Evolutionary Sequence of Spacetime and Intrinsic Spacetime and Associated Sequence of Geometries in Metric Force Fields I

Two classes of three-dimensional metric spaces are identified. They are the conventional three-dimensional metric space and a new ‘three-dimensional’ absolute intrinsic metric space. Whereas an initial flat conventional proper metric space IE′3 can transform into a curved three-dimensionalRiemannian metric space IM′3 without any of its dimension spanning the time dimension (or in the absence of the time dimension), in conventional Riemann geometry, an initial flat ‘three-dimensional’ absolute intrinsic metric space ∅IˆE3 (as a flat hyper-surface) along the horizontal, evolves into a curved ‘three-dimensional’ absolute intrinsic metric space ∅IˆM3, which is curved (as a curved hyper-surface) toward the absolute intrinsic metric time ‘dimension’ along the vertical, and it is identified as ‘three-dimensional’ absolute intrinsic Riemannian metric space. It invariantly projects a flat ‘three-dimensional’ absolute proper intrinsic metric space ∅IE′3ab along the horizontal, which is made manifested outwardly in flat ‘three-dimensional’ absolute proper metric space IE′3ab, overlying it, both as flat hyper-surfaces along the horizontal. The flat conventional three-dimensional relative proper metric space IE′3 and its underlying flat three-dimensional relative proper intrinsic metric space ∅IE′3 remain unchanged. The observers are located in IE′3. The projective ∅IE′3ab is imperceptibly embedded in ∅IE′3 and IE′3ab in IE′3. The corresponding absolute intrinsic metric time ‘dimension’ is not curved from its vertical position simultaneously with ‘three-dimensional’ absolute intrinsic metric space. The development of absolute intrinsic Riemannian geometry is commenced and the conclusion that the resulting geometry is more all-encompassing then the conventional Riemannian geometry on curved conventional metric space IM′3 only is reached.

Fatigue Detection for Human Aware Adaptation in Human-Robot Collaboration

Knowledge about human cognitive and physical state is a key factor in physical Human-robot collaboration (pHRC). Such information benefits the robot in planning an adaptive control strategy to prevent or mitigate human fatigue. In this paper, we present a method to detect upper limb muscle fatigue during pHRC using a low-cost myoelectric sensor. We used Riemann geometry to extract robust features from the time-series data and designed a classifier to detect the binary state of human fatigue i.e. fatigued vs not fatigued. We evaluated the method using a fine-motor coordination task for the human to guide an industrial robot along a virtual path for sometime followed by a muscle curl exercise until it induces fatigue in the muscles, and then repeat the robot experiment. We recruited nine participants for the study and recorded muscle activity from their dominant upper limb using the myoelectric sensor and used the data to develop a classifier. We compared the accuracy and robustness of the classifier against conventional time-domain and wavelet-based features and showed that Riemann geometry-based features yield higher classification accuracy (∼ 91%) compared to conventional features and require less computational effort. Such classifier can be used in real-time to develop a human-aware adaptation strategy to prevent fatigue.

Analisando o impacto do espectro do sinal de EEG na abordagem via Geometria Riemanniana

In this work we investigate the use of a Gaussian membership function as a technique to improve the steps of feature extraction and classification in brain-computer interface (BCI) systems based on motor imagery (IM). The main idea of this approach is to filter the spectral information of the electroencephalogram (EEG) signal via parameterized covariance matrices to highlight features that contribute to signal classification through a classifier based on Riemann’s distance. The results, in relation to the accuracy performance, acquired in this work arevalidated from dataset 2a of the IV International ICM Competition. The results obtained suggest that the spectral filtering performed using the Riemann Geometry approach can positively affect the performance of the ICM system, increasing its flexibility.

Application of Brain-computer Interfaces in Assistive Technologies

In the paper issues of brain-computer interface applications in assistive technologies are considered in particular for robotic devices control. Noninvasive brain-computer interfaces are built based on the classification of electroencephalographic signals, which show bioelectrical activity in different zones of the brain. Such brain-computer interfaces after training are able to decode electroencephalographic patterns corresponding to different imaginary movements and patterns corresponding to different audio-visual stimulus. The requirements which must be met by brain-computer interfaces operating in real time, so that biological feedback is effective and the user's brain can correctly associate responses with events are formulated. The process of electroencephalographic signal processing in noninvasive brain-computer interface is examined including spatial and temporal filtering, artifact removal, feature selection, and classification. Descriptions and comparison of classifiers based on support vector machines, artificial neural networks, and Riemann geometry are presented. It was shown that such classifiers can provide accuracy at the level of 60-80% for recognition of imaginary movements from two to four classes. Examples of application of the classifiers to control robotic devices were presented. The approach is intended both to help healthy users to perform daily functions better and to increase the quality of life of people with movement disabilities. Tasks to increase the efficiency of technology application are formulated.

A Note on the Representation of Clifford Algebra

In this note we construct explicit complex and real matrix representations for the generators of real Clifford algebra $C\ell_{p,q}$. The representation is based on Pauli matrices and has an elegant structure similar to the fractal geometry. We find two classes of representation, the normal representation and exceptional one. The normal representation is a large class of representation which can only be expanded into $4m+1$ dimension, but the exceptional representation can be expanded as generators of the next period. In the cases $p+q=4m$, the representation is unique in equivalent sense. These results are helpful for both theoretical analysis and practical calculation. The generators of Clifford algebra are the faithful basis of $p+q$ dimensional Minkowski space-time or Riemann space, and Clifford algebra converts the complicated relations in geometry into simple and concise algebraic operations, so the Riemann geometry expressed in Clifford algebra will be much simple and clear.

Some Applications of Clifford Algebra in Geometry

In this paper, we provide some enlightening examples of the application of Clifford algebra in geometry, which show the concise representation, simple calculation and profound insight of this algebra. The definition of Clifford algebra implies geometric concepts such as vector, length, angle, area and volume, and unifies the calculus of scalar, spinor, vector and tensor, so that it is able to naturally describe all variables and calculus in geometry and physics. Clifford algebra unifies and generalizes real number, complex, quaternion and vector algebra, converts complicated relations and operations into intuitive matrix algebra independent of coordinate systems. By localizing the basis or frame of space-time and introducing differential and connection operators, Clifford algebra also contains Riemann geometry. Clifford algebra provides a unified, standard, elegant and open language and tools for numerous complicated mathematical and physical theories. Clifford algebra calculus is an arithmetic-like operation that can be well understood by everyone. This feature is very useful for teaching purposes, and popularizing Clifford algebra in high schools and universities will greatly improve the efficiency of students to learn fundamental knowledge of mathematics and physics. So Clifford algebra can be expected to complete a new big synthesis of scientific knowledge.

Inequality in the Universe, Imaginary Numbers and a Brief Solution to P=NP? Problem

While I was working about some basic physical phenomena, I discovered some geometric relations that also interest mathematics. In this work, I applied the rules I have been proven to P=NP? problem over impossibility of perpendicularity in the universe. It also brings out extremely interesting results out like imaginary numbers which are known as real numbers currently. Also it seems that Euclidean Geometry is impossible. The actual geometry is Riemann Geometry and complex numbers are real.