scholarly journals Quantum Polar Duality and the Symplectic Camel: A New Geometric Approach to Quantization

2021 ◽  
Vol 51 (3) ◽  
Author(s):  
Maurice A. de Gosson
Keyword(s):  
Uncertainty Principle ◽  
Convex Geometry ◽  
Geometric Approach ◽  
Point Of View ◽  
Reconstruction Problem ◽  
Orthogonal Projections ◽  
Topological Version ◽  
Mahler Conjecture ◽  
Quantum Indeterminacy ◽  
Dual Quantum

AbstractWe define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong interplay between the uncertainty principle and symplectic and convex geometry and our approach could therefore pave the way for a geometric and topological version of quantum indeterminacy. We relate our results to the Blaschke–Santaló inequality and to the Mahler conjecture. We also discuss the Hardy uncertainty principle and the less-known Donoho–Stark principle from the point of view of quantum polarity.

Geometric approach to the revitalization process of medieval Serbian monasteries

2015 ◽  
Author(s):  
Magdalena Slavko Dragović ◽  
Aleksandar Čučaković ◽  
Milesa Srećković
Keyword(s):  
Geometric Approach ◽  
Geometric Design ◽  
Point Of View ◽  
Archaeological Sites ◽  
Architectural Style ◽  
15Th Century ◽  
History Of ◽  
History Of Architecture ◽  
Cultural Heritage Preservation

Among the standard approaches concerning cultural heritage preservation, the architectural point of view deserves particular attention. The special place in medieval Serbian history of architecture belongs to the world famous monastery complexes Studenica, Dečani and Gračanica. Beside them numerous significant monuments (churches and monasteries) exist as witnesses of the national testimony, currently in the state of ruins, archaeological sites, or damaged ones. A lot of them have adequate needs for revitalisation, where the start point is engineering documentation. The focus of the research is on the role of specific geometric and engineering graphics tasks when these areas are concerning. Monastery church devoted to Introduction of Holy Theotokos in village Slavkovica (near town Ljig), with three old sarcophaguses, dated back to 15th century, is presented and analysed from several aspects:measuring, architectural style characteristics - geometric design, 3D modelling (classical-CAD and terrestrial photogrammetric) with visualization and presentation.The attention was paid on preservation of authentic architectural style and medieval building techniques, which allow imperfections in realization.The opinion of experienced scientists and specialists involved in all the phases of monument's revitalisation has been followed as a guideline to the final result – a proposed geometric design of the revitalised church in Slavkovica.

A Geometric Approach to a Quantum Counterpart of a Saddle-Node Bifurcation in a 1:1 Resonant Perturbed Oscillator

1998 ◽  
Vol 08 (05) ◽  
pp. 941-950 ◽  
Author(s):  
Yoshio Uwano
Keyword(s):  
Energy Levels ◽  
Geometric Approach ◽  
Pitchfork Bifurcation ◽  
Point Of View ◽  
Saddle Node Bifurcation ◽  
Bifurcation Set ◽  
Saddle Node ◽  
Geometric Point ◽  
Two Parameters ◽  
Quantum Counterpart

In previous papers [Uwano, 1994, 1995], it was shown that a degeneracy of energy levels is a quantum counterpart of a Hamiltonian pitchfork bifurcation of periodic trajectories of a certain 1:1 resonant oscillator with two parameters. As a continuation of those papers, a quantum study is made from a geometric point of view in order to find a quantum counterpart of a saddle-node bifurcation taking place in a certain 1:1 resonant perturbed oscillator with three parameters. The torus quantization method is applied to the perturbed oscillator to show that a degeneracy of energy levels is a quantum counterpart of that bifurcation: The bifurcation set for the saddle-node bifurcation in classical theory is viewed as a "bifurcation set" for the degeneracy of energy levels in quantum theory.

scholarly journals Virial theorem in quasi-coordinates and Lie algebroid formalism

2014 ◽  
Vol 11 (09) ◽  
pp. 1450055 ◽  
Author(s):  
José F. Cariñena ◽  
Irina Gheorghiu ◽  
Eduardo Martínez ◽  
Patrícia Santos
Keyword(s):  
Virial Theorem ◽  
Mechanical Systems ◽  
Geometric Approach ◽  
Point Of View ◽  
Lie Algebroids ◽  
Hamiltonian Mechanics ◽  
Lie Algebroid ◽  
Lagrangian And Hamiltonian Mechanics ◽  
Generalized Virial Theorem

In this paper, the geometric approach to the virial theorem (VT) developed in [J. F. Cariñena, F. Falceto and M. F. Rañada, A geometric approach to a generalized virial theorem, J. Phys. A: Math. Theor. 45 (2012) 395210, 19 pp.] is written in terms of quasi-velocities (see [J. F. Cariñena, J. Nunes da Costa and P. Santos, Quasi-coordinates from the point of view of Lie algebroid structures, J. Phys. A: Math. Theor. 40 (2007) 10031–10048]). A generalization of the VT for mechanical systems on Lie algebroids is also given, using the geometric tools of Lagrangian and Hamiltonian mechanics on the prolongation of the Lie algebroid.

