy Performance

We report the results of an in-depth study of 15 variants of five different Convolutional Neural Network (CNN) architectures for the classification of seeds of seven different grass species that possess symmetry properties. The performance metrics of the nets are investigated in relation to the computational load and the number of parameters. The results indicate that the relation between the accuracy performance and operation count or number of parameters is linear in the same family of nets but that there is no relation between the two when comparing different CNN architectures. Using default pre-trained weights of the CNNs was found to increase the classification accuracy by ≈3% compared with training from scratch. The best performing CNN was found to be DenseNet201 with a 99.42% test accuracy for the highest resolution image set.

Nets of quadrics and deformations of Σ3〈3〉 singularities

1989 ◽  
Vol 105 (1) ◽  
pp. 109-115
Author(s):  
S. A. Edwards ◽  
C. T. C. Wall
Keyword(s):  
Parameter Space ◽  
General Type ◽  
Single Parameter ◽  
Connected Components ◽  
Quadratic Maps ◽  
Nets Of Quadrics

The 2-jet of a Σ3 map-germ f:(3, 0) → (3, 0) determines a net of quadratic maps from 3 to 3; for nets of general type this jet is sufficient for equivalence. The classification of such nets involves a single parameter c. It is shown in [7], also in [3], that the versai unfolding of f is topologically trivial over the parameter space. However, there are 4 connected components of this space of nets. The main object of this paper is to show that the corresponding unfolded maps are of different topological types.

Noether symmetries of vacuum classes of pp-waves and the wave equation

2016 ◽  
Vol 13 (09) ◽  
pp. 1650109 ◽  
Author(s):  
Sameerah Jamal ◽  
Ghulam Shabbir
Keyword(s):  
Wave Equation ◽  
Gravitational Waves ◽  
Wave Equations ◽  
Noether Symmetry ◽  
Symmetry Groups ◽  
Noether Symmetries ◽  
Noether’S Theorem ◽  
Metric Functions ◽  
Symmetry Algebras

The Noether symmetry algebras admitted by wave equations on plane-fronted gravitational waves with parallel rays are determined. We apply the classification of different metric functions to determine generators for the wave equation, and also adopt Noether's theorem to derive conserved forms. For the possible cases considered, there exist symmetry groups with dimensions two, three, five, six and eight. These symmetry groups contain the homothetic symmetries of the spacetime.

Sign in / Sign up

Export Citation Format

Share Document