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Symmetry
Volume 11
Issue 4
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Symmetry, Volume 11, Issue 4 (April 2019) – 156 articles

Cover Story (view full-size image): The symmetry of molecular arrangement was used in recognition of supramolecular synthons, with topological similarities of crystal structures allowed to identify a supramolecular polymer with contacts above the semi-coordination distance limit. Stabilizing cooperation and destabilizing competition between coordination and hydrogen bonds were observed. View this paper.
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27 pages, 16382 KiB  
Article
Identification of Determinants of the Speed-Reducing Effect of Pedestrian Refuges in Villages Located on a Chosen Regional Road
by Alicja Sołowczuk and Dominik Kacprzak
Symmetry 2019, 11(4), 597; https://doi.org/10.3390/sym11040597 - 25 Apr 2019
Cited by 5 | Viewed by 4558
Abstract
Traffic calming, as a traffic engineering discipline, is becoming an increasingly important aspect of the road engineering process. One of the traffic calming treatments are pedestrian refuges—raised islands located on or at the road centreline. This paper presents factors relevant to the performance [...] Read more.
Traffic calming, as a traffic engineering discipline, is becoming an increasingly important aspect of the road engineering process. One of the traffic calming treatments are pedestrian refuges—raised islands located on or at the road centreline. This paper presents factors relevant to the performance of this kind of traffic calming devices retrofitted on the stretches of regional roads in village areas. To this end, speed surveys were carried out before and after the islands in each direction on purposefully chosen test sections. In order to identify the determinants, each test section was characterised by features including the symmetry of the road layout geometry, surrounding features and the existing traffic signs and, last but not least, visibility of the road ahead. The survey data were used by the authors to perform analyses in order to group the speeds at the pedestrian refuges and relate them to specific factors and, finally, identify the determinants of speed reduction. In this way, the authors arrived at a conclusion that the performance of pedestrian refuges depends on a number of factors rather than solely on their geometric parameters. The analyses showed that the pedestrian refuge geometric parameters, features located in its proximity that influence the driver’s perception and placement of appropriate marking, can, in combination, result in achieving the desired speed reduction and ensure safety of non-motorised users. These hypotheses were tested on a stretch of a regional road in village area at three points of the process: before upgrading, after installation of pedestrian refuges, and after retrofitting of enhancements. Full article
(This article belongs to the Special Issue Multi-Criteria Decision-Making Techniques for Improvement Sustainability Engineering Processes)
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<p>Diagrams of the analysed pedestrian refuges: (<b>a</b>) symmetric, 2 m wide (entry zone); (<b>b</b>) symmetric, 2 m wide (entry zone; after centre island); (<b>c</b>) symmetric, 2 m wide (between two centre islands, both positioned on one side of the centreline); (<b>d</b>) 2.5 m wide pedestrian refuge on one side of the centreline (end of entry zone/beginning of the village centre area); (<b>e</b>) asymmetric, 4 m wide (village centre, asymmetric shift of the lane alignments by 1 m and 3 m respectively). The test sections P1, P2, … P<span class="html-italic">n</span>, are marked on the respective travel lanes before the pedestrian refuges, in this way indicating the traffic direction under analysis.</p> Full article ">Figure 2
<p>Test sections P1 and P2 with the pedestrian refuge positioned between the end of entry zone and beginning of the village centre area: (<b>a</b>) P1—entry zone; (<b>b</b>) P2—exit zone.</p> Full article ">Figure 3
<p>Test sections P3 and P4, pedestrian refuge placed 170 m after the gateway island symmetrical about the road centreline: (<b>a</b>) P3—exit zone; (<b>b</b>) P4—entry zone.</p> Full article ">Figure 4
<p>Test sections P5 and P6, pedestrian refuge placed 170 m after gateway island positioned on one side of the road centreline: (<b>a</b>) P5—entry zone; (<b>b</b>) P6—exit zone.</p> Full article ">Figure 5
<p>Test sections P7 and P8 between the end of entry zone and beginning of the village centre area: (<b>a</b>) P7—pedestrian refuge on one side of the centreline imposing large lateral shift by 2.5 m; (<b>b</b>) P8—departure lane without any imposed lateral shift.</p> Full article ">Figure 6
<p>Test sections P9 and P10 in the village centre area: (<b>a</b>) P9—small lateral shift (1 m); (<b>b</b>) P10—big lateral shift (3 m).</p> Full article ">Figure 7
<p>Distribution of speed parameters on the test sections in the range of percentile speeds: <span class="html-italic">v</span><sub>85</sub>–<span class="html-italic">v</span><sub>50</sub> and <span class="html-italic">v</span><sub>50</sub>–<span class="html-italic">v</span><sub>25</sub>.</p> Full article ">Figure 8
<p>Method of analysis and general and detailed factors under analysis.</p> Full article ">Figure 9
<p>Percentages of the speed ranges before the pedestrian refuges.</p> Full article ">Figure 10
<p>Determinants of the approach speed depending on the pedestrian refuge location: (<b>a</b>) in the entry zone or in the village centre; (<b>b</b>) in the exit zone.</p> Full article ">Figure 11
<p>Relationship between the speed parameters ∆<span class="html-italic">v</span><sub>85</sub> and <span class="html-italic">v</span><sub>85</sub> after the gateway island and the lateral shift (redrawn from Wirksamkeit geschwindigkeitsdämpfender Maßnahmen außerorts, 1997 [<a href="#B14-symmetry-11-00597" class="html-bibr">14</a>], p. 9).</p> Full article ">Figure 12
<p>Relationship between the speed parameters: ∆<span class="html-italic">v<sub>85</sub></span> and <span class="html-italic">v<sub>85</sub></span> before the pedestrian refuge and the lateral shift.</p> Full article ">Figure 13
<p>Percentages of a speed ranges after the pedestrian refuges.</p> Full article ">Figure 14
<p>Determinants of the departure speed depending on the pedestrian refuge location: (<b>a</b>) in the entry zone or in the village centre; (<b>b</b>) in the exit zone.</p> Full article ">Figure 15
<p>Relationship between the speed parameters: ∆<span class="html-italic">v</span><sub>85</sub> and <span class="html-italic">v</span><sub>85</sub> after the pedestrian refuge and the lateral shift.</p> Full article ">Figure 16
<p>Speed variation on the road section alongside the pedestrian refuge (the sight distance is given in relation to the pedestrian refuge axis).</p> Full article ">Figure 17
<p>Determinants of the variation of speed at pedestrian refuges located: (<b>a</b>) in the entry zone or in the village centre; (<b>b</b>) in the exit zone.</p> Full article ">Figure 18
<p>Stretch of the regional road lined with post-and-chain barriers: (<b>a</b>) in the village centre; (<b>b</b>) at the pedestrian crossing.</p> Full article ">Figure 19
<p>Free-flow speed profiles before and after upgrading of the regional road section in the village area (drawings by D. Kacprzak and A. Sołowczuk [<a href="#B15-symmetry-11-00597" class="html-bibr">15</a>]): (<b>a</b>) <span class="html-italic">v</span><sub>85</sub> speed profile; (<b>b</b>) <span class="html-italic">v<sub>av</sub></span> speed profile.</p> Full article ">Figure 20
<p>The speed distribution parameters before and after the pedestrian refuges on the test sections with sight distances of: (<b>a</b>) 200 m; (<b>b</b>) 140 m.</p> Full article ">Figure 21
<p>Calculated percentages of ≤50 km/h and &gt;50 km/h speeds on test sections under analysis with sight distances of: (<b>a</b>) 200 m; (<b>b</b>) 140 m.</p> Full article ">Figure 22
<p>Speed differences on the analysed test sections.</p> Full article ">Figure 23
<p>Sight distance after the pedestrian refuge: (<b>a</b>) 200 m; (<b>b</b>) 140 m.</p> Full article ">Figure 24
<p>Flow chart for designing pedestrian refuges on sections of regional roads in villages. <sup>a</sup> interactive road signs; <sup>b</sup> barrier bollards; <sup>c</sup> additional plantings; <sup>d</sup> pedestrian refuges with post-and-chain barriers or barrier bollards (<a href="#symmetry-11-00597-f026" class="html-fig">Figure 26</a> and <a href="#symmetry-11-00597-f027" class="html-fig">Figure 27</a>); <sup>e</sup> pedestrian refuges combined with kerb buildouts (bulb-outs) or pinchpoints (<a href="#symmetry-11-00597-f028" class="html-fig">Figure 28</a> and <a href="#symmetry-11-00597-f029" class="html-fig">Figure 29</a>).</p> Full article ">Figure 25
<p>Procedure for designing pedestrian asylums in entry zone: (<b>a</b>) rural area; (<b>b</b>) forest. <sup>b</sup> barrier bollards U-12c; <sup>c</sup> additional plantings.</p> Full article ">Figure 26
<p>Schematic design of pedestrian refuge enhanced by post-and-chain barriers U12-a: (<b>a</b>) plan view; (<b>b</b>) visualisation of the proposed design (drawings by Agata Misztal).</p> Full article ">Figure 27
<p>Schematic design of pedestrian refuge enhanced by barrier bollards U12-c: (<b>a</b>) plan view; (<b>b</b>) visualisation of the proposed design (drawings by Agata Misztal).</p> Full article ">Figure 28
<p>Schematic design of pedestrian refuge enhanced by bulb-outs: (<b>a</b>) plan view; (<b>b</b>) visualisation of the proposed design (drawings by Agata Misztal).</p> Full article ">Figure 29
<p>Schematic design of pedestrian refuge enhanced by a pinchpoint: (<b>a</b>) plan view; (<b>b</b>) visualisation of the proposed design (drawings by Agata Misztal).</p> Full article ">
15 pages, 3487 KiB  
Article
Hydrogen Power Plant Site Selection Under Fuzzy Multicriteria Decision-Making (FMCDM) Environment Conditions
by Chia-Nan Wang, Ming-Hsien Hsueh and Da-Fu Lin
Symmetry 2019, 11(4), 596; https://doi.org/10.3390/sym11040596 - 25 Apr 2019
Cited by 25 | Viewed by 5699
Abstract
Fuel and energy are basic resources necessary to meet a country’s socioeconomic development needs; further, countries rich in these resources have the best premise for meeting the inputs of an economic system; however, this also poses many political challenges and threats to national [...] Read more.
Fuel and energy are basic resources necessary to meet a country’s socioeconomic development needs; further, countries rich in these resources have the best premise for meeting the inputs of an economic system; however, this also poses many political challenges and threats to national security. Vietnam is located in the Southeast Asian monsoon-humid tropical region and has diverse fuel-energy resources such as coal, petroleum, and hydropower, along with renewable energy sources such as solar energy, biomass energy, and geothermal energy. However, the reality of economic development in recent years shows complex fluctuations in fuel and energy usage, i.e., besides the export of coal and crude oil, Vietnam still has imported processed oil products. To overcome this issue, many hydrogen power plants will be built in the future. This is why we propose fuzzy multicriteria decision-making (FMCDM) for hydrogen power plant site selection in this research. All criteria affecting location selection are determined by experts and literature reviews, and the weight of all criteria are defined by a fuzzy analytic hierarchy process (FAHP). The technique for order of preference by similarity to an ideal solution (TOPSIS) is a multicriteria decision analysis method, which is used for ranking potential locations in the final stage. As a result, the decision-making unit, DMU010 (DMU010), has become the optimal solution for building hydrogen power plants in Vietnam. A multicriteria decision-making (MCDM) model for hydrogen power plant site selection in Vietnam under fuzzy environment conditions is a contribution of this study. This research also provides useful tools for other types of renewable energies in Vietnam and other countries. Full article
(This article belongs to the Special Issue Multi-Criteria Decision-Making Techniques for Improvement Sustainability Engineering Processes)
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<p>The general process of site selection [<a href="#B6-symmetry-11-00596" class="html-bibr">6</a>,<a href="#B7-symmetry-11-00596" class="html-bibr">7</a>].</p> Full article ">Figure 2
<p>Research methodology.</p> Full article ">Figure 3
<p>Vietnam power generation mix (TWh). Source: Wook Mackenie.</p> Full article ">Figure 4
<p>Ten potential locations on the map of Vietnam.</p> Full article ">Figure 5
<p>The objectives hierarchy.</p> Full article ">Figure 6
<p>Negative ideal solution (NIS) and positive ideal solution (PIS) value.</p> Full article ">Figure 7
<p>Final ranking score.</p> Full article ">
11 pages, 273 KiB  
Article
Extended Degenerate r-Central Factorial Numbers of the Second Kind and Extended Degenerate r-Central Bell Polynomials
by Dae San Kim, Dmitry V. Dolgy, Taekyun Kim and Dojin Kim
Symmetry 2019, 11(4), 595; https://doi.org/10.3390/sym11040595 - 24 Apr 2019
Cited by 8 | Viewed by 2697
Abstract
In this paper, we introduce the extended degenerate r-central factorial numbers of the second kind and the extended degenerate r-central Bell polynomials. They are extended versions of the degenerate central factorial numbers of the second kind and the degenerate central Bell [...] Read more.
In this paper, we introduce the extended degenerate r-central factorial numbers of the second kind and the extended degenerate r-central Bell polynomials. They are extended versions of the degenerate central factorial numbers of the second kind and the degenerate central Bell polynomials, and also degenerate versions of the extended r-central factorial numbers of the second kind and the extended r-central Bell polynomials, all of which have been studied by Kim and Kim. We study various properties and identities concerning those numbers and polynomials and also their connections. Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with Their Applications Ⅱ)
17 pages, 332 KiB  
Article
Extended Rectangular b-Metric Spaces and Some Fixed Point Theorems for Contractive Mappings
by Zead Mustafa, Vahid Parvaneh, Mohammed M.M. Jaradat and Zoran Kadelburg
Symmetry 2019, 11(4), 594; https://doi.org/10.3390/sym11040594 - 24 Apr 2019
Cited by 25 | Viewed by 4410
Abstract
In this paper, we introduce the class of extended rectangular b-metric spaces as a generalization of both rectangular metric and rectangular b-metric spaces. In addition, some fixed point results connected with certain contractions are obtained and examples are given to illustrate [...] Read more.
In this paper, we introduce the class of extended rectangular b-metric spaces as a generalization of both rectangular metric and rectangular b-metric spaces. In addition, some fixed point results connected with certain contractions are obtained and examples are given to illustrate these results. Full article
(This article belongs to the Special Issue Symmetric and non-symmetric contractions in various abstract spaces)
16 pages, 5265 KiB  
Article
A Device Performance and Data Analytics Concept for Smartphones’ IoT Services and Machine-Type Communication in Cellular Networks
by Kingsley A. Ogudo, Dahj Muwawa Jean Nestor, Osamah Ibrahim Khalaf and Hamed Daei Kasmaei
Symmetry 2019, 11(4), 593; https://doi.org/10.3390/sym11040593 - 24 Apr 2019
Cited by 82 | Viewed by 5779
Abstract
With the advancement of new technologies, the number of connected devices, the amount of data generated, and the need to build an intelligently connected network of things to improve and enrich the human ecosystem open new doors to modifications and adaptations of current [...] Read more.
With the advancement of new technologies, the number of connected devices, the amount of data generated, and the need to build an intelligently connected network of things to improve and enrich the human ecosystem open new doors to modifications and adaptations of current cellular network infrastructures. While more focus is given to low power wide area (LPWA) applications and devices, a significant challenge is the definition of Internet of Things (IoT) use cases and the value generation of applications on already existing IoT devices. Smartphones and related devices are currently manufactured with a wide range of smart sensors such as accelerometers, video sensors, compasses, gyros, proximity sensors, fingerprint sensors, temperature sensors, and biometric sensors used for various purposes. Many of these sensors can be automatically expanded to monitor a user’s daily activities (e.g., fitness workouts), locations, movements, and real-time body temperatures. Mobile network operators (MNOs) play a substantial role in providing IoT communications platforms, as they manage traffic flow in the network. In this paper, we discuss the global concept of IoT and machine-type communication (MTC), and we conduct device performance analytics based on data traffic collected from a cellular network. The experiment equips service providers with a model and framework to monitor device performance in a network. Full article
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<p>Scheme of a simplified device analytics model.</p> Full article ">Figure 2
<p>Data processing model for traffic capturing analytics.</p> Full article ">Figure 3
<p>Process methodology.</p> Full article ">Figure 4
<p>Three-layer horizontal and three-layer vertical analytics model.</p> Full article ">Figure 5
<p>Use Case 1: Top 10 device manufacturers with high video throughput.</p> Full article ">Figure 6
<p>The top 10 device manufacturers with high video throughput, further grouped by technology.</p> Full article ">Figure 7
<p>The Layer 3 use case result.</p> Full article ">Figure 8
<p>The Layer 3 use case result based on packet loss.</p> Full article ">Figure 9
<p>The Layer 3 use case based on device adoption.</p> Full article ">Figure 10
<p>The Layer 3 use case based on device class adoption.</p> Full article ">Figure 11
