scholarly journals Invariant classification of second-order conformally flat superintegrable systems

2014 ◽  
Vol 47 (49) ◽  
pp. 495202 ◽  
Author(s):  
J J Capel ◽  
J M Kress
Keyword(s):  
Second Order ◽  
Conformally Flat ◽  
Superintegrable Systems

Helicoidal surfaces with prescribed curvatures in a conformally flat 3-space

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Chul Woo Lee ◽  
Jae Won Lee ◽  
Dae Won Yoon
Keyword(s):  
Mean Curvature ◽  
Extrinsic Curvature ◽  
Conformal Factor ◽  
Second Order ◽  
Conformally Flat ◽  
Non Linear ◽  
Linear Ode ◽  
Helicoidal Surfaces ◽  
Conformally Flat Metric

Abstract In this paper, we study a conformally flat 3-space 𝔽 3 {\mathbb{F}_{3}} which is an Euclidean 3-space with a conformally flat metric with the conformal factor 1 F 2 {\frac{1}{F^{2}}} , where F ⁢ ( x ) = e - x 1 2 - x 2 2 {F(x)=e^{-x_{1}^{2}-x_{2}^{2}}} for x = ( x 1 , x 2 , x 3 ) ∈ ℝ 3 {x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}} . In particular, we construct all helicoidal surfaces in 𝔽 3 {\mathbb{F}_{3}} by solving the second-order non-linear ODE with extrinsic curvature and mean curvature functions. As a result, we give classification of minimal helicoidal surfaces as well as examples for helicoidal surfaces with some extrinsic curvature and mean curvature functions in 𝔽 3 {\mathbb{F}_{3}} .

scholarly journals Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators

2020 ◽  
Author(s):  
Bjorn K. Berntson ◽  
◽  
Ernest G. Kalnins ◽  
Willard Miller ◽  
◽  
...  
Keyword(s):  
Euclidean Space ◽  
Flat Space ◽  
Structure Theory ◽  
Conformally Flat ◽  
3 Dimensional ◽  
Superintegrable Systems ◽  
Complex Euclidean Space ◽  
Spatial Coordinates ◽  
Symmetry Generators

We make significant progress toward the classification of 2nd order superintegrable systems on 3-dimensional conformally flat space that have functionally linearly dependent (FLD) symmetry generators, with special emphasis on complex Euclidean space. The symmetries for these systems are linearly dependent only when the coefficients are allowed to depend on the spatial coordinates. The Calogero-Moser system with 3 bodies on a line and 2-parameter rational potential is the best known example of an FLD superintegrable system. We work out the structure theory for these FLD systems on 3D conformally flat space and show, for example, that they always admit a 1st order symmetry. A partial classification of FLD systems on complex 3D Euclidean space is given. This is part of a project to classify all 3D 2nd order superintegrable systems on conformally flat spaces.

SATANIC AND THELEMIC CIRCLES ON KNOTS

2013 ◽  
Vol 22 (05) ◽  
pp. 1350017 ◽  
Author(s):  
G. FLOWERS
Keyword(s):  
Geometric Interpretation ◽  
Second Order ◽  
Geometric Properties ◽  
Vassiliev Invariants ◽  
Vassiliev Invariant ◽  
Knot Diagrams

While Vassiliev invariants have proved to be a useful tool in the classification of knots, they are frequently defined through knot diagrams, and fail to illuminate any significant geometric properties the knots themselves may possess. Here, we provide a geometric interpretation of the second-order Vassiliev invariant by examining five-point cocircularities of knots, extending some of the results obtained in [R. Budney, J. Conant, K. P. Scannell and D. Sinha, New perspectives on self-linking, Adv. Math. 191(1) (2005) 78–113]. Additionally, an analysis on the behavior of other cocircularities on knots is given.

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