Symmetry

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25 pages, 3854 KiB

Open AccessArticle

Omnidimensional Convex Polytopes

by Szymon Łukaszyk and Andrzej Tomski

Symmetry 2023, 15(3), 755; https://doi.org/10.3390/sym15030755 - 19 Mar 2023

Cited by 2 | Viewed by 2228

Abstract

The study shows that the volumes and surfaces of n-balls, n-simplices, and n-orthoplices are holomorphic functions of n, which makes those objects omnidimensional, that is well defined in any complex dimension. Applications of these formulas to the omnidimensional polytopes [...] Read more.

The study shows that the volumes and surfaces of n-balls, n-simplices, and n-orthoplices are holomorphic functions of n, which makes those objects omnidimensional, that is well defined in any complex dimension. Applications of these formulas to the omnidimensional polytopes inscribed in and circumscribed about n-balls reveal previously unknown properties of these geometric objects. In particular, for

0 < n < 1

, the volumes of the omnidimensional polytopes are larger than those of circumscribing n-balls, and both their volumes and surfaces are smaller than those of inscribed n-balls. The surface of an n-simplex circumscribing a unit diameter n-ball is spirally convergent to zero with real n approaching negative infinity but first has a local maximum at

n = - 3.5

. The surface of an n-orthoplex circumscribing a unit diameter n-ball is spirally divergent with real n approaching negative infinity but first has a local minimum at

n = - 1.5

, where its real and imaginary parts are equal to each other; similarly, is its volume, where the similar local minimum occurs at

n = - 3.5

. Reflection functions for volumes and surfaces of these polytopes inscribed in and circumscribed about n-balls are proposed. Symmetries of products and quotients of the volumes in complex dimensions n and

- n

and of the surfaces in complex dimensions n and

2 - n

are shown to be independent of the metric factor and the gamma function. Specific symmetries also hold between the volumes and surfaces in dimensions

n = - 1 / 2

and

n = 1 / 2

. Full article

(This article belongs to the Special Issue Mathematical Modelling of Physical Systems 2021)

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67 pages, 702 KiB

Open AccessArticle

Modelling Cosmic Springs with Finsler and Generalised Finsler Geometries

by Matthew J. Lake

Symmetry 2022, 14(10), 2166; https://doi.org/10.3390/sym14102166 - 16 Oct 2022

Cited by 1 | Viewed by 1424

Abstract

We show that the equations of motion governing the dynamics of strings in a compact internal space can be written as dispersion relations, with a local speed that depends on the velocity and curvature of the string in the large dimensions. From a [...] Read more.

We show that the equations of motion governing the dynamics of strings in a compact internal space can be written as dispersion relations, with a local speed that depends on the velocity and curvature of the string in the large dimensions. From a (3+1)-dimensional perspective these can be viewed as dispersion relations for waves propagating in the string interior and are analogous to those for current-carrying topological defects. This allows us to construct a unified framework with which to study and interpret the internal structure of various field-theoretic and fundamental string species, in a simple physically intuitive coordinate system, without the need for dimensional reduction or approximate effective actions. This, in turn, allows us to identify the precise conditions under which higher-dimensional strings and current-carrying defects are observationally indistinguishable, for macroscopic observers. Our approach naturally incorporates the description of so-called ‘cosmic springs’, whose dynamics are expressed in terms of an effective Finsler geometry, for circular loops, or generalised Finsler geometry, for non-circular configurations. This demonstrates the importance of these novel geometric structures and their utility in modelling complex physical phenomena in cosmology and astrophysics. Full article

(This article belongs to the Special Issue Mathematical Modelling of Physical Systems 2021)

41 pages, 479 KiB

Open AccessFeature PaperArticle

Topological Quantization of Fractional Quantum Hall Conductivity

by J. Miller and M. A. Zubkov

Symmetry 2022, 14(10), 2095; https://doi.org/10.3390/sym14102095 - 8 Oct 2022

Cited by 1 | Viewed by 1249

Abstract

We derive a novel topological expression for the Hall conductivity. To that degree we consider the quantum Hall effect (QHE) in a system of interacting electrons. Our formalism is valid for systems in the presence of an external magnetic field, as well as [...] Read more.