<p>Percentage of well and poorly performing 3G4G devices in the dataset.</p> Full article ">Figure 12
<p>Training result for the support vector machine (SVM) algorithm.</p> Full article ">Figure 13
<p>Training Result for the C50 algorithm.</p> Full article ">Figure 14
<p>Machine learning (ML) algorithm performance comparison on the training set.</p> Full article ">Figure 15
<p>C50 algorithm performance on the device testing dataset.</p> Full article ">Figure 16
<p>SVM algorithm performance on the device testing data set.</p> Full article ">
26 pages, 329 KiB  
Article
Hamilton–Jacobi Wave Theory in Manifestly-Covariant Classical and Quantum Gravity
by Claudio Cremaschini and Massimo Tessarotto
Symmetry 2019, 11(4), 592; https://doi.org/10.3390/sym11040592 - 24 Apr 2019
Cited by 3 | Viewed by 3300
Abstract
The axiomatic geometric structure which lays at the basis of Covariant Classical and Quantum Gravity Theory is investigated. This refers specifically to fundamental aspects of the manifestly-covariant Hamiltonian representation of General Relativity which has recently been developed in the framework of a synchronous [...] Read more.
The axiomatic geometric structure which lays at the basis of Covariant Classical and Quantum Gravity Theory is investigated. This refers specifically to fundamental aspects of the manifestly-covariant Hamiltonian representation of General Relativity which has recently been developed in the framework of a synchronous deDonder–Weyl variational formulation (2015–2019). In such a setting, the canonical variables defining the canonical state acquire different tensorial orders, with the momentum conjugate to the field variable g μ ν being realized by the third-order 4-tensor Π μ ν α . It is shown that this generates a corresponding Hamilton–Jacobi theory in which the Hamilton principal function is a 4-tensor S α . However, in order to express the Hamilton equations as evolution equations and apply standard quantization methods, the canonical variables must have the same tensorial dimension. This can be achieved by projection of the canonical momentum field along prescribed tensorial directions associated with geodesic trajectories defined with respect to the background space-time for either classical test particles or raylights. It is proved that this permits to recover a Hamilton principal function in the appropriate form of 4-scalar type. The corresponding Hamilton–Jacobi wave theory is studied and implications for the manifestly-covariant quantum gravity theory are discussed. This concerns in particular the possibility of achieving at quantum level physical solutions describing massive or massless quanta of the gravitational field. Full article
(This article belongs to the Special Issue Space-Time and Symmetry Properties: Classical and Quantum Descriptions)
12 pages, 3899 KiB  
Article
Corn Classification System based on Computer Vision
by Xiaoming Li, Baisheng Dai, Hongmin Sun and Weina Li
Symmetry 2019, 11(4), 591; https://doi.org/10.3390/sym11040591 - 24 Apr 2019
Cited by 38 | Viewed by 7990
Abstract
Automated classification of corn is important for corn sorting in intelligent agriculture. This paper presents a reliable corn classification method based on techniques of computer vision and machine learning. To discriminate different damaged types of corns, a line profile segmentation method is firstly [...] Read more.
Automated classification of corn is important for corn sorting in intelligent agriculture. This paper presents a reliable corn classification method based on techniques of computer vision and machine learning. To discriminate different damaged types of corns, a line profile segmentation method is firstly used to segment and separate a group of touching corns. Then, twelve color features and five shape features are extracted for each individual corn object. Finally, a maximum likelihood estimator is trained to classify normal and damaged corns. To evaluate the performance of the proposed method, a private dataset consisting of images of normal corn and six kinds of damage corns, including heat-damaged, germ-damaged, cob-rot-damaged, blue eye mold-damaged, insect-damaged, and surface mold-damaged, were collected in this work. The proposed method achieved an accuracy of 96.67% for the classification between normal corns and the first four common damaged corns, and an accuracy of 74.76% was achieved for the classification between normal corns and six kinds of damaged corns. The experimental results demonstrated the effectiveness of the proposed corn classification system. Full article
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<p>Corn image collection system.</p> Full article ">Figure 2
<p>Illustrations of images captured with different lighting systems. (<b>a</b>) corn image captured using lower fluorescent lamps, and (<b>b</b>) corn image captured using upper fluorescent lamps.</p> Full article ">Figure 3
<p>Flowchart of the proposed corn classification method.</p> Full article ">Figure 4
<p>Schematic diagram of the line profile-based segmentation algorithm.</p> Full article ">Figure 5
<p>Corn segmentation results with the line profile-based segmentation algorithm.</p> Full article ">Figure 6
<p>The feature extraction tree for each corn kernel.</p> Full article ">Figure 7
<p>Illustration of two faces of corn. (<b>a</b>) the face up of corn and (<b>b</b>) the face down of corn.</p> Full article ">Figure 8
<p>Illustration of germ area detection.</p> Full article ">Figure 9
<p>ROC curves for classification results of three types of damaged corns and normal corn.</p> Full article ">Figure 10
<p>Classification results for a group of corns.</p> Full article ">
11 pages, 2134 KiB  
Article
Supervised Reinforcement Learning via Value Function
by Yaozong Pan, Jian Zhang, Chunhui Yuan and Haitao Yang
Symmetry 2019, 11(4), 590; https://doi.org/10.3390/sym11040590 - 24 Apr 2019
Cited by 2 | Viewed by 4358
Abstract
Using expert samples to improve the performance of reinforcement learning (RL) algorithms has become one of the focuses of research nowadays. However, in different application scenarios, it is hard to guarantee both the quantity and quality of expert samples, which prohibits the practical [...] Read more.
Using expert samples to improve the performance of reinforcement learning (RL) algorithms has become one of the focuses of research nowadays. However, in different application scenarios, it is hard to guarantee both the quantity and quality of expert samples, which prohibits the practical application and performance of such algorithms. In this paper, a novel RL decision optimization method is proposed. The proposed method is capable of reducing the dependence on expert samples via incorporating the decision-making evaluation mechanism. By introducing supervised learning (SL), our method optimizes the decision making of the RL algorithm by using demonstrations or expert samples. Experiments are conducted in Pendulum and Puckworld scenarios to test the proposed method, and we use representative algorithms such as deep Q-network (DQN) and Double DQN (DDQN) as benchmarks. The results demonstrate that the method adopted in this paper can effectively improve the decision-making performance of agents even when the expert samples are not available. Full article
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<p>The principle of DQN.</p> Full article ">Figure 2
<p>The framework of SRLVF.</p> Full article ">Figure 3
<p>Pendulum (<b>a</b>) and Puckworld (<b>b</b>).</p> Full article ">Figure 4
<p>(<b>a</b>) Average rewards on DQN and SDQN in Puckworld. (<b>b</b>) Average rewards on DDQN and SDDQN in Puckworld too.</p> Full article ">Figure 5
<p>(<b>a</b>) Average rewards on DQN and SDQN in Pendulum. (<b>b</b>) Average rewards on DDQN and SDDQN in Pendulum too.</p> Full article ">Figure 6
<p>(<b>a</b>) Average rewards on SDDQN using different <math display="inline"><semantics> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> in Pendulum. (<b>b</b>) Average rewards on SDDQN using different <math display="inline"><semantics> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> in Puckworld. (<b>c</b>) Average rewards on SDQN using different <math display="inline"><semantics> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> in Puckworld too.</p> Full article ">
16 pages, 4383 KiB  
Article
The Hay Inclined Plane in Coalbrookdale (Shropshire, England): Geometric Modeling and Virtual Reconstruction
by José Ignacio Rojas-Sola and Eduardo De la Morena-De la Fuente
Symmetry 2019, 11(4), 589; https://doi.org/10.3390/sym11040589 - 24 Apr 2019
Cited by 17 | Viewed by 4069
Abstract
This article shows the geometric modeling and virtual reconstruction of the inclined plane of Coalbrookdale (Shropshire, England) that was in operation from 1792 to 1894. This historical invention, work of the Englishman William Reynolds, allowed the transportation of boats through channels located at [...] Read more.
This article shows the geometric modeling and virtual reconstruction of the inclined plane of Coalbrookdale (Shropshire, England) that was in operation from 1792 to 1894. This historical invention, work of the Englishman William Reynolds, allowed the transportation of boats through channels located at different levels. Autodesk Inventor Professional software has been used to obtain the 3D CAD model of this historical invention and its geometric documentation. The material for the research is available on the website of the Betancourt Project of the Canary Orotava Foundation for the History of Science. Also, because the single sheet does not have a scale, it has been necessary to adopt a graphic scale so that the dimensions of the different elements are coherent. Furthermore, it has been necessary to establish some dimensional, geometric, and movement restrictions (degrees of freedom) so that the set will work properly. One of the main conclusions is that William Reynolds designed a mechanism seeking a longitudinal symmetry so that, from a single continuous movement, the mechanism allows two vessels to ascend and descend simultaneously. This engineering solution facilitated a doubling of the working capacity of the device, as well as a reduction of the energy needs of the system. Full article
(This article belongs to the Special Issue Symmetry in Engineering Sciences)
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<p>Hay inclined plane drawn by Agustín de Betancourt [<a href="#B15-symmetry-11-00589" class="html-bibr">15</a>] (Courtesy of Fundación Canaria Orotava de Historia de la Ciencia).</p> Full article ">Figure 2
<p>Sheet drawn by Agustín de Betancourt in the Recicourt memory [<a href="#B17-symmetry-11-00589" class="html-bibr">17</a>] (Courtesy of Fundación Canaria Orotava de Historia de la Ciencia).</p> Full article ">Figure 3
<p>Summative scheme of the methodology followed in the 3D modeling process.</p> Full article ">Figure 4
<p>Rendered isometric view of the 3D CAD model (above) and opposite (below).</p> Full article ">Figure 5
<p>Exploded view of the 3D CAD model.</p> Full article ">Figure 6
<p>Plan of the ensemble of the Hay inclined plane with an indicative list of the elements.</p> Full article ">Figure 7
<p>Longitudinal section of the Hay inclined plane.</p> Full article ">Figure 8
<p>Profile view of the Hay inclined plane.</p> Full article ">Figure 9
<p>Direction of rotation when the transmission system is activated.</p> Full article ">Figure 10
<p>Sequence of movements in the ascent of the boats.</p> Full article ">Figure 11
<p>Braking system of the drive shafts.</p> Full article ">
23 pages, 950 KiB  
Article
Product Operations on q-Rung Orthopair Fuzzy Graphs
by Songyi Yin, Hongxu Li and Yang Yang
Symmetry 2019, 11(4), 588; https://doi.org/10.3390/sym11040588 - 23 Apr 2019
Cited by 17 | Viewed by 2896
Abstract
The q-rung orthopair fuzzy graph is an extension of intuitionistic fuzzy graph and Pythagorean fuzzy graph. In this paper, the degree and total degree of a vertex in q-rung orthopair fuzzy graphs are firstly defined. Then, some product operations on q [...] Read more.
The q-rung orthopair fuzzy graph is an extension of intuitionistic fuzzy graph and Pythagorean fuzzy graph. In this paper, the degree and total degree of a vertex in q-rung orthopair fuzzy graphs are firstly defined. Then, some product operations on q-rung orthopair fuzzy graphs, including direct product, Cartesian product, semi-strong product, strong product, and lexicographic product, are defined. Furthermore, some theorems about the degree and total degree under these product operations are put forward and elaborated with several examples. In particular, these theorems improve the similar results in single-valued neutrosophic graphs and Pythagorean fuzzy graphs. Full article
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<p>A road network using <italic>q</italic>-rung orthopair fuzzy graph (<italic>q</italic>-ROFG) with <italic>q</italic> = 4.</p> Full article ">Figure 2
<p>Two <italic>q</italic>-ROFGs with <italic>q</italic> = 3.</p> Full article ">Figure 3
<p>Direct product of two <italic>q</italic>-ROFGs.</p> Full article ">Figure 4
<p>Cartesian product of two <italic>q</italic>-ROFGs.</p> Full article ">Figure 5
<p>Semi-strong product of two <italic>q</italic>-ROFGs.</p> Full article ">Figure 6
<p>Strong product of two <italic>q</italic>-ROFGs.</p> Full article ">Figure 7
<p>Lexicographic product of two <italic>q</italic>-ROFGs.</p> Full article ">
46 pages, 594 KiB  
Article
Hadronic and Hadron-Like Physics of Dark Matter
by Vitaly Beylin, Maxim Yu. Khlopov, Vladimir Kuksa and Nikolay Volchanskiy
Symmetry 2019, 11(4), 587; https://doi.org/10.3390/sym11040587 - 23 Apr 2019
Cited by 34 | Viewed by 4377
Abstract
The problems of simple elementary weakly interacting massive particles (WIMPs) appeal to extend the physical basis for nonbaryonic dark matter. Such extension involves more sophisticated dark matter candidates from physics beyond the Standard Model (BSM) of elementary particles. We discuss several models of [...] Read more.
The problems of simple elementary weakly interacting massive particles (WIMPs) appeal to extend the physical basis for nonbaryonic dark matter. Such extension involves more sophisticated dark matter candidates from physics beyond the Standard Model (BSM) of elementary particles. We discuss several models of dark matter, predicting new colored, hyper-colored or techni-colored particles and their accelerator and non-accelerator probes. The nontrivial properties of the proposed dark matter candidates can shed new light on the dark matter physics. They provide interesting solutions for the puzzles of direct and indirect dark matter search. Full article
(This article belongs to the Special Issue Cosmological Inflation, Dark Matter and Dark Energy)
9 pages, 265 KiB  
Article
Derivative Free Fourth Order Solvers of Equations with Applications in Applied Disciplines
by Ramandeep Behl, Ioannis K. Argyros, Fouad Othman Mallawi and J. A. Tenreiro Machado
Symmetry 2019, 11(4), 586; https://doi.org/10.3390/sym11040586 - 23 Apr 2019
Cited by 1 | Viewed by 2447
Abstract
This paper develops efficient equation solvers for real- and complex-valued functions. An earlier study by Lee and Kim, used the Taylor-type expansions and hypotheses on higher than first order derivatives, but no derivatives appeared in the suggested method. However, we have many cases [...] Read more.
This paper develops efficient equation solvers for real- and complex-valued functions. An earlier study by Lee and Kim, used the Taylor-type expansions and hypotheses on higher than first order derivatives, but no derivatives appeared in the suggested method. However, we have many cases where the calculations of the fourth derivative are expensive, or the result is unbounded, or even does not exist. We only use the first order derivative of function Ω in the proposed convergence analysis. Hence, we expand the utilization of the earlier scheme, and we study the computable radii of convergence and error bounds based on the Lipschitz constants. Furthermore, the range of starting points is also explored to know how close the initial guess should be considered for assuring convergence. Several numerical examples where earlier studies cannot be applied illustrate the new technique. Full article
(This article belongs to the Special Issue Symmetry in Complex Systems)
13 pages, 3284 KiB  
Article
Lightweight Architecture for Real-Time Hand Pose Estimation with Deep Supervision
by Yufei Wu, Xiaofei Ruan, Yu Zhang, Huang Zhou, Shengyu Du and Gang Wu
Symmetry 2019, 11(4), 585; https://doi.org/10.3390/sym11040585 - 23 Apr 2019
Cited by 3 | Viewed by 3695
Abstract
The high demand for computational resources severely hinders the deployment of deep learning applications in resource-limited devices. In this work, we investigate the under-studied but practically important network efficiency problem and present a new, lightweight architecture for hand pose estimation. Our architecture is [...] Read more.
The high demand for computational resources severely hinders the deployment of deep learning applications in resource-limited devices. In this work, we investigate the under-studied but practically important network efficiency problem and present a new, lightweight architecture for hand pose estimation. Our architecture is essentially a deeply-supervised pruned network in which less important layers and branches are removed to achieve a higher real-time inference target on resource-constrained devices without much accuracy compromise. We further make deployment optimization to facilitate the parallel execution capability of central processing units (CPUs). We conduct experiments on NYU and ICVL datasets and develop a demo1 using the RealSense camera. Experimental results show our lightweight network achieves an average running time of 32 ms (31.3 FPS, the original is 22.7 FPS) before deployment optimization. Meanwhile, the model is only about half parameters size of the original one with 11.9 mm mean joint error. After the further optimization with OpenVINO, the optimized model can run at 56 FPS on CPUs in contrast to 44 FPS running on a graphics processing unit (GPU) (Tensorflow) and it can achieve the real-time goal. Full article
►▼ Show Figures

