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Symmetry
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Symmetry and Complexity
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Symmetry and Complexity

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (30 June 2018) | Viewed by 26351

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Prof. Dr. Carlo Cattani

E-Mail Website
Guest Editor
Engineering School (DEIM), University of Tuscia, Largo dell'Università, 01100 Viterbo, Italy
Interests: wavelets; fractals; fractional and stochastic equations; numerical and computational methods; mathematical physics; nonlinear systems; artificial intelligence
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Symmetry and complexity are two fundamental features of almost all phenomena in nature and science. Any complex physical model is characterized by the existence of some symmetry groups at different scales. On the other hand, breaking the symmetry of a scientific model has been always considered as the most challenging direction for new discoveries. Modeling complexity has recently become an increasingly popular subject, with an impressive growth concerning applications. The main goal of modeling complexity is the search for hidden or broken symmetries.

Usually, complexity is modeled by dealing with big data or dynamical systems, depending on a large number of parameters. Nonlinear dynamical systems and chaotic dynamical systems are also used for modeling complexity. Complex models are often represented by un-smoth objects, non-differentiable objects, fractals, pseudo-random phenomena, and stochastic process.

The discovery of complexity and symmetry in mathematics, physics, engineering, economics, biology and medicine have opened new challenging fields of research. Therefore, new mathematical tools were developed in order to obtain quantitative information from models, newly reformulated in terms of nonlinear differential equations.

This Special Issue focuses on the most recent advances in calculus, applied to dynamical problems, linear and nonlinear (fractional, stochastic) ordinary and partial differential equations, integral differential equations and stochastic integral problems, arising in all fields of science, engineering applications, and other applied fields dealing with complexity.

We are soliciting contributions covering a broad range of topics on symmetry and complexity in:

  • Mathematics
  • Chemistry
  • Physics
  • Fluid-dynamics and aero-dynamics
  • Unified physical theory
  • Biology
  • Nonlinear dynamical systems
  • Nonlinear science
  • Chaos
  • Fractals
  • Image and data analysis
  • Computational systems
  • Artificial intelligence, neural networks
  • History of science, philosophy, semantic structures

Prof. Dr. Carlo Cattani
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • mathematics
  • physics
  • mathematical physics
  • mechanics
  • fractal
  • differential equations
  • dynamical systems
  • chaos
  • computational methods
  • stochastic process
  • stochastic differential equations
  • pattern recognition
  • data analysis

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Published Papers (6 papers)

