Jacobi fields in nonholonomic mechanics

In this paper, we define Jacobi fields for nonholonomic mechanics using a similar characterization than in Riemannian geometry. We give explicit conditions to find Jacobi fields and finally we find the nonholonomic Jacobi fields in two equivalent ways: the first one, using an appropriate complete lift of the nonholonomic system and, in the second one, using the curvature and torsion of the associated nonholonomic connection.

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.

Mental State Detection Using Riemannian Geometry on Electroencephalogram Brain Signals

The goal of this study was to implement a Riemannian geometry (RG)-based algorithm to detect high mental workload (MWL) and mental fatigue (MF) using task-induced electroencephalogram (EEG) signals. In order to elicit high MWL and MF, the participants performed a cognitively demanding task in the form of the letter n-back task. We analyzed the time-varying characteristics of the EEG band power (BP) features in the theta and alpha frequency band at different task conditions and cortical areas by employing a RG-based framework. MWL and MF were considered as too high, when the Riemannian distances of the task-run EEG reached or surpassed the threshold of the baseline EEG. The results of this study showed a BP increase in the theta and alpha frequency bands with increasing experiment duration, indicating elevated MWL and MF that impedes/hinders the task performance of the participants. High MWL and MF was detected in 8 out of 20 participants. The Riemannian distances also showed a steady increase toward the threshold with increasing experiment duration, with the most detections occurring toward the end of the experiment. To support our findings, subjective ratings (questionnaires concerning fatigue and workload levels) and behavioral measures (performance accuracies and response times) were also considered.

On Riemannian Geometries of Visual Space

This is a study of the phenomenal geometries constructed in the Riemannian geometry framework from simulated iso-disparity conics in the horizontal visual plane of the binocular system with the asymmetric eyes (AEs). The iso-disparity conic's arcs in the Cyclopean direction are the frontal visual geodesics. For the eyes' resting vergence posture, which depends on parameters of the AE, the iso-disparity conics are frontal straight lines in physical space. For all other fixations, the iso-disparity conics consist of families of the ellipses or hyperbolas depending on both the AE's parameters and the bifoveal fixation. An assumption underlying the relevant architecture of the human visual system is combined with results from simulated iso-disparity straight lines, giving the relative depth as a function of the distance. This establishes the metric tensor in binocular space of fixations for the eyes' resting vergence posture. The resulting geodesics in the gaze direction, give the distance to the horizon and zero curvature. For all other fixations, only the sign of the curvature can be inferred from the global behavior of the simulated iso-disparity conics.

How smooth is quantum complexity?

The “quantum complexity” of a unitary operator measures the difficulty of its construction from a set of elementary quantum gates. While the notion of quantum complexity was first introduced as a quantum generalization of the classical computational complexity, it has since been argued to hold a fundamental significance in its own right, as a physical quantity analogous to the thermodynamic entropy. In this paper, we present a unified perspective on various notions of quantum complexity, viewed as functions on the space of unitary operators. One striking feature of these functions is that they can exhibit non-smooth and even fractal behaviour. We use ideas from Diophantine approximation theory and sub-Riemannian geometry to rigorously quantify this lack of smoothness. Implications for the physical meaning of quantum complexity are discussed.