Figure 1

Figure 1
<p>Compact network architecture. (<b>a</b>) Layer pruning: the second stack is pruned. Post-processing module includes coordinate reconstruction from three heat maps (yellow blocks). Blue and green blocks stand for a series of convolutional and pool operations which are omitted by limited space (refer to Reference [<a href="#B5-symmetry-11-00585" class="html-bibr">5</a>] for details). (<b>b</b>) Level pruning: varying modes (<math display="inline"><semantics> <mrow> <msup> <mi mathvariant="normal">L</mi> <mn>1</mn> </msup> </mrow> </semantics></math>~<math display="inline"><semantics> <mrow> <msup> <mi mathvariant="normal">L</mi> <mn>4</mn> </msup> </mrow> </semantics></math>) can be selected for different complexity. Supervision modules are added to every decoder (red lines).</p> Full article ">Figure 2
<p>Workflow of OpenVINO toolkit: the trained model firstly optimized by the model optimizer to <span class="html-italic">IR</span> and then executed by the inference engine.</p> Full article ">Figure 3
<p>L1-norm of two stacks, block id stands for the id of skip and ResNet blocks.</p> Full article ">Figure 4
<p>Loss descent comparison of w/ and w/o deep supervision (NYU dataset [<a href="#B1-symmetry-11-00585" class="html-bibr">1</a>]).</p> Full article ">Figure 5
<p>Comparison of the original network on NYU [<a href="#B1-symmetry-11-00585" class="html-bibr">1</a>]. Percentage of frames in which all joints are below a threshold is plotted.</p> Full article ">Figure 6
<p>Qualitative results. Hand pose estimation results on NYU [<a href="#B1-symmetry-11-00585" class="html-bibr">1</a>] dataset. (<b>a</b>) Successful samples (top row) (<b>b</b>) Failed samples (top row) and the corresponding ground-truth (bottom row).</p> Full article ">Figure 7
<p>Qualitative results. Hand pose estimation results on ICVL dataset [<a href="#B13-symmetry-11-00585" class="html-bibr">13</a>]. (<b>a</b>) Successful samples (top row) (<b>b</b>) Failed samples (top row) and the corresponding ground-truth (bottom row).</p> Full article ">Figure 8
<p>Comparison with state-of-the-art on ICVL [<a href="#B13-symmetry-11-00585" class="html-bibr">13</a>]. Percentage of frames in which all joints are below a threshold is plotted.</p> Full article ">
30 pages, 7472 KiB  
Article
Edge Even Graceful Labeling of Cylinder Grid Graph
by Ahmed A. Elsonbaty and Salama Nagy Daoud
Symmetry 2019, 11(4), 584; https://doi.org/10.3390/sym11040584 - 22 Apr 2019
Cited by 6 | Viewed by 3795
Abstract
Edge even graceful labeling (e.e.g., l.) of graphs is a modular technique of edge labeling of graphs, introduced in 2017. An e.e.g., l. of simple finite undirected graph G = ( V ( G ) , E ( G ) ) of order [...] Read more.
Edge even graceful labeling (e.e.g., l.) of graphs is a modular technique of edge labeling of graphs, introduced in 2017. An e.e.g., l. of simple finite undirected graph G = ( V ( G ) , E ( G ) ) of order P = | ( V ( G ) | and size q = | E ( G ) | is a bijection f : E ( G ) → { 2 , 4 , … , 2 q } , such that when each vertex v ∈ V ( G ) is assigned the modular sum of the labels (images of f ) of the edges incident to v , the resulting vertex labels are distinct mod 2 r , where r = max ( p , q ) . In this work, the family of cylinder grid graphs are studied. Explicit formulas of e.e.g., l. for all of the cases of each member of this family have been proven. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry)
►▼ Show Figures

Figure 1

Figure 1
<p>Cylinder grid graph <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>The cylinder grid graph <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mi>m</mi> </semantics></math> is even and <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>An edge even graceful labeling (e.e.g., l.) of the cylinder grid graphs <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>8</mn> <mo>,</mo> <mn>11</mn> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>8</mn> <mo>,</mo> <mn>12</mn> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 3 Cont.
<p>An edge even graceful labeling (e.e.g., l.) of the cylinder grid graphs <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>8</mn> <mo>,</mo> <mn>11</mn> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>8</mn> <mo>,</mo> <mn>12</mn> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>The cylinder grid graph <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>3</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>≡</mo> <mn>1</mn> <mi>mod</mi> <mn>6</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>The cylinder grid graph <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>3</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>≡</mo> <mn>3</mn> <mi>mod</mi> <mn>6</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>The cylinder grid graph <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>3</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>≡</mo> <mn>5</mn> <mi>mod</mi> <mn>6</mn> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>An e.e.g., l. of the cylinder grid graphs <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>25</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>27</mn> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>29</mn> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 7 Cont.
<p>An e.e.g., l. of the cylinder grid graphs <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>25</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>27</mn> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>29</mn> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>An e.e.g., l. of the cylinder grid graph <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>5</mn> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>The cylinder grid graph <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mrow> </semantics></math>,<math display="inline"><semantics> <mi>m</mi> </semantics></math> is odd greater than <math display="inline"><semantics> <mn>3</mn> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </semantics></math>.</p> Full article ">Figure 10
<p>The cylinder grid graph <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mi>m</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 11
<p>An e.e.g., l. of the cylinder grid graph <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mi>m</mi> <mo>,</mo> <mn>4</mn> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 12
<p>An e.e.g., l. of the cylinder grid graphs <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>9</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>9</mn> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>10</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>12</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>14</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>16</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>18</mn> </mrow> </msub> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>20</mn> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 12 Cont.
<p>An e.e.g., l. of the cylinder grid graphs <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>9</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>9</mn> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>10</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>12</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>14</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>16</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>18</mn> </mrow> </msub> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>20</mn> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 12 Cont.
<p>An e.e.g., l. of the cylinder grid graphs <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>9</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>9</mn> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>10</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>12</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>14</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>16</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>18</mn> </mrow> </msub> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>20</mn> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 12 Cont.
<p>An e.e.g., l. of the cylinder grid graphs <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>9</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>9</mn> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>10</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>12</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>14</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>16</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>18</mn> </mrow> </msub> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>20</mn> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 13
<p>The cylinder grid graph <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mi>m</mi> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mi>m</mi> </semantics></math> is odd, <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>≥</mo> <mn>3</mn> </mrow> </semantics></math>.</p> Full article ">Figure 14
<p>An e.e.g.l. of the cylinder grid graphs <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>9</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>9</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>11</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>13</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>15</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>17</mn> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>19</mn> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 14 Cont.
<p>An e.e.g.l. of the cylinder grid graphs <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>9</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>9</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>11</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>13</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>15</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>17</mn> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>19</mn> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 14 Cont.
<p>An e.e.g.l. of the cylinder grid graphs <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>9</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>9</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>11</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>13</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>15</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>17</mn> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>19</mn> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 14 Cont.
<p>An e.e.g.l. of the cylinder grid graphs <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>9</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>9</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>11</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>13</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>15</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>17</mn> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>19</mn> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">
14 pages, 659 KiB  
Article
A Scalable and Hybrid Intrusion Detection System Based on the Convolutional-LSTM Network
by Muhammad Ashfaq Khan, Md. Rezaul Karim and Yangwoo Kim
Symmetry 2019, 11(4), 583; https://doi.org/10.3390/sym11040583 - 22 Apr 2019
Cited by 139 | Viewed by 14116
Abstract
With the rapid advancements of ubiquitous information and communication technologies, a large number of trustworthy online systems and services have been deployed. However, cybersecurity threats are still mounting. An intrusion detection (ID) system can play a significant role in detecting such security threats. [...] Read more.
With the rapid advancements of ubiquitous information and communication technologies, a large number of trustworthy online systems and services have been deployed. However, cybersecurity threats are still mounting. An intrusion detection (ID) system can play a significant role in detecting such security threats. Thus, developing an intelligent and accurate ID system is a non-trivial research problem. Existing ID systems that are typically used in traditional network intrusion detection system often fail and cannot detect many known and new security threats, largely because those approaches are based on classical machine learning methods that provide less focus on accurate feature selection and classification. Consequently, many known signatures from the attack traffic remain unidentifiable and become latent. Furthermore, since a massive network infrastructure can produce large-scale data, these approaches often fail to handle them flexibly, hence are not scalable. To address these issues and improve the accuracy and scalability, we propose a scalable and hybrid IDS, which is based on Spark ML and the convolutional-LSTM (Conv-LSTM) network. This IDS is a two-stage ID system: the first stage employs the anomaly detection module, which is based on Spark ML. The second stage acts as a misuse detection module, which is based on the Conv-LSTM network, such that both global and local latent threat signatures can be addressed. Evaluations of several baseline models in the ISCX-UNB dataset show that our hybrid IDS can identify network misuses accurately in 97.29% of cases and outperforms state-of-the-art approaches during 10-fold cross-validation tests. Full article
(This article belongs to the Special Issue Symmetry-Adapted Machine Learning for Information Security)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>An overview of the proposed ID model.</p> Full article ">Figure 2
<p>A schematic representation of the Conv-LSTM network, which starts by measuring attack, traffic and passing that data to both the CNN and LSTM layers before getting a flattened vector, which was fed through dense and Softmax layers for predicting the malicious traffic.</p> Full article ">
15 pages, 2306 KiB  
Article
Analysis of Periodic Structures Made of Pins Inside a Parallel Plate Waveguide
by Nafsika Memeletzoglou, Carlos Sanchez-Cabello, Francisco Pizarro-Torres and Eva Rajo-Iglesias
Symmetry 2019, 11(4), 582; https://doi.org/10.3390/sym11040582 - 22 Apr 2019
Cited by 13 | Viewed by 4998
Abstract
In this work, we have analyzed different versions of periodic structures made with metallic pins located inside a parallel plate waveguide (PPWG), varying the symmetry and disposition of the pins. The analysis focuses on two main parameters related to wave propagation. On one [...] Read more.
In this work, we have analyzed different versions of periodic structures made with metallic pins located inside a parallel plate waveguide (PPWG), varying the symmetry and disposition of the pins. The analysis focuses on two main parameters related to wave propagation. On one hand, we have studied how the different proposed structures can create a stopband so that the parallel plate modes can be used in gap waveguide technology or filtering structures. On the other hand, we have analyzed the dispersion and equivalent refractive index of the first propagating transverse electromagnetic mode (TEM). The results show how the use of complex structures made with pins in the top and bottom plates of a PPWG have no advantages in terms of the achieved stopband size. However, for the case of the propagating mode, it is possible to find less dispersive modes and a higher range of equivalent refractive indices when using double-pin structures compared to a reference case with single pins. Full article
(This article belongs to the Special Issue Higher Symmetries and Its Application in Microwave Technology, Antennas and Metamaterials)
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Figure 1