We derive a novel topological expression for the Hall conductivity. To that degree we consider the quantum Hall effect (QHE) in a system of interacting electrons. Our formalism is valid for systems in the presence of an external magnetic field, as well as for systems with a nontrivial band topology. That is, the expressions for the conductivity derived are valid for both the ordinary QHE and for the intrinsic anomalous QHE. The expression for the conductivity applies to external fields that may vary in an arbitrary way, and takes into account disorder. Properties related to symmetry and topology are revealed in the fractional quantization of the Hall conductivity. It is assumed that the ground state of the system is degenerate. We represent the QHE conductivity as

\frac{e^{2}}{h} \times \frac{N}{K}

, where K is the degeneracy of the ground state, while

N

is the topological invariant composed of the Wigner-transformed multi-leg Green functions, which takes discrete values. Full article

(This article belongs to the Special Issue Mathematical Modelling of Physical Systems 2021)

15 pages, 373 KiB

Open AccessArticle

A Constructive Method of Solving Local Boundary Value Problems for Nonlinear Systems with Perturbations and Control Delays

by Alexander N. Kvitko and Alexey S. Eremin

Symmetry 2022, 14(8), 1595; https://doi.org/10.3390/sym14081595 - 3 Aug 2022

Viewed by 1225

Abstract

In this paper, a class of controllable nonlinear stationary systems of ordinary differential equations with an account of external perturbations is studied. The control satisfies given restrictions, and there is a fixed delay in it. An algorithm to construct a control transferring a [...] Read more.

In this paper, a class of controllable nonlinear stationary systems of ordinary differential equations with an account of external perturbations is studied. The control satisfies given restrictions, and there is a fixed delay in it. An algorithm to construct a control transferring a system from a certain initial state to an arbitrary neighborhood of the origin is proposed. The algorithm has both numerical and analytical stages and is easy to implement. A constructive sufficient Kalman-type condition of possibility of the transfer is derived. The algorithm efficiency is demonstrated by solving a robot manipulator controlling problem. Full article

(This article belongs to the Special Issue Mathematical Modelling of Physical Systems 2021)

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10 pages, 1208 KiB

Open AccessArticle

Impact of Brownian Motion on the Analytical Solutions of the Space-Fractional Stochastic Approximate Long Water Wave Equation

by Farah M. Al-Askar, Wael W. Mohammed and Mohammad Alshammari

Symmetry 2022, 14(4), 740; https://doi.org/10.3390/sym14040740 - 4 Apr 2022

Cited by 17 | Viewed by 1977

Abstract

The space-fractional stochastic approximate long water wave equation (SFSALWWE) is considered in this work. The Riccati equation method is used to get analytical solutions of the SFSALWWE. This equation has never been examined with stochastic term and fractional space at the same time. [...] Read more.

The space-fractional stochastic approximate long water wave equation (SFSALWWE) is considered in this work. The Riccati equation method is used to get analytical solutions of the SFSALWWE. This equation has never been examined with stochastic term and fractional space at the same time. In general, the noise term that preserves the symmetry reduces the domain of instability. To check the impact of Brownian motion on these solutions, we use a MATLAB package to plot 3D and 2D graphs for some analytical fractional stochastic solutions. Full article

(This article belongs to the Special Issue Mathematical Modelling of Physical Systems 2021)

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34 pages, 6638 KiB

Open AccessArticle

Virtual Attractive-Repulsive Potentials Control Theory: A Review and an Extension to Riemannian Manifolds

by Luca Bigelli, Federico Polenta and Simone Fiori

Symmetry 2022, 14(2), 257; https://doi.org/10.3390/sym14020257 - 28 Jan 2022

Cited by 2 | Viewed by 2697

Abstract

The present paper is concerned with an instance of automatic control for autonomous vehicles based on the theory of virtual attractive-repulsive potentials (VARP). The first part of this paper presents a review of the VARP control theory as developed specifically by B. Nguyen, [...] Read more.