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Research

16 pages, 25520 KiB  
Article
Computing Zagreb Indices and Zagreb Polynomials for Symmetrical Nanotubes
by Zehui Shao, Muhammad Kamran Siddiqui and Mehwish Hussain Muhammad
Symmetry 2018, 10(7), 244; https://doi.org/10.3390/sym10070244 - 28 Jun 2018
Cited by 122 | Viewed by 4376
Abstract
Topological indices are numbers related to sub-atomic graphs to allow quantitative structure-movement/property/danger connections. These topological indices correspond to some specific physico-concoction properties such as breaking point, security, strain vitality of chemical compounds. The idea of topological indices were set up in compound graph [...] Read more.
Topological indices are numbers related to sub-atomic graphs to allow quantitative structure-movement/property/danger connections. These topological indices correspond to some specific physico-concoction properties such as breaking point, security, strain vitality of chemical compounds. The idea of topological indices were set up in compound graph hypothesis in view of vertex degrees. These indices are valuable in the investigation of mitigating exercises of specific Nanotubes and compound systems. In this paper, we discuss Zagreb types of indices and Zagreb polynomials for a few Nanotubes covered by cycles. Full article
(This article belongs to the Special Issue Symmetry and Complexity)
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27 pages, 3113 KiB  
Article
Intelligent Prognostics of Degradation Trajectories for Rotating Machinery Based on Asymmetric Penalty Sparse Decomposition Model
by Qing Li and Steven Y. Liang
Symmetry 2018, 10(6), 214; https://doi.org/10.3390/sym10060214 - 12 Jun 2018
Cited by 8 | Viewed by 3194
Abstract
The ability to accurately track the degradation trajectories of rotating machinery components is arguably one of the challenging problems in prognostics and health management (PHM). In this paper, an intelligent prediction approach based on asymmetric penalty sparse decomposition (APSD) algorithm combined with wavelet [...] Read more.
The ability to accurately track the degradation trajectories of rotating machinery components is arguably one of the challenging problems in prognostics and health management (PHM). In this paper, an intelligent prediction approach based on asymmetric penalty sparse decomposition (APSD) algorithm combined with wavelet neural network (WNN) and autoregressive moving average-recursive least squares algorithm (ARMA-RLS) is proposed for degradation prognostics of rotating machinery, taking the accelerated life test of rolling bearings as an example. Specifically, the health indicators time series (e.g., peak-to-peak value and Kurtosis) is firstly decomposed into low frequency component (LFC) and high frequency component (HFC) using the APSD algorithm; meanwhile, the resulting non-convex regularization problem can be efficiently solved using the majorization-minimization (MM) method. In particular, the HFC part corresponds to the stable change around the zero line of health indicators which most extensively occurs; in contrast, the LFC part is essentially related to the evolutionary trend of health indicators. Furthermore, the nonparametric-based method, i.e., WNN, and parametric-based method, i.e., ARMA-RLS, are respectively introduced to predict the LFC and HFC that focus on abrupt degradation regions (e.g., last 100 points). Lastly, the final predicted data could be correspondingly obtained by integrating the predicted LFC and predicted HFC. The proposed methodology is tested using degradation health indicator time series from four rolling bearings. The proposed approach performed favorably when compared to some state-of-the-art benchmarks such as WNN and largest Lyapunov (LLyap) methods. Full article
(This article belongs to the Special Issue Symmetry and Complexity)
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13 pages, 779 KiB  
Article
Laplacian Spectra for Categorical Product Networks and Its Applications
by Shin Min Kang, Muhammad Kamran Siddiqui, Najma Abdul Rehman, Muhammad Imran and Mehwish Hussain Muhammad
Symmetry 2018, 10(6), 206; https://doi.org/10.3390/sym10060206 - 7 Jun 2018
Cited by 12 | Viewed by 3379
Abstract
The Kirchhoff index, global mean-first passage time, average path length and number of spanning trees are of great importance in the field of networking. The “Kirchhoff index” is known as a structure descriptor index. The “global mean-first passage time” is known as a [...] Read more.
The Kirchhoff index, global mean-first passage time, average path length and number of spanning trees are of great importance in the field of networking. The “Kirchhoff index” is known as a structure descriptor index. The “global mean-first passage time” is known as a measure for nodes that are quickly reachable from the whole network. The “average path length” is a measure of the efficiency of information or mass transport on a network, and the “number of spanning trees” is used to minimize the cost of power networks, wiring connections, etc. In this paper, we have selected a complex network based on a categorical product and have used the spectrum approach to find the Kirchhoff index, global mean-first passage time, average path length and number of spanning trees. We find the expressions for the product and sum of reciprocals of all nonzero eigenvalues of a categorical product network with the help of the eigenvalues of the path and cycles. Full article
(This article belongs to the Special Issue Symmetry and Complexity)
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13 pages, 881 KiB  
Article
Dynamics of Trapped Solitary Waves for the Forced KdV Equation
by Sunmi Lee
Symmetry 2018, 10(5), 129; https://doi.org/10.3390/sym10050129 - 24 Apr 2018
Cited by 24 | Viewed by 3960
Abstract