Figure 1
<p>Pin geometries considered in the study. * In these cases, we will consider two cases of shift in one direction (<span class="html-italic">X</span>) and two directions (<span class="html-italic">XY</span>) shifts.</p> Full article ">Figure 2
<p>Unit cell description including axes. Top view of the translations in only X (<b>b</b>) or XY (<b>c</b>) directions.</p> Full article ">Figure 3
<p>Dispersion diagrams for cases <span class="html-italic">a</span>, <span class="html-italic">b</span> and <span class="html-italic">c</span> according to <a href="#symmetry-11-00582-f001" class="html-fig">Figure 1</a>. The frequency is normalized to the frequency corresponding to the single-pin height <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mrow> <mi>p</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>λ</mi> <mo>/</mo> <mn>4</mn> </mrow> </semantics></math>. Blue lines correspond to the first mode and red lines to the second mode.</p> Full article ">Figure 4
<p>Dispersion diagrams for case <span class="html-italic">d</span> in <a href="#symmetry-11-00582-f001" class="html-fig">Figure 1</a> for different pin heights, where <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mrow> <mi>t</mi> <mi>o</mi> <mi>p</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>h</mi> <mrow> <mi>b</mi> <mi>o</mi> <mi>t</mi> <mi>t</mi> <mi>o</mi> <mi>m</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>h</mi> <mrow> <mi>p</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mrow> </semantics></math>. Blue lines represent the first mode and red lines the second mode.</p> Full article ">Figure 5
<p>Dispersion diagrams for case <span class="html-italic">e</span> in <a href="#symmetry-11-00582-f001" class="html-fig">Figure 1</a> for different pin heights, where <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mrow> <mi>t</mi> <mi>o</mi> <mi>p</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>h</mi> <mrow> <mi>b</mi> <mi>o</mi> <mi>t</mi> <mi>t</mi> <mi>o</mi> <mi>m</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>h</mi> <mrow> <mi>p</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mrow> </semantics></math>. Blue lines represent the first mode and red lines the second mode.</p> Full article ">Figure 6
<p>Dispersion diagrams for pins in case <span class="html-italic">f</span> according to <a href="#symmetry-11-00582-f001" class="html-fig">Figure 1</a>. Different colors represent different modes.</p> Full article ">Figure 7
<p>Parametric study for the geometry described as <span class="html-italic">d</span> in <a href="#symmetry-11-00582-f001" class="html-fig">Figure 1</a> for <span class="html-italic">h<sub>bottom</sub></span> = 1.5 mm. Red lines represent the start frequency of the stopband whilst blue lines represent its end frequency.</p> Full article ">Figure 8
<p>Parametric study for the geometry described as <span class="html-italic">d</span> in <a href="#symmetry-11-00582-f001" class="html-fig">Figure 1</a> for <span class="html-italic">h<sub>bottom</sub></span> = 2.0 mm. Red lines represent the start frequency of the stopband whilst blue lines represent its end frequency.</p> Full article ">Figure 9
<p>Parametric study for the X-shifted geometry with <span class="html-italic">h<sub>top</sub></span> = <span class="html-italic">h<sub>bottom</sub></span>, <span class="html-italic">p</span><sub>1</sub> = 0.125<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">p</span><sub>2</sub> = 0.1875<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">p</span><sub>3</sub> = 0.25<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">p</span><sub>4</sub> = 0.375<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">p</span><sub>5</sub> = 0.5<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">g</span><sub>1</sub> = 0.0125<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">g</span><sub>2</sub> = 0.0625<span class="html-italic">λ</span><sub>0</sub>, and <span class="html-italic">g</span><sub>3</sub> = 0.125<span class="html-italic">λ</span><sub>0</sub>.</p> Full article ">Figure 10
<p>Parametric study for the X-shifted geometry with <span class="html-italic">h<sub>bottom</sub></span> = 2 mm, <span class="html-italic">p</span><sub>1</sub> = 0.125<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">p</span><sub>2</sub> = 0.1875<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">p</span><sub>3</sub> = 0.25<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">p</span><sub>4</sub> = 0.375<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">p</span><sub>5</sub> = 0.5<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">g</span><sub>1</sub> = 0.0125<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">g</span><sub>2</sub> = 0.0625<span class="html-italic">λ</span><sub>0</sub>, and <span class="html-italic">g</span><sub>3</sub> = 0.125<span class="html-italic">λ</span><sub>0</sub>.</p> Full article ">Figure 11
<p>Parametric study for the XY-shifted geometry with <span class="html-italic">h<sub>top</sub></span> = <span class="html-italic">h<sub>bottom</sub></span>, <span class="html-italic">p</span><sub>1</sub> = 0.125<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">p</span><sub>2</sub> = 0.1875<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">p</span><sub>3</sub> = 0.25<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">p</span><sub>4</sub> = 0.375<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">p</span><sub>5</sub> = 0.5<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">g</span><sub>1</sub> = 0.0125<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">g</span><sub>2</sub> = 0.0625<span class="html-italic">λ</span><sub>0</sub>, and <span class="html-italic">g</span><sub>3</sub> = 0.125<span class="html-italic">λ</span><sub>0</sub>.</p> Full article ">Figure 12
<p>Parametric study for the XY-shifted geometry with <span class="html-italic">h<sub>bottom</sub></span> = 2 mm, <span class="html-italic">p</span><sub>1</sub> = 0.125<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">p</span><sub>2</sub> = 0.1875<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">p</span><sub>3</sub> = 0.25<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">p</span><sub>4</sub> = 0.375<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">p</span><sub>5</sub> = 0.5<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">g</span><sub>1</sub> = 0.0125<span class="html-italic">λ</span><sub>0</sub>, <span class="html-italic">g</span><sub>2</sub> = 0.0625<span class="html-italic">λ</span><sub>0</sub>, and <span class="html-italic">g</span><sub>3</sub> = 0.125<span class="html-italic">λ</span><sub>0</sub>.</p> Full article ">Figure 13
<p>Equivalent refractive indices for different cases from <a href="#symmetry-11-00582-f001" class="html-fig">Figure 1</a>: in particular cases, <span class="html-italic">a</span>, <span class="html-italic">b</span>, <span class="html-italic">d</span> and <span class="html-italic">f</span>.</p> Full article ">Figure 14
<p>Equivalent refractive indices for different cases in <a href="#symmetry-11-00582-f001" class="html-fig">Figure 1</a> after shifting the pins.</p> Full article ">Figure 15
<p>Effect of varying different parameters in the geometries <span class="html-italic">a</span>, <span class="html-italic">b</span> and <span class="html-italic">d</span> from <a href="#symmetry-11-00582-f001" class="html-fig">Figure 1</a>. The reference case has a gap of 0.5 mm, a period of 2 mm and a width of 1 mm.</p> Full article ">Figure 15 Cont.
<p>Effect of varying different parameters in the geometries <span class="html-italic">a</span>, <span class="html-italic">b</span> and <span class="html-italic">d</span> from <a href="#symmetry-11-00582-f001" class="html-fig">Figure 1</a>. The reference case has a gap of 0.5 mm, a period of 2 mm and a width of 1 mm.</p> Full article ">Figure 16
<p>Effect of varying different parameters in the geometry <span class="html-italic">f</span> (interleaved pins) from <a href="#symmetry-11-00582-f001" class="html-fig">Figure 1</a>. The reference case has a gap of 0.5 mm, a period of 4 mm and a width of 1 mm.</p> Full article ">Figure 16 Cont.
<p>Effect of varying different parameters in the geometry <span class="html-italic">f</span> (interleaved pins) from <a href="#symmetry-11-00582-f001" class="html-fig">Figure 1</a>. The reference case has a gap of 0.5 mm, a period of 4 mm and a width of 1 mm.</p> Full article ">Figure 17
<p>Effect of varying different parameters in the geometry <span class="html-italic">c</span> from <a href="#symmetry-11-00582-f001" class="html-fig">Figure 1</a>. The reference case has a gap of 0.5 mm, a period of 4 mm and a width of 1 mm.</p> Full article ">Figure 18
<p>Effect of varying different parameters in the geometry <span class="html-italic">e</span> from <a href="#symmetry-11-00582-f001" class="html-fig">Figure 1</a>. The reference case has a gap of 0.5 mm, a period of 4 mm and a width of 1 mm.</p> Full article ">
8 pages, 4083 KiB  
Article
Fully Metallic Flat Lens Based on Locally Twist-Symmetric Array of Complementary Split-Ring Resonators
by Oskar Dahlberg, Guido Valerio and Oscar Quevedo-Teruel
Symmetry 2019, 11(4), 581; https://doi.org/10.3390/sym11040581 - 22 Apr 2019
Cited by 16 | Viewed by 4531
Abstract
In this article, we demonstrate how twist symmetries can be employed in the design of flat lenses. A lens design is proposed, consisting of 13 perforated metallic sheets separated by an air gap. The perforation in the metal is a two-dimensional array of [...] Read more.
In this article, we demonstrate how twist symmetries can be employed in the design of flat lenses. A lens design is proposed, consisting of 13 perforated metallic sheets separated by an air gap. The perforation in the metal is a two-dimensional array of complementary split-ring resonators. In this specific design, the twist symmetry is local, as it is only applied to the unit cell of the array. Moreover, the twist symmetry is an approximation, as it is only applied to part of the unit cell. First, we demonstrate that, by varying the order of twist symmetry, the phase delay experienced by a wave propagating through the array can be accurately controlled. Secondly, a lens is designed by tailoring the unit cells throughout the aperture of the lens in order to obtain the desired phase delay. Simulation and measurement results demonstrate that the lens successfully transforms a spherical wave emanating from the focal point into a plane wave at the opposite side of the lens. The demonstrated concepts find application in future wireless communication networks where fully-metallic directive antennas are desired. Full article
(This article belongs to the Special Issue Higher Symmetries and Its Application in Microwave Technology, Antennas and Metamaterials)
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Figure 1

Figure 1
<p>Simulated CSRRs with local twist symmetry. Studied structures: (<b>a</b>) purely periodic (<math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>), (<b>b</b>) 3-fold, (<b>c</b>) 4-fold, and (<b>d</b>) 6-fold.</p> Full article ">Figure 2
<p>(<b>a</b>) Simulated propagation constant with dimensions: <math display="inline"><semantics> <mrow> <mi>u</mi> <mi>w</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> mm, <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math> mm, <math display="inline"><semantics> <mrow> <mi>s</mi> <mi>w</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>3</mn> </mrow> </semantics></math> mm, <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>=</mo> </mrow> </semantics></math> 1.5 mm, <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> mm, and <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> mm. (<b>b</b>) Simulated effective refractive index at 11 GHz with dimensions: <math display="inline"><semantics> <mrow> <mi>u</mi> <mi>w</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> mm, <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math> mm, <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>=</mo> </mrow> </semantics></math> 1.5 mm, <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> mm, and <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> mm.</p> Full article ">Figure 3
<p>(<b>a</b>–<b>b</b>) Layout of the lens: (<b>a</b>) front view, and (<b>b</b>) perspective view. The distance between the metallic sheets is exaggerated for increased clarity. (<b>c</b>) Measurement setup in the anechoic chamber.</p> Full article ">Figure 4
<p>Simulated E-field and far-field of the lens. (<b>a</b>–<b>b</b>) <span class="html-italic">x</span>-component of the electric field excited by an <span class="html-italic">x</span>-oriented half-wavelength dipole placed in the focal point of the lens: (<b>a</b>) E-plane and (<b>b</b>) H-plane. (<b>c</b>–<b>d</b>) Normalized radiation pattern of the lens excited with a rectangular waveguide (WR90) placed at the focal point of the lens: (<b>c</b>) E-plane and (<b>d</b>) H-plane. The measurement of the H-plane cut is included as well. The width, <span class="html-italic">w</span>, of the lens is 220 mm, and the focal point is 130 mm from the first layer. The separation between the sheets is 4 mm, and the thickness of the metallic sheets is 1 mm.</p> Full article ">
19 pages, 478 KiB  
Article
Green Simulation of Pandemic Disease Propagation
by Spencer Wilson, Abdullah Alabdulkarim and David Goldsman
Symmetry 2019, 11(4), 580; https://doi.org/10.3390/sym11040580 - 22 Apr 2019
Viewed by 3203
Abstract
This paper is concerned with the efficient stochastic simulation of multiple scenarios of an infectious disease as it propagates through a population. In particular, we propose a simple “green” method to speed up the simulation of disease transmission as we vary the probability [...] Read more.
This paper is concerned with the efficient stochastic simulation of multiple scenarios of an infectious disease as it propagates through a population. In particular, we propose a simple “green” method to speed up the simulation of disease transmission as we vary the probability of infection of the disease from scenario to scenario. After running a baseline scenario, we incrementally increase the probability of infection, and use the common random numbers variance reduction technique to avoid re-simulating certain events in the new scenario that would not otherwise have changed from the previous scenario. A set of Monte Carlo experiments illustrates the effectiveness of the procedure. We also propose various extensions of the method, including its use to estimate the sensitivity of propagation characteristics in response to small changes in the infection probability. Full article
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Figure 1