Modeling and Path-Tracking for a Load-Haul-Dump Mining Vehicle

1997 ◽  
Vol 119 (1) ◽  
pp. 40-47 ◽  
Author(s):  
R. M. DeSantis
Keyword(s):  
Geometric Approach ◽  
Path Tracking ◽  
Point Of View ◽  
Practical Implementation ◽  
Dynamic Component ◽  
Economic Point ◽  
Tracking Controller

A path-tracking controller for a load-haul-dump mining vehicle is designed using a geometric approach recently developed in the context of car- and tractor-trailer-like vehicles. This controller is made up of a kinematic component computing the velocities required of the vehicle for satisfactory path-tracking, and a dynamic component determining the propulsion and steering that are necessary to acquire these velocities. Prospects for practical implementation appear to be promising from an operational, a technological, and an economic point of view.

TAKING IT ON TRUST: REASON AND OBSERVABILITY IN QUANTUM COMPUTING

2005 ◽  
Vol 03 (01) ◽  
pp. 49-56
Author(s):  
ROSSELLA LUPACCHINI
Keyword(s):  
Quantum Computing ◽  
Uncertainty Principle ◽  
Natural Process ◽  
Point Of View ◽  
Basic Question ◽  
Logical Point ◽  
Characteristic Features ◽  
Definition Of ◽  
Locality Conditions ◽  
Theoretical Computing

If quantum computing is located somewhere between physics and theoretical computing, a basic question concerns which characteristic features are derived from the latter. From a logical point of view, the concept of computation provides a definition of the natural process of calculare. It rests on trust that a procedure of reason can be reproduced mechanically. Turing argues for the adequacy of the concept by introducing a requirement of "observability," which is expressed through finiteness and locality conditions. However, according to the uncertainty principle, no computational path can be observed. How does quantum computing contend with Turing's constraints? What observables are relevant to the computation? This is an attempt to sharpen such questions.

scholarly journals The Pauli Problem for Gaussian Quantum States: Geometric Interpretation

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2578
Author(s):  
Maurice A. de Gosson
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Convex Geometry ◽  
Schur Complement ◽  
Geometric Interpretation ◽  
Quantum States ◽  
Pauli Problem ◽  
Polar Duality

We solve the Pauli tomography problem for Gaussian signals using the notion of Schur complement. We relate our results and method to a notion from convex geometry, polar duality. In our context polar duality can be seen as a sort of geometric Fourier transform and allows a geometric interpretation of the uncertainty principle and allows to apprehend the Pauli problem in a rather simple way.

scholarly journals GALILEO GALILEI´S THESIS EXPANDED

2021 ◽  
Vol 5 (1) ◽  
pp. 26-31
Author(s):  
Walter GOMIDE ◽  
Keyword(s):  
Uncertainty Principle ◽  
Point Of View ◽  
Real Numbers ◽  
Short Article ◽  
Galileo Galilei ◽  
Mathematical Point ◽  
Physical Laws

In this short article, I try to show alternative maths to real numbers in such a way that these maths (especially Transreal Numbers by James Anderson and Arithmetic of Infinity by Yaroslav Sergeyev) can also be considered as legitimate instruments for presenting the structure of reality. I call this thesis of expanding the possibilities of understanding Nature mathematically the "Galileo Galilei´s thesis extended". As an example of the application of the thesis that the mathematics that is at the base of Nature must be extended to a better assessment of the scope of physical laws, here we present the Heisenberg´s Uncertainty Principle, approached in an alternative way from a mathematical point of view.

A Computational Geometric Approach to Visual Hulls

1998 ◽  
Vol 08 (04) ◽  
pp. 407-436 ◽  
Author(s):  
Sylvain Petitjean
Keyword(s):  
Computer Vision ◽  
Geometric Approach ◽  
Point Of View ◽  
Visual Hull ◽  
3 Dimensional ◽  
3D Objects ◽  
The Best Approximation ◽  
Recognition Of Objects ◽  
Visual Hulls ◽  
Volume Intersection

Recognizing 3D objects from their 2D silhouettes is a popular topic in computer vision. Object reconstruction can be performed using the volume intersection approach. The visual hull of an object is the best approximation of an object that can be obtained by volume intersection. From the point of view of recognition from silhouettes, the visual hull can not be distinguished from the original object. In this paper, we present efficient algorithms for computing visual hulls. We start with the case of planar figures (polygons and curved objects) and base our approach on an efficient algorithm for computing the visibility graph of planar figures. We present and tackle many topics related to the query of visual hulls and to the recognition of objects equal to their visual hulls. We then move on to the 3-dimensional case and give a flavor of how it may be approached.

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