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AI for Theory of Everything
Published in Volker Knecht, AI for Physics, 2023
The bottom-line is that each six-dimensional manifold M gives a different four-dimensional universe, with its characteristic elementary particles and cosmology.‡ For each manifold M, there is a topological quantity yielding the corresponding number of quark and lepton generations (which is three in the Standard Model of particle physics, as described in Chapter 4 and shown in Figure 4.1). Overall, there is a dictionary between the topology and geometry of M and the particle physics and gravity of this universe. Establishing this dictionary, performing the requisite geometric computations of M, and seeing whether the resulting universe is anything akin to ours, is the field of string phenomenology.6 This is undoubtedly one of the most important pursuits in the theoretical sciences and a perfect realization of that old Keplerian dream of matter from geometry! This field of theoretical physics, where one uses the geometry (and more generally, algebra, arithmetic, and combinatorics) of manifolds to “create” physical worlds in terms of quantum field theories and gravity, has been affectionately dubbed Geometrical Engineering.7
Pertinent Properties of Euclidean Space
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
The most important topological spaces are manifolds. A topological space X is called an n-dimensional manifold if X is a Hausdorff space and each x∈X has a neighborhood N(x) that is homeomorphic to Euclidean n-space ℝn. Bernhard Riemann, a former student of Carl Gauss, was the first to coin the term ”manifold” in his extensive studies of surfaces in higher dimensions. In modern physics, our universe is a manifold. Einstein's general relative theory relies on a four-dimensional spacetime manifold. The string theory model of our universe is based on a ten-dimensional manifold R×M, where R denotes the four-dimensional spacetime manifold and M denotes the six-dimensional spacial Calabi-Yau manifold. Einstein's spacetime manifold is a generalized Riemannian manifold that uses the Ricci curvature tensor [32]. We do not expect the reader to be familiar with manifold theory and associated geometries. One reason for mentioning the subject of manifolds is that even Einstein's manifold, which tells us that that space and the gravitational field are one and the same, needs the locality of ℝ4 at each point of the manifold.
Control of Robotic Systems in Contact Tasks
Published in Osita D. I. Nwokah, Yildirim Hurmuzlu, The Mechanical Systems Design Handbook, 2017
Dragoljub Šurdilović, Miomir Vukobratović
The above dynamical model can be transformed into an equivalent form that is more convenient for analysis and synthesis of a robot controller for contact tasks. When the manipulator interacts with the environment, it is convenient to describe its dynamics in the space where manipulation task is described, rather than in joint coordinate space (also termed configuration space). The end effector position and orientation with respect to a reference coordinate system can be described by a six-dimensional vector x. The reference system is chosen to suit a particular robot application. Most frequently, a fixed coordinate frame attached to the manipulator base is considered as the reference system. Using the Jacobian matrix, we can transform the dynamic models (23.4) from the joint into the end effector coordinate system: Λ(x)x¨+B(x)x˙+μ(x,x˙)+p(x)=τ+F
Electromagnetic articulography (EMA) for real-time feedback application: computational techniques
Published in Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2019
B. Haworth, E. Kearney, P. Faloutsos, M. Baljko, Y. Yunusova
Early methods for tracking the tongue were primarily limited to two-dimensional (2D) data outputs, and the devices required lengthy calibration processes (Schönle et al. 1987; Perkell et al. 1992). Later methods provided full six-dimensional (6D) tracking of the position and rotation of sensors on the tongue (Zierdt 1993; Kaburagi et al. 2005). Commercial speech research solutions are now readily available, such as the Carstens AG500 line of products (Carstens Medizinelektronik GmbH, Bovenden) and the Wave Speech Research (NDI, Waterloo) systems. These commercial systems have been tested for their accuracy and demonstrate adequate performance (Yunusova et al. 2009; Berry 2011; Kroos 2012; Savariaux et al. 2017). Most of these current commercial systems, such as the wave, produce audio-aligned six-dimensional (6D) kinematic time-series information, and they come with recording and data transformation software, as well as Application Programmers Interfaces (APIs) for developing external applications.
A modelling and predictive control approach to linear two-stage inverted pendulum based on RBF-ARX model
Published in International Journal of Control, 2021
Xiaoying Tian, Hui Peng, Xiaoyong Zeng, Feng Zhou, Wenquan Xu, Xiaoyan Peng
Model (9) is a continuous-time model. To design a discrete-time LQR, we need to derive the discrete-time model of (9). By discretizing the continuous-time model (9), a discrete model can be derived as where k is the sample instant, , T denotes the system sampling period, , is a six-dimensional unit matrix, , , and .
Lamb waves in stratified and functionally graded plates: discrepancy, similarity, and convergence
Published in Waves in Random and Complex Media, 2021
Introducing a new vectorial variable and a six-dimensional vector equations of motion (3) can be rewritten in terms of the vector : where is the six-dimensional matrix