Figure 1
<p>Exact expected length of pandemic (in days) for the case <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> with a one-day infectiousness period (Example 1).</p> Full article ">Figure 2
<p>Exact expected number of infected individuals for the case <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> with a one-day infectiousness period (Example 1).</p> Full article ">Figure 3
<p>Exact expected length of pandemic (in days) for the case <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> with a two-day infectiousness period (Example 2).</p> Full article ">Figure 4
<p>Exact expected number of infected individuals for the case <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> with a two-day infectiousness period (Example 2).</p> Full article ">Figure 5
<p>Histogram of length of pandemic (in days) for the case <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, one initial infective, and a one-day infectiousness period (based on 1,000,000 replications).</p> Full article ">Figure 6
<p>Histogram of number of individuals ultimately infected for the case <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, one initial infective, and a one-day infectiousness period (based on 1,000,000 replications).</p> Full article ">Figure 7
<p>Histogram of the total number of PRNs used for the case <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, one initial infective, and a one-day infectiousness period (based on 1,000,000 replications).</p> Full article ">Figure 8
<p>Histogram of length of pandemic (in days) for the case <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>, one initial infective, and a one-day infectiousness period (based on 1,000,000 replications).</p> Full article ">Figure 9
<p>Histogram of number of individuals ultimately infected for the case <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>, one initial infective, and a one-day infectiousness period (based on 1,000,000 replications).</p> Full article ">Figure 10
<p>Histogram of the total number of PRNs used for the case <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>, one initial infective, and a one-day infectiousness period (based on 1,000,000 replications).</p> Full article ">Figure 11
<p>Histogram of length of pandemic (in days) for the case <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.02</mn> </mrow> </semantics></math>, one initial infective, and a three-day infectiousness period, as described in Example 4 (based on 200 replications).</p> Full article ">Figure 12
<p>Histogram of number of individuals ultimately infected for the case <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.02</mn> </mrow> </semantics></math>, one initial infective, and a three-day infectiousness period, as described in Example 4 (based on 200 replications).</p> Full article ">Figure 13
<p>Histogram of the total number of PRNs used for the case <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.02</mn> </mrow> </semantics></math>, one initial infective, and a three-day infectiousness period, as described in Example 4 (based on 200 replications).</p> Full article ">
13 pages, 3709 KiB  
Article
Experimental Investigation on the Cooling and Inerting Effects of Liquid Nitrogen Injected into a Confined Space
by Huaijun Ji, Yunzhuo Li, Hetao Su, Wuyi Cheng and Xiang Wu
Symmetry 2019, 11(4), 579; https://doi.org/10.3390/sym11040579 - 22 Apr 2019
Cited by 18 | Viewed by 3281
Abstract
As a highly effective and environmentally benign suppression agent, liquid nitrogen (LN2) has been widely used for fire extinguishing in plants, dwellings, enclosed underground tunnels, and other confined spaces through cooling and inerting. It is of great significance to understand the [...] Read more.
As a highly effective and environmentally benign suppression agent, liquid nitrogen (LN2) has been widely used for fire extinguishing in plants, dwellings, enclosed underground tunnels, and other confined spaces through cooling and inerting. It is of great significance to understand the cooling and inerting effects of LN2 injected into a confined space. A confined-space experimental platform was developed to study the injecting LN2 into the platform with different injection parameters, such as mass flux, pipe diameter, and inclination angle. In addition, a mathematical model of quantitatively assessing cooling and inerting effects was proposed by using heat transfer capacity, inerting coefficient, and cooling rate. Results showed that the inerting effect was gradually enhanced with a mass flux increasing from 0.014 to 0.026 kg/s and then tended to level off; an appropriate pipe diameter of 12 mm was optimal for the cooling and inerting effects in this experiment. In addition, a positively increasing inclination angle could contribute to the cooling and inerting effects. However, there was little effect on the cooling and inerting with an inclination angle less than 0°. This study can provide technical guidances for environmentally friendly fire extinguishing with LN2 in a confined space. Full article
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Figure 1

Figure 1
<p>Sketch view of the experimental setup.</p> Full article ">Figure 2
<p>Temperatures (<b>a</b>) and oxygen concentrations (<b>b</b>) on six slices of <span class="html-italic">x</span> = 0.0, 1.0, 2.0 m, <span class="html-italic">y</span> = 1.0 m, <span class="html-italic">z</span> = 0.3, 1.0 m at 0, 90, 180, 360, 720, and 1440 s.</p> Full article ">Figure 3
<p>Heat transfer capacity versus the time of LN<sub>2</sub> injection with different mass fluxes.</p> Full article ">Figure 4
<p>Average oxygen concentrations in the confined space versus time (≥1140 s).</p> Full article ">Figure 5
<p>Temperatures (<b>a</b>) and oxygen concentrations (<b>b</b>) at measuring points #13, #14, and #15 versus mass flux at 180 s.</p> Full article ">Figure 6
<p>Cooling rate and heat transfer capacity at 180 s (<b>a</b>), inerting coefficient at 180 s (<b>b</b>) versus mass flux.</p> Full article ">Figure 7
<p>Temperatures (<b>a</b>) and oxygen concentrations (<b>b</b>) at measuring points #13, #14, and #15 versus pipe diameter at 180 s.</p> Full article ">Figure 8
<p>Cooling rate and heat transfer capacity at 180 s (<b>a</b>), inerting coefficient at 180 s (<b>b</b>) versus pipe diameter.</p> Full article ">Figure 9
<p>Temperatures (<b>a</b>) and oxygen concentrations (<b>b</b>) at measuring points #13, #14, and #15 versus inclination angle at 180 s.</p> Full article ">Figure 10
<p>Cooling rate and heat transfer capacity at 180 s (<b>a</b>), inerting coefficient at 180 s (<b>b</b>) versus inclination angle.</p> Full article ">
10 pages, 1495 KiB  
Article
Convenient Asymmetric Synthesis of Fmoc-(S)-6,6,6-Trifluoro-Norleucine
by Haibo Mei, Zizhen Yin, Toshio Miwa, Hiroki Moriwaki, Hidenori Abe, Jianlin Han and Vadim A. Soloshonok
Symmetry 2019, 11(4), 578; https://doi.org/10.3390/sym11040578 - 21 Apr 2019
Cited by 25 | Viewed by 7821
Abstract
In this work we report a convenient asymmetric synthesis of Fmoc-(S)-6,6,6-trifluoro-norleucine via alkylation reaction of chiral glycine equivalent. The target amino acid of 99% enantiomeric purity was prepared with 82.4% total yield (three steps). Full article
(This article belongs to the Special Issue Amino Acid Chirality)
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Scheme 1