The forced Korteweg-de Vries equation is considered to investigate the impact of bottom configurations on the free surface waves in a two-dimensional channel flow. In the study of shallow water waves, the bottom topography plays a critical role, which can determine the characteristics [...] Read more.
The forced Korteweg-de Vries equation is considered to investigate the impact of bottom configurations on the free surface waves in a two-dimensional channel flow. In the study of shallow water waves, the bottom topography plays a critical role, which can determine the characteristics of wave motions significantly. The interplay between solitary waves and the bottom topography can exhibit more interesting dynamics of the free surface waves when the bottom configuration is more complex. In the presence of two bumps, there are multiple trapped solitary wave solutions, which remain stable between two bumps up to a finite time when they evolve in time. In this work, various stationary trapped wave solutions of the forced KdV equation are explored as the bump sizes and the distance between two bumps are varied. Moreover, the semi-implicit finite difference method is employed to study their time evolutions in the presence of two-bump configurations. Our numerical results show that the interplay between trapped solitary waves and two bumps is the key determinant which influences the time evolution of those wave solutions. The trapped solitary waves tend to remain between two bumps for a longer time period as the distance between two bumps increases. Interestingly, there exists a nontrivial relationship between the bump size and the time until trapped solitary waves remain stable between two bumps. Full article
(This article belongs to the Special Issue Symmetry and Complexity)
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11 pages, 1754 KiB  
Article
A Framework for Circular Multilevel Systems in the Frequency Domain
by Guomin Sun, Jinsong Leng and Carlo Cattani
Symmetry 2018, 10(4), 101; https://doi.org/10.3390/sym10040101 - 8 Apr 2018
Cited by 2 | Viewed by 3277
Abstract
In this paper, we will construct a new multilevel system in the Fourier domain using the harmonic wavelet. The main advantages of harmonic wavelet are that its frequency spectrum is confined exactly to an octave band, and its simple definition just as Haar [...] Read more.
In this paper, we will construct a new multilevel system in the Fourier domain using the harmonic wavelet. The main advantages of harmonic wavelet are that its frequency spectrum is confined exactly to an octave band, and its simple definition just as Haar wavelet. The constructed multilevel system has the circular shape, which forms a partition of the frequency domain by shifting and scaling the basic wavelet functions. To possess the circular shape, a new type of sampling grid, the circular-polar grid (CPG), is defined and also the corresponding modified Fourier transform. The CPG consists of equal space along rays, where different rays are equally angled. The main difference between the classic polar grid and CPG is the even sampling on polar coordinates. Another obvious difference is that the modified Fourier transform has a circular shape in the frequency domain while the polar transform has a square shape. The proposed sampling grid and the new defined Fourier transform constitute a completely Fourier transform system, more importantly, the harmonic wavelet based multilevel system defined on the proposed sampling grid is more suitable for the distribution of general images in the Fourier domain. Full article
(This article belongs to the Special Issue Symmetry and Complexity)
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91 pages, 19423 KiB  
Article
Three Classes of Fractional Oscillators
by Ming Li
Symmetry 2018, 10(2), 40; https://doi.org/10.3390/sym10020040 - 30 Jan 2018
Cited by 54 | Viewed by 5262
Abstract
This article addresses three classes of fractional oscillators named Class I, II and III. It is known that the solutions to fractional oscillators of Class I type are represented by the Mittag-Leffler functions. However, closed form solutions to fractional oscillators in Classes II [...] Read more.
This article addresses three classes of fractional oscillators named Class I, II and III. It is known that the solutions to fractional oscillators of Class I type are represented by the Mittag-Leffler functions. However, closed form solutions to fractional oscillators in Classes II and III are unknown. In this article, we present a theory of equivalent systems with respect to three classes of fractional oscillators. In methodology, we first transform fractional oscillators with constant coefficients to be linear 2-order oscillators with variable coefficients (variable mass and damping). Then, we derive the closed form solutions to three classes of fractional oscillators using elementary functions. The present theory of equivalent oscillators consists of the main highlights as follows. (1) Proposing three equivalent 2-order oscillation equations corresponding to three classes of fractional oscillators; (2) Presenting the closed form expressions of equivalent mass, equivalent damping, equivalent natural frequencies, equivalent damping ratio for each class of fractional oscillators; (3) Putting forward the closed form formulas of responses (free, impulse, unit step, frequency, sinusoidal) to each class of fractional oscillators; (4) Revealing the power laws of equivalent mass and equivalent damping for each class of fractional oscillators in terms of oscillation frequency; (5) Giving analytic expressions of the logarithmic decrements of three classes of fractional oscillators; (6) Representing the closed form representations of some of the generalized Mittag-Leffler functions with elementary functions. The present results suggest a novel theory of fractional oscillators. This may facilitate the application of the theory of fractional oscillators to practice. Full article
(This article belongs to the Special Issue Symmetry and Complexity)
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