The present paper is concerned with an instance of automatic control for autonomous vehicles based on the theory of virtual attractive-repulsive potentials (VARP). The first part of this paper presents a review of the VARP control theory as developed specifically by B. Nguyen, Y.-L. Chuang, D. Tung, C. Hsieh, Z. Jin, L. Shi, D. Marthaler, A. Bertozzi and R. Murray, in the paper ‘Virtual attractive-repulsive potentials for cooperative control of second order dynamic vehicles on the Caltech MVWT’, which appeared in the Proceedings of the 2005 American Control Conference, (Portland, OR, USA) held in June 2005 (pp. 1084–1089). The aim of the first part of the present paper is to recall the mathematical and logical steps that lead to controlling an autonomous robot by a VARP-based control theory. The concepts recalled in the first part of the present paper, with special reference to the physical interpretation of the terms in the developed control field, serve as the starting point to develop a more convoluted control theory for (second-order) dynamical systems whose state spaces are (possibly high-dimensional) curved manifolds. The second part of this paper is, in fact, devoted to extending the classical VARP control theory to regulate dynamical systems whose state spaces possess the mathematical structure of smooth manifolds through manifold calculus. Manifold-type state spaces present a high degree of symmetry, due to mutual non-linear constraints between single physical variables. A comprehensive set of numerical experiments complements the review of the VARP theory and the theoretical developments towards its extension to smooth manifolds. Full article

(This article belongs to the Special Issue Mathematical Modelling of Physical Systems 2021)

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13 pages, 286 KiB

Open AccessArticle

Cylindrical Gravitational Wave: Source and Resonance

by Yu-Zhu Chen, Shi-Lin Li, Yu-Jie Chen and Wu-Sheng Dai

Symmetry 2021, 13(8), 1425; https://doi.org/10.3390/sym13081425 - 4 Aug 2021

Cited by 1 | Viewed by 1641

Abstract

Gravitational waves are regarded as linear waves in the weak field approximation, which ignores the spacetime singularity. In this paper, we analyze singularities in exact gravitational wave solutions. We provide an exact general solution of the gravitational wave with cylindrical symmetry. The general [...] Read more.

Gravitational waves are regarded as linear waves in the weak field approximation, which ignores the spacetime singularity. In this paper, we analyze singularities in exact gravitational wave solutions. We provide an exact general solution of the gravitational wave with cylindrical symmetry. The general solution includes some known cylindrical wave solutions as special cases. We show that there are two kinds of singularities in the cylindrical gravitational wave solution. The first kind of singularity corresponds to a singular source. The second kind of singularity corresponds to a resonance between different gravitational waves. When two gravitational waves coexist, the interference term in the source may vanish in the sense of time averaging. Full article

(This article belongs to the Special Issue Mathematical Modelling of Physical Systems 2021)

Review

Jump to: Research

28 pages, 593 KiB

Open AccessReview

A Critical Review of Works Pertinent to the Einstein-Bohr Debate and Bell’s Theorem

by Karl Hess

Symmetry 2022, 14(1), 163; https://doi.org/10.3390/sym14010163 - 14 Jan 2022

Cited by 13 | Viewed by 2147

Abstract

This review is related to the Einstein-Bohr debate and to Einstein–Podolsky–Rosen’s (EPR) and Bohm’s (EPRB) Gedanken-experiments as well as their realization in actual experiments. I examine a significant number of papers, from my minority point of view and conclude that the well-known theorems [...] Read more.

This review is related to the Einstein-Bohr debate and to Einstein–Podolsky–Rosen’s (EPR) and Bohm’s (EPRB) Gedanken-experiments as well as their realization in actual experiments. I examine a significant number of papers, from my minority point of view and conclude that the well-known theorems of Bell and Clauser, Horne, Shimony and Holt (CHSH) deal with mathematical abstractions that have only a tenuous relation to quantum theory and the actual EPRB experiments. It is also shown that, therefore, Bell-CHSH cannot be used to assess the nature of quantum entanglement, nor can physical features of entanglement be used to prove Bell-CHSH. Their proofs are, among other factors, based on a statistical sampling argument that is invalid for general physical entities and processes and only applicable for finite “populations”; not for elements of physical reality that are linked, for example, to a time-like continuum. Bell-CHSH have, furthermore, neglected the subtleties of the theorem of Vorob’ev that includes their theorems as special cases. Vorob’ev found that certain combinatorial-topological cyclicities of classical random variables form a necessary and sufficient condition for the constraints that are now known as Bell-CHSH inequalities. These constraints, however, must not be linked to the observables of quantum theory nor to the actual EPRB experiments for a variety of reasons, including the existence of continuum-related variables and appropriate considerations of symmetry. Full article

(This article belongs to the Special Issue Mathematical Modelling of Physical Systems 2021)

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