Scheme 1
<p>Literature methods for synthesis of 6,6,6-trifluoro-norleucine (<b>1</b>) via functional group elaborations (FGE) in <b>2</b> and alkyl halide alkylations (AHA) of glycine derivatives (<b>3</b>). Application of chiral Ni(II) complexes (<b>4</b>) for preparation of <b>1</b> via AHA.</p> Full article ">Scheme 2
<p>Synthesis of chiral Ni(II) complex of glycine Schiff base <b>4</b>.</p> Full article ">Scheme 3
<p>Alkylation of (<span class="html-italic">S</span>)-<b>3</b> with CF<sub>3</sub>(CH<sub>2</sub>)<sub>3</sub>I under homogeneous conditions.</p> Full article ">Scheme 4
<p>Disassembly of diastereomerically pure (<span class="html-italic">S</span>,2<span class="html-italic">S</span>)-<b>6</b>, recovery of chiral ligand (<span class="html-italic">S</span>)-<b>5</b>, and isolation of Fmoc-(<span class="html-italic">S</span>)-6,6,6-trifluoro-norleucine <b>9</b>.</p> Full article ">
10 pages, 4078 KiB  
Article
The Symmetry in the Noise-Perturbed Mandelbrot Set
by Tianwen Sun and Da Wang
Symmetry 2019, 11(4), 577; https://doi.org/10.3390/sym11040577 - 21 Apr 2019
Cited by 3 | Viewed by 5409
Abstract
This paper investigates the destruction of the symmetrical structure of the noise-perturbed Mandelbrot set (M-set). By applying the “symmetry criterion” method, we quantitatively compare the damages to the symmetry of the noise-perturbed Mandelbrot set resulting from additive and multiplicative noises. Because of the [...] Read more.
This paper investigates the destruction of the symmetrical structure of the noise-perturbed Mandelbrot set (M-set). By applying the “symmetry criterion” method, we quantitatively compare the damages to the symmetry of the noise-perturbed Mandelbrot set resulting from additive and multiplicative noises. Because of the uneven distribution between the core positions and the edge positions of the noise-perturbed Mandelbrot set, the comparison results reveal a paradox between the visual sense and quantified result. Thus, we propose a new “visual symmetry criterion” method that is more suitable for the measurement of visual asymmetry. Full article
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Figure 1
<p>The Mandelbrot set <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </semantics></math> of the map (<a href="#FD1-symmetry-11-00577" class="html-disp-formula">1</a>) without noise.</p> Full article ">Figure 2
<p>The noise-perturbed Mandelbrot set: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <msubsup> <mi>f</mi> <mi>u</mi> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </msubsup> <mo>)</mo> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <msubsup> <mi>f</mi> <mi>u</mi> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <msubsup> <mi>f</mi> <mi>u</mi> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </msubsup> <mo>)</mo> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <msubsup> <mi>f</mi> <mi>u</mi> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </msubsup> <mo>)</mo> </mrow> </semantics></math>; (<b>e</b>) <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <msubsup> <mi>f</mi> <mi>u</mi> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> </semantics></math>; (<b>f</b>) <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <msubsup> <mi>f</mi> <mi>u</mi> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </msubsup> <mo>)</mo> </mrow> </semantics></math>; (<b>g</b>) <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <msubsup> <mi>f</mi> <mi>n</mi> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </msubsup> <mo>)</mo> </mrow> </semantics></math>; (<b>h</b>) <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <msubsup> <mi>f</mi> <mi>n</mi> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> </semantics></math>; (<b>i</b>) <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <msubsup> <mi>f</mi> <mi>n</mi> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </msubsup> <mo>)</mo> </mrow> </semantics></math>; (<b>j</b>) <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <msubsup> <mi>f</mi> <mi>n</mi> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </msubsup> <mo>)</mo> </mrow> </semantics></math>; (<b>k</b>) <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <msubsup> <mi>f</mi> <mi>n</mi> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> </semantics></math>; (<b>l</b>) <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <msubsup> <mi>f</mi> <mi>n</mi> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </msubsup> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>The flowchart of <a href="#sec2-symmetry-11-00577" class="html-sec">Section 2</a> and <a href="#sec3-symmetry-11-00577" class="html-sec">Section 3</a>. <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>C</mi> </mrow> </semantics></math>, Symmetry Criterion; <math display="inline"><semantics> <mrow> <mi>V</mi> <mi>S</mi> <mi>C</mi> </mrow> </semantics></math>, Visual Symmetry Criterion.</p> Full article ">Figure 4
<p>(<b>a</b>) The <math display="inline"><semantics> <mrow> <mi>S</mi> <msub> <mi>C</mi> <msub> <mi>c</mi> <mn>1</mn> </msub> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> curve with <math display="inline"><semantics> <mrow> <msub> <mi>w</mi> <mi>n</mi> </msub> <mo>∼</mo> <mi mathvariant="script">U</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> as the strength of noise increases: additive noise is represented by cool colors, and multiplicative noise is represented by warm colors. (<b>b</b>) The <math display="inline"><semantics> <mrow> <mi>S</mi> <msub> <mi>C</mi> <msub> <mi>c</mi> <mn>1</mn> </msub> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> curve with <math display="inline"><semantics> <mrow> <msub> <mi>w</mi> <mi>n</mi> </msub> <mo>∼</mo> <mi mathvariant="script">N</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> as the strength of noise increases: additive noise is represented by cool colors, and multiplicative noise is represented by warm colors.</p> Full article ">Figure 5
<p><math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <msubsup> <mi>f</mi> <mi>u</mi> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </msubsup> <mo>)</mo> </mrow> </semantics></math> and three partially-enlarged details of it.</p> Full article ">Figure 6
<p>(<b>a</b>) The <math display="inline"><semantics> <mrow> <mi>S</mi> <msubsup> <mi>C</mi> <msub> <mi>c</mi> <mn>1</mn> </msub> <mrow> <mi>V</mi> <mi>i</mi> <mi>s</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> curve with <math display="inline"><semantics> <mrow> <msub> <mi>w</mi> <mi>n</mi> </msub> <mo>∼</mo> <mi mathvariant="script">U</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> as the strength of the noise increases: additive noise is represented by cool colors, and multiplicative noise is represented by warm colors. (<b>b</b>) The <math display="inline"><semantics> <mrow> <mi>S</mi> <msubsup> <mi>C</mi> <msub> <mi>c</mi> <mn>1</mn> </msub> <mrow> <mi>V</mi> <mi>i</mi> <mi>s</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> curve with <math display="inline"><semantics> <mrow> <msub> <mi>w</mi> <mi>n</mi> </msub> <mo>∼</mo> <mi mathvariant="script">N</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> as the strength of the noise increases: additive noise is represented by cool colors, and multiplicative noise is represented by warm colors.</p> Full article ">
17 pages, 5117 KiB  
Article
A Method to Determine Core Design Problems and a Corresponding Solution Strategy
by Yuanming Xie, Wenqiang Li, Yin Luo, Yan Li and Song Li
Symmetry 2019, 11(4), 576; https://doi.org/10.3390/sym11040576 - 19 Apr 2019
Cited by 3 | Viewed by 3231
Abstract
The lack of information on the correlation between root causes and corresponding control criteria in the importance calculation of root causes of design problems results in less accurate determinations of core problems. Based on the interaction between customer needs, bad product parameters, and [...] Read more.
The lack of information on the correlation between root causes and corresponding control criteria in the importance calculation of root causes of design problems results in less accurate determinations of core problems. Based on the interaction between customer needs, bad product parameters, and root causes, a hierarchical representation model of the design problem is established in this paper. A network layer of bad parameters, including various types of correlations, and a control layer, including technical feasibility and cost, are constructed. Then, a method based on the network analytic hierarchy process is proposed to rank the importance of root causes of the design problem and determine the core problems. Finally, a product design process based on the core problem solving is established to assist designers with improving design quality and efficiency. The design for the coolant flow distribution device in the lower chamber of a third-generation pressurized water reactor is employed as an example to demonstrate the effectiveness of the proposed method. Full article
(This article belongs to the Special Issue Symmetry in Mechanical Engineering)
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<p>Hierarchical representation model of design problem.</p> Full article ">Figure 2
<p>Product hierarchy model.</p> Full article ">Figure 3
<p>Product design process based on solving the core problem.</p> Full article ">Figure 4
<p>CRT of bad parameters of the coolant flow distribution device.</p> Full article ">Figure 5
<p>Hierarchical structure model of coolant flow distribution device.</p> Full article ">Figure 6
<p>CRD diagram of core problem <math display="inline"><semantics> <mrow> <mi>C</mi> <msub> <mi>p</mi> <mn>2</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Conceptual design scheme of coolant flow distribution device: (1) core support plate; (2) radial support key; (3) flow equalizing plate; (4) flow distribution cylinder; (5) supporting column; (6) energy-absorbing device; (7) pressure vessel bottom head; (I) inverted cone structure; (II) cap structure.</p> Full article ">Figure 8
<p>CRD diagram of core problem <math display="inline"><semantics> <mrow> <mi>C</mi> <msub> <mi>p</mi> <mn>1</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>CRD diagram of the core problem <math display="inline"><semantics> <mrow> <mi>C</mi> <msub> <mi>p</mi> <mn>3</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 10
<p>CRD diagram of the root cause <math display="inline"><semantics> <mrow> <mi>R</mi> <msubsup> <mi>c</mi> <mn>1</mn> <mn>1</mn> </msubsup> </mrow> </semantics></math>.</p> Full article ">Figure 11
<p>Velocity vectors of cross section of lower chamber. The velocity of the coolant around the upper surface of the core support plate is slightly greater than that in the middle. Although the velocity is not exactly the same on the upper surface, the velocity of the coolant in the middle is basically the same. What is more, the change of velocity consistency from the middle to the periphery is very small. This proves the effectiveness of the flow distribution effect of the design scheme. The direction of coolant flow in the lower chamber is stable, the variation degree of the flow direction is slow, and there is only a few of vortices. Structurally, the structure of the lower chamber is simplified by reducing the number of supporting columns and detachable connectors. Therefore, the design scheme has the characteristics of good flow distribution effect, a few of vortices, and simple structures.</p> Full article ">
26 pages, 2852 KiB  
Article
When Considering More Elements: Attribute Correlation in Unsupervised Data Cleaning under Blocking
by Pei Li, Chaofan Dai and Wenqian Wang
Symmetry 2019, 11(4), 575; https://doi.org/10.3390/sym11040575 - 19 Apr 2019
Cited by 3 | Viewed by 2657
Abstract
In banks, governments, and internet companies, due to the increasing demand for data in various information systems and continuously shortening of the cycle for data collection and update, there may be a variety of data quality issues in a database. As the expansion [...] Read more.
In banks, governments, and internet companies, due to the increasing demand for data in various information systems and continuously shortening of the cycle for data collection and update, there may be a variety of data quality issues in a database. As the expansion of data scales, methods such as pre-specifying business rules or introducing expert experience into a repair process are no longer applicable to some information systems requiring rapid responses. In this case, we divided data cleaning into supervised and unsupervised forms according to whether there were interventions in the repair processes and put forward a new dimension suitable for unsupervised cleaning in this paper. For weak logic errors in unsupervised data cleaning, we proposed an attribute correlation-based (ACB)-Framework under blocking, and designed three different data blocking methods to reduce the time complexity and test the impact of clustering accuracy on data cleaning. The experiments showed that the blocking methods could effectively reduce the repair time by maintaining the repair validity. Moreover, we concluded that the blocking methods with a too high clustering accuracy tended to put tuples with the same elements into a data block, which reduced the cleaning ability. In summary, the ACB-Framework with blocking can reduce the corresponding time cost and does not need the guidance of domain knowledge or interventions in repair, which can be applied in information systems requiring rapid responses, such as internet web pages, network servers, and sensor information acquisition. Full article
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<p>Flowchart of the attribute correlation-based (ACB)-Framework.</p> Full article ">Figure 2
<p>Illustration of the random blocking algorithm (RBA) method.</p> Full article ">Figure 3
<p>The experimental results of the control group WOB-Repair method on the MHTS dataset. (<b>a</b>–<b>c</b>) The validity, satisfaction, and runtime of the WOB-Repair, respectively, with fixed <span class="html-italic">amount</span> and EDS, and the <span class="html-italic">n</span> is 2, 7, 12, 17, …, 57 in the experiment. (<b>d</b>–<b>f</b>) The validity, satisfaction, and runtime of the WOB-Repair, respectively, with fixed <span class="html-italic">n</span>, and the <span class="html-italic">amount</span> is 10, 12, 14, …, 22 in the experiment.</p> Full article ">Figure 4
<p>The experimental results on validity (<b>a</b>), satisfaction (<b>b</b>), and runtime (<b>c</b>) of the RBA method on the MHTS dataset with fixed <span class="html-italic">amount</span> and <span class="html-italic">n</span>, and the <span class="html-italic">amount</span> and <span class="html-italic">n</span> are the same with the WOB-Repair method. The <span class="html-italic">k</span> is 2, 3, 4, …, 9 in the experiment.</p> Full article ">Figure 5
<p>The experimental results on validity (<b>a</b>), satisfaction (<b>b</b>), and runtime (<b>c</b>) of the SBA method on the MHTS dataset with a fixed <span class="html-italic">amount</span> and <span class="html-italic">n</span>, and the <span class="html-italic">amount</span> and <span class="html-italic">n</span> were the same as the RBA and WOB-Repair methods. In the experiment, we found that the <math display="inline"><semantics> <mrow> <msub> <mi>I</mi> <mrow> <mi>c</mi> <mi>f</mi> </mrow> </msub> </mrow> </semantics></math> would be divided into many blocks when the <math display="inline"><semantics> <mrow> <mi>t</mi> <mi>h</mi> <msub> <mi>d</mi> <mi>s</mi> </msub> </mrow> </semantics></math> was larger than 0.5, so we set the threshold <math display="inline"><semantics> <mrow> <mi>t</mi> <mi>h</mi> <msub> <mi>d</mi> <mi>s</mi> </msub> </mrow> </semantics></math> of the SBA as 0.3, 0.33, 0.36, …, 0.51.</p> Full article ">Figure 6
<p>The experimental results on validity (<b>a</b>), satisfaction (<b>b</b>), and blocking amount (<b>c</b>) of the RWBA method on the MHTS dataset with a fixed <span class="html-italic">amount</span> and <span class="html-italic">n</span>, and the <span class="html-italic">amount</span> and <span class="html-italic">n</span> were the same as the RBA, SBA, and WOB-Repair methods.</p> Full article ">Figure 7
<p>The experimental results on validity (<b>a</b>), satisfaction (<b>b</b>), and runtime (<b>c</b>) of the WOB-Repair, RBA, SBA, and RWBA methods on the MHTS dataset with the same <span class="html-italic">k</span>. We set the values of the <math display="inline"><semantics> <mrow> <mi>t</mi> <mi>h</mi> <msub> <mi>d</mi> <mi>s</mi> </msub> </mrow> </semantics></math> of the SBA and the <math display="inline"><semantics> <mrow> <mi>t</mi> <mi>h</mi> <msub> <mi>d</mi> <mi>r</mi> </msub> </mrow> </semantics></math> of the RWBA to obtain a specific <span class="html-italic">k</span> in the experiment. Because it costs too much time to establish a similarity graph for the RWBA, we did not compare the runtime index of the RWBA with the other three methods in (<b>c</b>).</p> Full article ">Figure 8
<p>The experimental results on validity (<b>a</b>), satisfaction (<b>b</b>), and runtime (<b>c</b>) of the ACB-Repair, RBM, and IBM methods on the TCC dataset with the same EDS. We set the values of <math display="inline"><semantics> <mrow> <mi>t</mi> <mi>h</mi> <msub> <mi>d</mi> <mi>s</mi> </msub> </mrow> </semantics></math> to 0.36 and the <span class="html-italic">amount</span> was 20, 22, 24, …, 40.</p> Full article ">
17 pages, 7822 KiB  
Article
Fractional-Order Fusion Model for Low-Light Image Enhancement
by Qiang Dai, Yi-Fei Pu, Ziaur Rahman and Muhammad Aamir
Symmetry 2019, 11(4), 574; https://doi.org/10.3390/sym11040574 - 19 Apr 2019
Cited by 70 | Viewed by 5426
Abstract
In this paper, a novel fractional-order fusion model (FFM) is presented for low-light image enhancement. Existing image enhancement methods don’t adequately extract contents from low-light areas, suppress noise, and preserve naturalness. To solve these problems, the main contributions of this paper are using [...] Read more.
In this paper, a novel fractional-order fusion model (FFM) is presented for low-light image enhancement. Existing image enhancement methods don’t adequately extract contents from low-light areas, suppress noise, and preserve naturalness. To solve these problems, the main contributions of this paper are using fractional-order mask and the fusion framework to enhance the low-light image. Firstly, the fractional mask is utilized to extract illumination from the input image. Secondly, image exposure adjusts to visible the dark regions. Finally, the fusion approach adopts the extracting of more hidden contents from dim areas. Depending on the experimental results, the fractional-order differential is much better for preserving the visual appearance as compared to traditional integer-order methods. The FFM works well for images having complex or normal low-light conditions. It also shows a trade-off among contrast improvement, detail enhancement, and preservation of the natural feel of the image. Experimental results reveal that the proposed model achieves promising results, and extracts more invisible contents in dark areas. The qualitative and quantitative comparison of several recent and advance state-of-the-art algorithms shows that the proposed model is robust and efficient. Full article
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<p>The flow chart of the fusion model.</p> Full article ">Figure 2
<p>Comparison of different illumination adjustment strategy. (<b>a</b>) the under-exposure image; (<b>b</b>) the Gamma transformation according to Equation (<a href="#FD8-symmetry-11-00574" class="html-disp-formula">8</a>); (<b>c</b>) the original camera response function (CRF) according to Equation (<a href="#FD9-symmetry-11-00574" class="html-disp-formula">9</a>); (<b>d</b>) the modified result of CRF; (<b>e</b>) the well-exposure image. (<b>f</b>), (<b>g</b>), (<b>h</b>), (<b>i</b>) and (<b>j</b>) are the three color channels (R, G, B) histograms of (<b>a</b>), (<b>b</b>), (<b>c</b>), (<b>d</b>) and (<b>e</b>), respectively.</p> Full article ">Figure 3
<p>Multi-scale fusion process weights setting. (<b>a</b>) the original image; (<b>b</b>) the first enhanced image; (<b>c</b>) the second enhanced image; (<b>d</b>) the result of fusion; (<b>e</b>) the weight function.</p> Full article ">Figure 4
<p>One-dimensional signal processing. (<b>a</b>) the original signal; (<b>b</b>) the signal with Gaussian noise added; (<b>c</b>) the result from Equation (<a href="#FD25-symmetry-11-00574" class="html-disp-formula">25</a>), <span class="html-italic">v</span> is 2.4 and <math display="inline"><semantics> <msub> <mi>λ</mi> <mn>3</mn> </msub> </semantics></math> is <math display="inline"><semantics> <mrow> <mn>1.125</mn> </mrow> </semantics></math>; (<b>d</b>–<b>f</b>) the results are from Equation (<a href="#FD23-symmetry-11-00574" class="html-disp-formula">23</a>), <math display="inline"><semantics> <msub> <mi>λ</mi> <mn>1</mn> </msub> </semantics></math> is 0.6, 0.8, 1 respectively; (<b>g</b>–<b>i</b>) the outputs are from Equation (<a href="#FD24-symmetry-11-00574" class="html-disp-formula">24</a>), <math display="inline"><semantics> <msub> <mi>λ</mi> <mn>2</mn> </msub> </semantics></math> is 1.6, 2.4, 3.2, respectively.</p> Full article ">Figure 5
<p>The impact of fractional-order. (<b>a</b>) the original image; (<b>b</b>) the reflectance from first-order differential is more prominent than fractional-order differential; (<b>c</b>) the reflectance measured with fractional-order differential is more prominent than first-order differential.</p> Full article ">Figure 6
<p>Comparison of visual contrast in the low-light images which the contents are very dark. (<b>a</b>,<b>i</b>) are original images; (<b>b</b>,<b>j</b>) MF [<a href="#B13-symmetry-11-00574" class="html-bibr">13</a>]; (<b>c</b>,<b>k</b>) LightenNet [<a href="#B17-symmetry-11-00574" class="html-bibr">17</a>]; (<b>d</b>,<b>l</b>) CRM [<a href="#B11-symmetry-11-00574" class="html-bibr">11</a>]; (<b>e</b>,<b>m</b>) NPE [<a href="#B12-symmetry-11-00574" class="html-bibr">12</a>]; (<b>f</b>,<b>n</b>) JIEP [<a href="#B9-symmetry-11-00574" class="html-bibr">9</a>]; (<b>g</b>,<b>o</b>) are enhanced images of FFM (<b>1</b>); (<b>h</b>,<b>p</b>) are final results of the proposed model.</p> Full article ">Figure 7
<p>Comparison of enhancement schemes for images which are slightly degraded from low-light conditions. (<b>a</b>,<b>i</b>,<b>q</b>) are original images; (<b>b</b>,<b>j</b>,<b>r</b>) MF [<a href="#B13-symmetry-11-00574" class="html-bibr">13</a>]; (<b>c</b>,<b>k</b>,<b>s</b>) LightenNet [<a href="#B17-symmetry-11-00574" class="html-bibr">17</a>]; (<b>d</b>,<b>l</b>,<b>t</b>) CRM [<a href="#B11-symmetry-11-00574" class="html-bibr">11</a>]; (<b>e</b>,<b>m</b>,<b>u</b>) NPE [<a href="#B12-symmetry-11-00574" class="html-bibr">12</a>]; (<b>f</b>,<b>n</b>,<b>v</b>) JIEP [<a href="#B9-symmetry-11-00574" class="html-bibr">9</a>]; (<b>g</b>,<b>o</b>,<b>w</b>) Our enhanced images of FFM(1); (<b>h</b>,<b>p</b>,<b>x</b>) are final results.</p> Full article ">Figure 7 Cont.
<p>Comparison of enhancement schemes for images which are slightly degraded from low-light conditions. (<b>a</b>,<b>i</b>,<b>q</b>) are original images; (<b>b</b>,<b>j</b>,<b>r</b>) MF [<a href="#B13-symmetry-11-00574" class="html-bibr">13</a>]; (<b>c</b>,<b>k</b>,<b>s</b>) LightenNet [<a href="#B17-symmetry-11-00574" class="html-bibr">17</a>]; (<b>d</b>,<b>l</b>,<b>t</b>) CRM [<a href="#B11-symmetry-11-00574" class="html-bibr">11</a>]; (<b>e</b>,<b>m</b>,<b>u</b>) NPE [<a href="#B12-symmetry-11-00574" class="html-bibr">12</a>]; (<b>f</b>,<b>n</b>,<b>v</b>) JIEP [<a href="#B9-symmetry-11-00574" class="html-bibr">9</a>]; (<b>g</b>,<b>o</b>,<b>w</b>) Our enhanced images of FFM(1); (<b>h</b>,<b>p</b>,<b>x</b>) are final results.</p> Full article ">Figure 8
<p>An example of a failure case. (<b>a</b>) the original image; (<b>b</b>) the effect of FFM(1); (<b>c</b>) the result of FFM(2).</p> Full article ">
12 pages, 253 KiB  
Article
Thermoelasticity of Initially Stressed Bodies with Voids: A Domain of Influence
by Marin Marin, Mohamed I. A. Othman, Sorin Vlase and Lavinia Codarcea-Munteanu
Symmetry 2019, 11(4), 573; https://doi.org/10.3390/sym11040573 - 19 Apr 2019
Cited by 13 | Viewed by 2439
Abstract
In our study, we will extend the domain of influence in order to cover the thermoelasticity of initially stressed bodies with voids. In what follows, we prove that, for a finite time t > 0 , the displacement field u i , the [...] Read more.
In our study, we will extend the domain of influence in order to cover the thermoelasticity of initially stressed bodies with voids. In what follows, we prove that, for a finite time t > 0 , the displacement field u i , the dipolar displacement field φ j k , the temperature θ and the change in volume fraction ϕ generate no disturbance outside a bounded domain B. Full article
(This article belongs to the Special Issue Symmetry in Applied Continuous Mechanics)
14 pages, 1577 KiB  
Article
A Dynamic Simulation of the Immune System Response to Inhibit and Eliminate Abnormal Cells
by S. A. Alharbi and A. S. Rambely
Symmetry 2019, 11(4), 572; https://doi.org/10.3390/sym11040572 - 19 Apr 2019
Cited by 15 | Viewed by 4277
Abstract
Diet has long been considered a risk factor related to an increased risk of cancer. This challenges us to understand the relationship between the immune system and diet when abnormal cells appear in a tissue. In this paper, we propose and analyze a [...] Read more.
Diet has long been considered a risk factor related to an increased risk of cancer. This challenges us to understand the relationship between the immune system and diet when abnormal cells appear in a tissue. In this paper, we propose and analyze a model from the point of view of a person who follows a healthy diet, i.e., one correlated to the food pyramid, and a person who follows an unhealthy diet. Normal cells and immune cells are used in the design of the model, which aims to describe how the immune system functions when abnormal cells appear in a tissue. The results show that the immune system is able to inhibit and eliminate abnormal cells through the three following stages: the response stage, the interaction stage, and the recovery stage. Specifically, the failure of the immune system to accomplish the interaction stage occurs when a person follows an unhealthy diet. According to the analysis and simulation of our model, we can deduce that dietary pattern has a significant impact on the functioning of the immune system. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Analysis and Applications)
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Figure 1

