Topic
Hilbert plane
About: Hilbert plane is a research topic. Over the lifetime, 7 publications have been published within this topic receiving 50 citations.
Papers
Patent•
17 Sep 2007
TL;DR: In this paper, a method for a Movement and Vibration Analyzer (MVA) based on Fast Fourier Transform spectral analysis, and empirical mode decomposition (EMD) for Hilbert transform of a timeseries recorded with an accelerometer attached to a human being or an object is described.
Abstract: The present patent describes a method for a Movement and Vibration Analyzer (MVA) based on Fast Fourier Transform spectral analysis, and empirical mode decomposition (EMD) for Hilbert transform of a timeseries recorded with an accelerometer attached to a human being or an object. The medical application is the detection of Parkinson's disease (PD) and other neurological motor disorders (Dystonias, Dyskinesias, Huntington's disease, Essential Tremor, Multiple System Atrophy (MSA), etc), which affects worldwide more than 5 million persons, where the highest percentage is in the ageing population. The industrial application is the study of vibration and maintenance of rotational devices (motors, turbines, and others which have an intrinsic sinusoidal likewise movement). An EMD is carried out on the acceleration signal which produces a collection of intrinsic mode functions (IMF), on which the Hilbert transform is carried out. A set of parameters extracted from the Hilbert Transformed signal gives information of the deviation of the discontinuities. (1) Number of peaks of the derivative of the Hilbert phase higher than a threshold and normalized to time length of the signal and sampling frequency. (2) Variance or standard deviation of the derivative of the Hubert phase, φ' H(t). (3) Fractal dimension (DF) of the curve (HR(t), H1(t)), Hilbert plane. From the power spectrum estimate of the acceleration signal, the parameters used are: (4) Mean frequency. (5) Frequencies of the N main components. These five parameters are combined using fuzzy logic or an ordinal multiple logistic regression to define the movement index (MI), an index from 0 to 100, where 0 indicates no deviation from the sinusoidal movement while increasing numbers indicate larger deviation from the sinusoidal movement.
21 citations
1 Jan 2017
TL;DR: A survey of projective spaces endowed with an orthogonality and the associated Cayley-Klein geometries can be found in this paper, where the authors present a survey of these absolute geometric structures and their first-order axiomatizations.
Abstract: In his 1854 Habilitationsvortrag Riemann presented a new concept of space endowed with a metric of great generality, which, through specification of the metric, gave rise to the spaces of constant curvature. In a different vein, yet with a similar aim, J. Hjelmslev, A. Schmidt, and F. Bachmann, introduced axiomatically a very general notion of plane geometry, which provides the foundation for the elementary versions of the geometries of spaces of constant curvature. We present a survey of these absolute geometric structures and their first-order axiomatizations, as well as of higher-dimensional variants thereof. In the 2-dimensional case, these structures were called metric planes by F. Bachmann, and they can be seen as the common substratum for the classical plane geometries: Euclidean, hyperbolic, and elliptic. They are endowed with a very general notion of orthogonality or reflection that can be specialized into that of the classical geometries by means of additional axioms. By looking at all the possible ways in which orthogonality can be introduced in terms of polarities, defined on (the intervals of a chain of subspaces of) projective spaces, one obtains a further generalization: the Cayley-Klein geometries. We present a survey of projective spaces endowed with an orthogonality and the associated Cayley-Klein geometries.
13 citations
TL;DR: It is shown here using elementary model theoretic tools that the universal first order consequences of any geometric theory T of Pappian planes which is consistent with the analytic geometry of the reals is decidable.
Abstract: We survey the status of decidability of the first order consequences in various axiomatizations of Hilbert-style Euclidean geometry. We draw attention to a widely overlooked result by Martin Ziegler from 1980, which proves Tarski’s conjecture on the undecidability of finitely axiomatizable theories of fields. We elaborate on how to use Ziegler’s theorem to show that the consequence relations for the first order theory of the Hilbert plane and the Euclidean plane are undecidable. As new results we add: It was already known that the universal theory of Hilbert planes and Wu’s orthogonal geometry is decidable. We show here using elementary model theoretic tools that The techniques used were all known to experts in mathematical logic and geometry in the past but no detailed proofs are easily accessible for practitioners of symbolic computation or automated theorem proving.
10 citations
Posted Content•
TL;DR: It is shown here using elementary model theoretic tools that the universal first order consequences of any geometric theory of Pappian planes which is consistent with the analytic geometry of the reals is decidable.
Abstract: We survey the status of decidabilty of the consequence relation in various axiomatizations of Euclidean geometry. We draw attention to a widely overlooked result by Martin Ziegler from 1980, which proves Tarski's conjecture on the undecidability of finitely axiomatizable theories of fields. We elaborate on how to use Ziegler's theorem to show that the consequence relations for the first order theory of the Hilbert plane and the Euclidean plane are undecidable. As new results we add: (A) The first order consequence relations for Wu's orthogonal and metric geometries (Wen-Tsun Wu, 1984), and for the axiomatization of Origami geometry (J. Justin 1986, H. Huzita 1991)are undecidable.
It was already known that the universal theory of Hilbert planes and Wu's orthogonal geometry is decidable. We show here using elementary model theoretic tools that (B) the universal first order consequences of any geometric theory $T$ of Pappian planes which is consistent with the analytic geometry of the reals is decidable.
8 citations
Posted Content•
Abstract: We give an affine proof of Feuerbach's theorem, by constructing an explicit affine map which takes the nine-point circle of any given Euclidean triangle to the incircle and fixes the Feuerbach point. The proof is shown to be valid in any Hilbert plane which satisfies the parallel postulate.
5 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2020 | 1 |
| 2019 | 1 |
| 2018 | 1 |
| 2017 | 3 |
| 2007 | 1 |