Menger curvature as a knot energy

2013 ◽  
Vol 530 (3) ◽  
pp. 257-290 ◽  
Author(s):  
Paweł Strzelecki ◽  
Heiko von der Mosel
Keyword(s):  
Menger Curvature ◽  
Knot Energy

Feature selection method based on Menger curvature and LDA theory for a P300 brain-computer interface

2021 ◽  
Author(s):  
ShuRui Li ◽  
Jing Jin ◽  
Ian Daly ◽  
Chang Liu ◽  
Andrzej Cichocki
Keyword(s):  
Feature Selection ◽  
Brain Computer Interface ◽  
Feature Selection Method ◽  
Event Related Potentials ◽  
Selection Method ◽  
Computer Interface ◽  
Feature Subset ◽  
Linear Discriminant ◽  
Related Potentials ◽  
Menger Curvature

Abstract Brain–computer interface (BCI) systems decode electroencephalogram signals to establish a channel for direct interaction between the human brain and the external world without the need for muscle or nerve control. The P300 speller, one of the most widely used BCI applications, presents a selection of characters to the user and performs character recognition by identifying P300 event-related potentials from the EEG. Such P300-based BCI systems can reach good levels of accuracy but are difficult to use in day-to-day life due to redundancy and noisy signal. A room for improvement should be considered. We propose a novel hybrid feature selection method for the P300-based BCI system to address the problem of feature redundancy, which combines the Menger curvature and linear discriminant analysis. First, selected strategies are applied separately to a given dataset to estimate the gain for application to each feature. Then, each generated value set is ranked in descending order and judged by a predefined criterion to be suitable in classification models. The intersection of the two approaches is then evaluated to identify an optimal feature subset. The proposed method is evaluated using three public datasets, i.e., BCI Competition III dataset II, BNCI Horizon dataset, and EPFL dataset. Experimental results indicate that compared with other typical feature selection and classification methods, our proposed method has better or comparable performance. Additionally, our proposed method can achieve the best classification accuracy after all epochs in three datasets. In summary, our proposed method provides a new way to enhance the performance of the P300-based BCI speller.

scholarly journals TANGENT-POINT SELF-AVOIDANCE ENERGIES FOR CURVES

2012 ◽  
Vol 21 (05) ◽  
pp. 1250044 ◽  
Author(s):  
PAWEŁ STRZELECKI ◽  
HEIKO VON DER MOSEL
Keyword(s):  
Upper Bound ◽  
Closed Interval ◽  
Local Energy ◽  
Triple Junctions ◽  
Tangent Point ◽  
Double Integral ◽  
Energy Formula ◽  
Knot Energy ◽  
Rectifiable Curves ◽  
Infinite Energy

We study a two-point self-avoidance energy [Formula: see text] which is defined for all rectifiable curves in ℝn as the double integral along the curve of 1/rq. Here r stands for the radius of the (smallest) circle that the is tangent to the curve at one point and passes through another point on the curve, with obvious natural modifications of this definition in the exceptional, non-generic cases. It turns out that finiteness of [Formula: see text] for q ≥ 2 guarantees that γ has no self-intersections or triple junctions and therefore must be homeomorphic to the unit circle 𝕊1 or to a closed interval I. For q > 2 the energy [Formula: see text] evaluated on curves in ℝ3 turns out to be a knot energy separating different knot types by infinite energy barriers and bounding the number of knot types below a given energy value. We also establish an explicit upper bound on the Hausdorff-distance of two curves in ℝ3 with finite [Formula: see text]-energy that guarantees that these curves are ambient isotopic. This bound depends only on q and the energy values of the curves. Moreover, for all q that are larger than the critical exponent q crit = 2, the arclength parametrization of γ is of class C1, 1-2/q, with Hölder norm of the unit tangent depending only on q, the length of γ, and the local energy. The exponent 1 - 2/q is optimal.

Universal growth law for knot energy of Faddeev type in general dimensions

2008 ◽  
Vol 464 (2098) ◽  
pp. 2741-2757 ◽  
Author(s):  
Fanghua Lin ◽  
Yisong Yang
Keyword(s):  
Fine Structure ◽  
Field Configuration ◽  
Hopf Fibration ◽  
Essential Characteristic ◽  
Knot Energy

The presence of a fractional-exponent growth law relating knot energy and knot topology is known to be an essential characteristic for the existence of ‘ideal’ knots. In this paper, we show that the energy infimum E N stratified at the Hopf charge N of the knot energy of the Faddeev type induced from the Hopf fibration ( n ≥1) in general dimensions obeys the sharp fractional-exponent growth law , where the exponent p is universally rendered as , which is independent of the detailed fine structure of the knot energy but determined completely by the dimensions of the domain and range spaces of the field configuration maps.

scholarly journals Minimal Hölder regularity implying finiteness of integral Menger curvature

2012 ◽  
Vol 141 (1-2) ◽  
pp. 125-147 ◽  
Author(s):  
Sławomir Kolasiński ◽  
Marta Szumańska
Keyword(s):  
Hölder Regularity ◽  
Menger Curvature ◽  
Holder Regularity

scholarly journals Regularizing and self-avoidance effects of integral Menger curvature

2010 ◽  
pp. 145-187
Author(s):  
Strzelecki Pawel ◽  
Szumańska Marta ◽  
von der Mosel Heiko
Keyword(s):  
Menger Curvature

scholarly journals A speed preserving Hilbert gradient flow for generalized integral Menger curvature

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Jan Knappmann ◽  
Henrik Schumacher ◽  
Daniel Steenebrügge ◽  
Heiko von der Mosel
Keyword(s):  
Gradient Descent ◽  
Gradient Flow ◽  
Optimization Methods ◽  
The Self ◽  
Initial Curve ◽  
Long Time Existence ◽  
Sobolev Gradient ◽  
Long Time ◽  
Menger Curvature ◽  
Time Existence

Abstract We establish long-time existence for a projected Sobolev gradient flow of generalized integral Menger curvature in the Hilbert case and provide C 1 , 1 C^{1,1} -bounds in time for the solution that only depend on the initial curve. The self-avoidance property of integral Menger curvature guarantees that the knot class of the initial curve is preserved under the flow, and the projection ensures that each curve along the flow is parametrized with the same speed as the initial configuration. Finally, we describe how to simulate this flow numerically with substantially higher efficiency than in the corresponding numerical L 2 L^{2} gradient descent or other optimization methods.

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