Figure 1
<p>The dietary management food pyramid according to the World Cancer Research Fund (WCRF) and American Institute for Cancer Research (AICR) where the amounts of food are estimated based on nutritional and practical considerations.</p> Full article ">Figure 2
<p>The phase portrait of the immune–healthy diet model (IHDM) and its solutions around the response and interaction equilibrium points.</p> Full article ">Figure 3
<p>The phase portrait of the IHDM and its solutions around the recovery equilibrium points.</p> Full article ">Figure 4
<p>The phase portrait of the immune-unhealthy diet model (IUNHDM) and its solutions around the response equilibrium point.</p> Full article ">Figure 5
<p>The phase portrait of the IUNHDM and its solutions around the coexistence equilibrium point.</p> Full article ">Figure 6
<p>The residual error at steps for the proposed numerical method for the IHDM.</p> Full article ">Figure 7
<p>The residual error at time t for the proposed numerical method for the IHDM.</p> Full article ">Figure 8
<p>The residual error at steps for the proposed numerical method for the IUNHDM.</p> Full article ">Figure 9
<p>The residual error at time t for the proposed numerical method for the IUNHDM.</p> Full article ">Figure 10
<p>The behavior of the IHDM where <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>0.431201</mn> <mo>,</mo> <mi>β</mi> <mo>=</mo> <mn>2.99</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mi>δ</mi> </semantics></math><math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mn>0.57</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.4787</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>0.2206</mn> <mo>,</mo> <mi>η</mi> <mo>=</mo> <mn>0.8791</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>0.6986</mn> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 11
<p>The behavior of the IUNHDM where <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>0.431201</mn> <mo>,</mo> <mi>β</mi> <mo>=</mo> <mn>2.99</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mo>,</mo> <mspace width="3.33333pt"/> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mi>δ</mi> </semantics></math><math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mn>0.57</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.3389</mn> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>0.2710</mn> <mo>,</mo> <mi>η</mi> <mo>=</mo> <mn>0.1379</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>0.8130</mn> <mo>.</mo> </mrow> </semantics></math></p> Full article ">
20 pages, 10751 KiB  
Article
Anomaly Detection Based on Mining Six Local Data Features and BP Neural Network
by Yu Zhang, Yuanpeng Zhu, Xuqiao Li, Xiaole Wang and Xutong Guo
Symmetry 2019, 11(4), 571; https://doi.org/10.3390/sym11040571 - 19 Apr 2019
Cited by 9 | Viewed by 4343
Abstract
Key performance indicators (KPIs) are time series with the format of (timestamp, value). The accuracy of KPIs anomaly detection is far beyond our initial expectations sometimes. The reasons include the unbalanced distribution between the normal data and the anomalies as well as the [...] Read more.
Key performance indicators (KPIs) are time series with the format of (timestamp, value). The accuracy of KPIs anomaly detection is far beyond our initial expectations sometimes. The reasons include the unbalanced distribution between the normal data and the anomalies as well as the existence of many different types of the KPIs data curves. In this paper, we propose a new anomaly detection model based on mining six local data features as the input of back-propagation (BP) neural network. By means of vectorization description on a normalized dataset innovatively, the local geometric characteristics of one time series curve could be well described in a precise mathematical way. Differing from some traditional statistics data characteristics describing the entire variation situation of one sequence, the six mined local data features give a subtle insight of local dynamics by describing the local monotonicity, the local convexity/concavity, the local inflection property and peaks distribution of one KPI time series. In order to demonstrate the validity of the proposed model, we applied our method on 14 classical KPIs time series datasets. Numerical results show that the new given scheme achieves an average F1-score over 90%. Comparison results show that the proposed model detects the anomaly more precisely. Full article
(This article belongs to the Special Issue Symmetry in Engineering Sciences)
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Figure 1

Figure 1
<p>Fourteen classical key performance indicators (KPIs). (<b>a</b>): Periodic time series; (<b>b</b>): Periodic and continuous fluctuation time series; (<b>c</b>): Unstable time series; (<b>d</b>): Unstable time series; (<b>e</b>): Stable time series; (<b>f</b>): Unstable time series; (<b>g</b>): Unstable time series; (<b>h</b>): Stable time series; (<b>i</b>): Unstable time series; (<b>j</b>): Continuous fluctuation time series; (<b>k</b>): Unstable time series; (<b>l</b>): Periodic and continuous fluctuation time series; (<b>m</b>): Stable time series; (<b>n</b>): Continuous fluctuation time series.</p> Full article ">Figure 1 Cont.
<p>Fourteen classical key performance indicators (KPIs). (<b>a</b>): Periodic time series; (<b>b</b>): Periodic and continuous fluctuation time series; (<b>c</b>): Unstable time series; (<b>d</b>): Unstable time series; (<b>e</b>): Stable time series; (<b>f</b>): Unstable time series; (<b>g</b>): Unstable time series; (<b>h</b>): Stable time series; (<b>i</b>): Unstable time series; (<b>j</b>): Continuous fluctuation time series; (<b>k</b>): Unstable time series; (<b>l</b>): Periodic and continuous fluctuation time series; (<b>m</b>): Stable time series; (<b>n</b>): Continuous fluctuation time series.</p> Full article ">Figure 2
<p>(<b>a</b>): The flowchart of the proposed approach for KPIs time series; (<b>b</b>): the semantic drawing of six local data feature space.</p> Full article ">Figure 3
<p>Schematic illustration of the feature <math display="inline"><semantics> <mrow> <msubsup> <mi>F</mi> <mi>i</mi> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Schematic illustration of the feature <math display="inline"><semantics> <mrow> <msubsup> <mi>F</mi> <mi>i</mi> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>F</mi> <mi>i</mi> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Six features mined of the KPIs. (<b>a</b>) Six features mined of the KPI1; (<b>b</b>) Six features mined of the KPI2; (<b>c</b>) Six features mined of the KPI3; (<b>d</b>) Six features mined of the KPI4; (<b>e</b>) Six features mined of the KPI5; (<b>f</b>) Six features mined of the KPI6; (<b>g</b>) Six features mined of the KPI7; (<b>h</b>) Six features mined of the KPI8; (<b>i</b>) Six features mined of the KPI9; (<b>j</b>) Six features mined of the KPI10; (<b>k</b>) Six features mined of the KPI11; (<b>l</b>) Six features mined of the KPI112; (<b>m</b>) Six features mined of the KPI13; (<b>n</b>) Six features mined of the KPI14.</p> Full article ">Figure 5 Cont.
<p>Six features mined of the KPIs. (<b>a</b>) Six features mined of the KPI1; (<b>b</b>) Six features mined of the KPI2; (<b>c</b>) Six features mined of the KPI3; (<b>d</b>) Six features mined of the KPI4; (<b>e</b>) Six features mined of the KPI5; (<b>f</b>) Six features mined of the KPI6; (<b>g</b>) Six features mined of the KPI7; (<b>h</b>) Six features mined of the KPI8; (<b>i</b>) Six features mined of the KPI9; (<b>j</b>) Six features mined of the KPI10; (<b>k</b>) Six features mined of the KPI11; (<b>l</b>) Six features mined of the KPI112; (<b>m</b>) Six features mined of the KPI13; (<b>n</b>) Six features mined of the KPI14.</p> Full article ">Figure 5 Cont.
<p>Six features mined of the KPIs. (<b>a</b>) Six features mined of the KPI1; (<b>b</b>) Six features mined of the KPI2; (<b>c</b>) Six features mined of the KPI3; (<b>d</b>) Six features mined of the KPI4; (<b>e</b>) Six features mined of the KPI5; (<b>f</b>) Six features mined of the KPI6; (<b>g</b>) Six features mined of the KPI7; (<b>h</b>) Six features mined of the KPI8; (<b>i</b>) Six features mined of the KPI9; (<b>j</b>) Six features mined of the KPI10; (<b>k</b>) Six features mined of the KPI11; (<b>l</b>) Six features mined of the KPI112; (<b>m</b>) Six features mined of the KPI13; (<b>n</b>) Six features mined of the KPI14.</p> Full article ">Figure 5 Cont.
<p>Six features mined of the KPIs. (<b>a</b>) Six features mined of the KPI1; (<b>b</b>) Six features mined of the KPI2; (<b>c</b>) Six features mined of the KPI3; (<b>d</b>) Six features mined of the KPI4; (<b>e</b>) Six features mined of the KPI5; (<b>f</b>) Six features mined of the KPI6; (<b>g</b>) Six features mined of the KPI7; (<b>h</b>) Six features mined of the KPI8; (<b>i</b>) Six features mined of the KPI9; (<b>j</b>) Six features mined of the KPI10; (<b>k</b>) Six features mined of the KPI11; (<b>l</b>) Six features mined of the KPI112; (<b>m</b>) Six features mined of the KPI13; (<b>n</b>) Six features mined of the KPI14.</p> Full article ">Figure 6
<p>F<sub>1</sub>-scores of different topology structures of BP network for each of 14 KPIs.</p> Full article ">Figure 7
<p>Anomaly detection results using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>. (<b>a</b>): Anomaly detection results of KPI1 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>b</b>): Anomaly detection results of KPI2 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>c</b>): Anomaly detection results of KPI3 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>d</b>): Anomaly detection results of KPI4 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>e</b>): Anomaly detection results of KPI5 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>f</b>): Anomaly detection results of KPI6 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>g</b>): Anomaly detection results of KPI7 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>h</b>): Anomaly detection results of KPI8 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>i</b>): Anomaly detection results of KPI9 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>j</b>): Anomaly detection results of KPI10 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>k</b>): Anomaly detection results of KPI11 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>l</b>): Anomaly detection results of KPI12 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>m</b>): Anomaly detection results of KPI13 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>n</b>): Anomaly detection results of KPI14 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 7 Cont.
<p>Anomaly detection results using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>. (<b>a</b>): Anomaly detection results of KPI1 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>b</b>): Anomaly detection results of KPI2 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>c</b>): Anomaly detection results of KPI3 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>d</b>): Anomaly detection results of KPI4 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>e</b>): Anomaly detection results of KPI5 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>f</b>): Anomaly detection results of KPI6 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>g</b>): Anomaly detection results of KPI7 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>h</b>): Anomaly detection results of KPI8 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>i</b>): Anomaly detection results of KPI9 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>j</b>): Anomaly detection results of KPI10 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>k</b>): Anomaly detection results of KPI11 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>l</b>): Anomaly detection results of KPI12 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>m</b>): Anomaly detection results of KPI13 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>n</b>): Anomaly detection results of KPI14 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 7 Cont.
<p>Anomaly detection results using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>. (<b>a</b>): Anomaly detection results of KPI1 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>b</b>): Anomaly detection results of KPI2 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>c</b>): Anomaly detection results of KPI3 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>d</b>): Anomaly detection results of KPI4 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>e</b>): Anomaly detection results of KPI5 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>f</b>): Anomaly detection results of KPI6 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>g</b>): Anomaly detection results of KPI7 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>h</b>): Anomaly detection results of KPI8 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>i</b>): Anomaly detection results of KPI9 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>j</b>): Anomaly detection results of KPI10 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>k</b>): Anomaly detection results of KPI11 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>l</b>): Anomaly detection results of KPI12 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>m</b>): Anomaly detection results of KPI13 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>n</b>): Anomaly detection results of KPI14 using the structure of <math display="inline"><semantics> <mrow> <mn>6</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>10</mn> <mo>→</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">
21 pages, 2641 KiB  
Article
Stereo Matching Methods for Imperfectly Rectified Stereo Images
by Phuc Hong Nguyen and Chang Wook Ahn
Symmetry 2019, 11(4), 570; https://doi.org/10.3390/sym11040570 - 19 Apr 2019
Cited by 14 | Viewed by 5438
Abstract
Stereo matching has been under development for decades and is an important process for many applications. Difficulties in stereo matching include textureless regions, occlusion, illumination variation, the fattening effect, and discontinuity. These challenges are effectively solved in recently developed stereo matching algorithms. A [...] Read more.
Stereo matching has been under development for decades and is an important process for many applications. Difficulties in stereo matching include textureless regions, occlusion, illumination variation, the fattening effect, and discontinuity. These challenges are effectively solved in recently developed stereo matching algorithms. A new imperfect rectification problem has recently been encountered in stereo matching, and the problem results from the high resolution of stereo images. State-of-the-art stereo matching algorithms fail to exactly reconstruct the depth information using stereo images with imperfect rectification, as the imperfectly rectified image problems are not explicitly taken into account. In this paper, we solve the imperfect rectification problems, and propose matching stereo matching methods that based on absolute differences, square differences, normalized cross correlation, zero-mean normalized cross correlation, and rank and census transforms. Finally, we conduct experiments to evaluate these stereo matching methods using the Middlebury datasets. The experimental results show the proposed stereo matching methods can reduce error rate significantly for stereo images with imperfect rectification. Full article
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Figure 1

Figure 1
<p>Results of the ImpCensus-based stereo matching algorithms with different <span class="html-italic">R</span> values using the Backpack stereo images with imperfect rectification. (<b>a</b>) Left image. (<b>b</b>) Right image. (<b>c</b>) Ground truth. (<b>d</b>) Disparity map of Census/Win (<math display="inline"><semantics> <mrow> <mi>E</mi> <mi>r</mi> <mi>r</mi> </mrow> </semantics></math> = <math display="inline"><semantics> <mrow> <mn>21.38</mn> <mo>%</mo> </mrow> </semantics></math>). (<b>e</b>) Disparity map of ImpCensus/Win/R1 (<math display="inline"><semantics> <mrow> <mi>E</mi> <mi>r</mi> <mi>r</mi> </mrow> </semantics></math> = <math display="inline"><semantics> <mrow> <mn mathvariant="bold">16.77</mn> <mo>%</mo> </mrow> </semantics></math>). (<b>f</b>) Disparity map of ImpCensus/Win/R2 (<math display="inline"><semantics> <mrow> <mi>E</mi> <mi>r</mi> <mi>r</mi> </mrow> </semantics></math> = <math display="inline"><semantics> <mrow> <mn>17.38</mn> <mo>%</mo> </mrow> </semantics></math>). (<b>g</b>) Disparity map of Census/GC (<math display="inline"><semantics> <mrow> <mi>E</mi> <mi>r</mi> <mi>r</mi> </mrow> </semantics></math> = <math display="inline"><semantics> <mrow> <mn>22.59</mn> <mo>%</mo> </mrow> </semantics></math>). (<b>h</b>) Disparity map of ImpCensus/GC/R1 (<math display="inline"><semantics> <mrow> <mi>E</mi> <mi>r</mi> <mi>r</mi> </mrow> </semantics></math> = <math display="inline"><semantics> <mrow> <mn>14.65</mn> <mo>%</mo> </mrow> </semantics></math>). (<b>i</b>) Disparity map of ImpCensus/GC/R2 (<math display="inline"><semantics> <mrow> <mi>E</mi> <mi>r</mi> <mi>r</mi> </mrow> </semantics></math> = <math display="inline"><semantics> <mrow> <mn mathvariant="bold">14.43</mn> <mo>%</mo> </mrow> </semantics></math>).</p> Full article ">Figure 2
<p>Results of the ImpZNCC-based stereo matching algorithms with different <span class="html-italic">R</span> values using the Motorcycle stereo images with imperfect rectification. (<b>a</b>) Left image. (<b>b</b>) Right image. (<b>c</b>) Ground truth. (<b>d</b>) Disparity map of ZNCC (<math display="inline"><semantics> <mrow> <mi>E</mi> <mi>r</mi> <mi>r</mi> </mrow> </semantics></math> = 35.97%). (<b>e</b>) Disparity map of ImpZNCC/R1 (<math display="inline"><semantics> <mrow> <mi>E</mi> <mi>r</mi> <mi>r</mi> </mrow> </semantics></math> = <math display="inline"><semantics> <mrow> <mn mathvariant="bold">25.78</mn> <mo>%</mo> </mrow> </semantics></math>). (<b>f</b>) Disparity map of ImpZNCC/R2 (<math display="inline"><semantics> <mrow> <mi>E</mi> <mi>r</mi> <mi>r</mi> </mrow> </semantics></math> = 26.30%).</p> Full article ">Figure 3
<p>Results of the Census-based stereo matching algorithms using the Sword1 stereo images of imperfect rectification and radiometric distortion. (<b>a</b>) Left image. (<b>b</b>) Right image with varying exposure. (<b>c</b>) Right image with varying illumination. (d–f) Disparity maps using the stereo pair (a,b). (<b>d</b>) Disparity map of Census/GC (<math display="inline"><semantics> <mrow> <mi>E</mi> <mi>r</mi> <mi>r</mi> </mrow> </semantics></math> = 18.87%). (<b>e</b>) Disparity map of ImpCensus/GC/R1 (<math display="inline"><semantics> <mrow> <mi>E</mi> <mi>r</mi> <mi>r</mi> </mrow> </semantics></math> = 14.55%). (<b>f</b>) Disparity map of ImpCensus/GC/R2 (<math display="inline"><semantics> <mrow> <mi>E</mi> <mi>r</mi> <mi>r</mi> </mrow> </semantics></math> = <math display="inline"><semantics> <mrow> <mn mathvariant="bold">14.39</mn> <mo>%</mo> </mrow> </semantics></math>). (<b>g</b>–<b>i</b>) Disparity maps using the stereo pair (a,c). (<b>g</b>) Disparity map of Census/GC (<math display="inline"><semantics> <mrow> <mi>E</mi> <mi>r</mi> <mi>r</mi> </mrow> </semantics></math> = 36.05%). (<b>h</b>) Disparity map of ImpCensus/GC/R1 (<math display="inline"><semantics> <mrow> <mi>E</mi> <mi>r</mi> <mi>r</mi> </mrow> </semantics></math> = 31.30%). (<b>i</b>) Disparity map of ImpCensus/GC/R2 (<math display="inline"><semantics> <mrow> <mi>E</mi> <mi>r</mi> <mi>r</mi> </mrow> </semantics></math> = <math display="inline"><semantics> <mrow> <mn mathvariant="bold">31.18</mn> <mo>%</mo> </mrow> </semantics></math>).</p> Full article ">Figure 4
<p>Results of the ImpCensus-based stereo matching algorithms with different <span class="html-italic">R</span> values using the perfectly rectified images. The first column is left images, and the second column is disparity maps for Census/Win. The next two columns are disparity maps for ImpCensus/Win/R1 and ImpCensus/Win/R2, respectively. The last column is ground truths.</p> Full article ">
10 pages, 378 KiB  
Article
Eigenvalue Based Approach for Assessment of Global Robustness of Nonlinear Dynamical Systems
by Robert Vrabel
Symmetry 2019, 11(4), 569; https://doi.org/10.3390/sym11040569 - 19 Apr 2019
Cited by 1 | Viewed by 2528
Abstract
In this paper we have established the sufficient conditions for asymptotic convergence of all solutions of nonlinear dynamical system (with potentially unknown and unbounded external disturbances) to zero with time t → ∞ . We showed here that the symmetric part of linear [...] Read more.
In this paper we have established the sufficient conditions for asymptotic convergence of all solutions of nonlinear dynamical system (with potentially unknown and unbounded external disturbances) to zero with time t → ∞ . We showed here that the symmetric part of linear part of nonlinear nominal system, or, to be more precise, its time-dependent eigenvalues, play important role in assessment of the robustness of systems. Full article
(This article belongs to the Special Issue Nonlinear Circuits and Systems in Symmetry)
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Figure 1

Figure 1
<p>Solution <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mfenced separators="" open="(" close=")"> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mi>T</mi> </msup> </mrow> </semantics></math> of (<a href="#FD7-symmetry-11-00569" class="html-disp-formula">7</a>) for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mspace width="0.166667em"/> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mfenced separators="" open="(" close=")"> <mfrac> <mrow> <mo form="prefix">arctan</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> <mo>,</mo> <mspace width="0.166667em"/> <mfrac> <mrow> <mrow> <mi>exp</mi> </mrow> <mo stretchy="false">[</mo> <mo>−</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mfenced> <mi>T</mi> </msup> </mrow> </semantics></math> and initial state <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mn>50</mn> <mo>,</mo> <mspace width="4pt"/> <mo>−</mo> <mn>20</mn> <mo stretchy="false">)</mo> </mrow> <mi>T</mi> </msup> <mo>.</mo> </mrow> </semantics></math> Obviously, <math display="inline"><semantics> <mrow> <msub> <mfenced separators="" open="&#x2225;" close="&#x2225;"> <mover accent="true"> <mi mathvariant="sans-serif">Δ</mi> <mo stretchy="false">˜</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mn>2</mn> </msub> <mo>=</mo> <msqrt> <mrow> <msup> <mfenced open="(" close=")"> <mfrac> <mrow> <mi>π</mi> <mo>/</mo> <mn>2</mn> </mrow> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mfenced> <mn>2</mn> </msup> <mo>+</mo> <mrow> <mi>exp</mi> </mrow> <mrow> <mo stretchy="false">[</mo> <mo>−</mo> <mn>2</mn> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </msqrt> <mo>=</mo> <mi>o</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> as <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>→</mo> <mo>∞</mo> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 2
<p>Solution <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mfenced separators="" open="(" close=")"> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mi>T</mi> </msup> </mrow> </semantics></math> of (<a href="#FD8-symmetry-11-00569" class="html-disp-formula">8</a>) for <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mspace width="0.166667em"/> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mfenced separators="" open="(" close=")"> <msup> <mi>t</mi> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="0.166667em"/> <mn>3</mn> <mo form="prefix">cos</mo> <mfenced separators="" open="(" close=")"> <mi>t</mi> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mfenced> </mfenced> <mi>T</mi> </msup> </mrow> </semantics></math> and initial state <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mn>10</mn> <mo>,</mo> <mspace width="4pt"/> <mo>−</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> <mi>T</mi> </msup> <mo>.</mo> </mrow> </semantics></math> Obviously, <math display="inline"><semantics> <mrow> <msub> <mfenced separators="" open="&#x2225;" close="&#x2225;"> <mover accent="true"> <mi mathvariant="sans-serif">Δ</mi> <mo stretchy="false">˜</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mn>2</mn> </msub> <mo>=</mo> <msqrt> <mrow> <msup> <mi>t</mi> <mn>3</mn> </msup> <mo>+</mo> <mn>9</mn> </mrow> </msqrt> <mo>=</mo> <mi>o</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> as <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>→</mo> <mo>∞</mo> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 3
<p>The solution component <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> of (<a href="#FD8-symmetry-11-00569" class="html-disp-formula">8</a>) for <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> </mrow> </semantics></math> initial state <math display="inline"><semantics> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and with (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mfenced separators="" open="(" close=")"> <mn>50</mn> <msup> <mi>t</mi> <mrow> <mn>1.95</mn> </mrow> </msup> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>δ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mi>T</mi> </msup> </mrow> </semantics></math> satisfying Assumption 3 of Theorem 1, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mfenced separators="" open="(" close=")"> <mn>50</mn> <msup> <mi>t</mi> <mrow> <mn>2.05</mn> </mrow> </msup> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>δ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mi>T</mi> </msup> </mrow> </semantics></math> that does not satisfy Assumption 3 of Theorem 1 and (<b>c</b>) the borderline case, <math display="inline"><semantics> <mrow> <mi>δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mfenced separators="" open="(" close=")"> <mn>50</mn> <msup> <mi>t</mi> <mrow> <mn>2.00</mn> </mrow> </msup> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>δ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mi>T</mi> </msup> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>δ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>t</mi> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>/</mo> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">
12 pages, 273 KiB  
Article
Tracking Control of a Class of Chaotic Systems
by Anqing Yang, Linshan Li, Zuoxun Wang and Rongwei Guo
Symmetry 2019, 11(4), 568; https://doi.org/10.3390/sym11040568 - 19 Apr 2019
Cited by 15 | Viewed by 2202
Abstract
This paper investigates the asymptotic tracking control problem of the chaotic system. Firstly, a reference system is presented, the output of which can asymptotically track a given command. Then, a both physically implementable and simple controller is designed, by which the given chaotic [...] Read more.
This paper investigates the asymptotic tracking control problem of the chaotic system. Firstly, a reference system is presented, the output of which can asymptotically track a given command. Then, a both physically implementable and simple controller is designed, by which the given chaotic system synchronizes the reference system, and thus the output of such chaotic systems can asymptotically track the given command. It should be pointed out that the output of the given chaotic system can asymptotically track arbitrary desired periodic orbits. Finally, several illustrative examples are taken as example to show the validity and effectiveness of the obtained results. Full article
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Figure 1

Figure 1
<p>Shows that the state <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> of the system (<a href="#FD15-symmetry-11-00568" class="html-disp-formula">15</a>) converges to <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> as <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Shows the state <math display="inline"><semantics> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> </semantics></math> of the system (<a href="#FD15-symmetry-11-00568" class="html-disp-formula">15</a>) converges to a constant as <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Shows the state <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> of the system (<a href="#FD22-symmetry-11-00568" class="html-disp-formula">22</a>) converges to <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>=</mo> <mo form="prefix">sin</mo> <mo>(</mo> <mn>3</mn> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> as <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Shows the state <math display="inline"><semantics> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> </semantics></math> of the system (<a href="#FD22-symmetry-11-00568" class="html-disp-formula">22</a>) converges to a constant as <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Shows the states <math display="inline"><semantics> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> </semantics></math> of the system (<a href="#FD22-symmetry-11-00568" class="html-disp-formula">22</a>) converges to <math display="inline"><semantics> <mrow> <mi>c</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mn>3</mn> <mo form="prefix">cos</mo> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <mn>3</mn> <mo form="prefix">sin</mo> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mrow> </semantics></math> as <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Shows phase portrait of the states <math display="inline"><semantics> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> </semantics></math> of the system (<a href="#FD29-symmetry-11-00568" class="html-disp-formula">29</a>) converges to a circle with radius 3 as <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Shows phase portrait of the states <math display="inline"><semantics> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>4</mn> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> </semantics></math> of the system (<a href="#FD29-symmetry-11-00568" class="html-disp-formula">29</a>).</p> Full article ">
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