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Symmetry
Volume 10
Issue 6
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Symmetry, Volume 10, Issue 6 (June 2018) – 50 articles

Cover Story (view full-size image): We reared Port Jackson shark eggs at present day temperatures and at plus 3 degrees to mimic future climate change scenarios. We found that fewer shark embryos survived at higher temperatures, but those that did survive had far stronger laterality and showed strong turn biases to the right. We suspect this increased cerebral lateralisation may be a compensatory mechanism to enhance cognitive capacity given a probable reduction in brain size due to developmental stress. View this paper.
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18 pages, 4317 KiB  
Article
Towards Real-Time Facial Landmark Detection in Depth Data Using Auxiliary Information
by Connah Kendrick, Kevin Tan, Kevin Walker and Moi Hoon Yap
Symmetry 2018, 10(6), 230; https://doi.org/10.3390/sym10060230 - 17 Jun 2018
Cited by 7 | Viewed by 4379
Abstract
Modern facial motion capture systems employ a two-pronged approach for capturing and rendering facial motion. Visual data (2D) is used for tracking the facial features and predicting facial expression, whereas Depth (3D) data is used to build a series of expressions on 3D [...] Read more.
Modern facial motion capture systems employ a two-pronged approach for capturing and rendering facial motion. Visual data (2D) is used for tracking the facial features and predicting facial expression, whereas Depth (3D) data is used to build a series of expressions on 3D face models. An issue with modern research approaches is the use of a single data stream that provides little indication of the 3D facial structure. We compare and analyse the performance of Convolutional Neural Networks (CNN) using visual, Depth and merged data to identify facial features in real-time using a Depth sensor. First, we review the facial landmarking algorithms and its datasets for Depth data. We address the limitation of the current datasets by introducing the Kinect One Expression Dataset (KOED). Then, we propose the use of CNNs for the single data stream and merged data streams for facial landmark detection. We contribute to existing work by performing a full evaluation on which streams are the most effective for the field of facial landmarking. Furthermore, we improve upon the existing work by extending neural networks to predict into 3D landmarks in real-time with additional observations on the impact of using 2D landmarks as auxiliary information. We evaluate the performance by using Mean Square Error (MSE) and Mean Average Error (MAE). We observe that the single data stream predicts accurate facial landmarks on Depth data when auxiliary information is used to train the network. The codes and dataset used in this paper will be made available. Full article
(This article belongs to the Special Issue Deep Learning for Facial Informatics)
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16 pages, 3132 KiB  
Review
Group-Theoretic Exploitations of Symmetry in Novel Prestressed Structures
by Yao Chen and Jian Feng
Symmetry 2018, 10(6), 229; https://doi.org/10.3390/sym10060229 - 15 Jun 2018
Cited by 10 | Viewed by 4932
Abstract
In recent years, group theory has been gradually adopted for computational problems of solid and structural mechanics. This paper reviews the advances made in the application of group theory in areas such as stability, form-finding, natural vibration and bifurcation of novel prestressed structures. [...] Read more.
In recent years, group theory has been gradually adopted for computational problems of solid and structural mechanics. This paper reviews the advances made in the application of group theory in areas such as stability, form-finding, natural vibration and bifurcation of novel prestressed structures. As initial prestress plays an important role in prestressed structures, its contribution to structural stiffness has been considered. General group-theoretic approaches for several problems are presented, where certain stiffness matrices and equilibrium matrices are expressed in symmetry-adapted coordinate system and block-diagonalized neatly. Illustrative examples on structural stability analysis, force-finding analysis, and generalized eigenvalue analysis on cable domes and cable-strut structures are drawn from recent studies by the authors. It shows how group theory, through symmetry spaces for irreducible representations and matrix decompositions, enables remarkable simplifications and reductions in the computational effort to be achieved. More importantly, before any numerical computations are performed, group theory allows valuable and effective insights on the behavior or intrinsic properties of a prestressed structure to be gained. Full article
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14 pages, 24071 KiB  
Article
Harmonic Principles of Elemental Crystals—From Atomic Interaction to Fundamental Symmetry
by Matthias Zschornak, Tilmann Leisegang, Falk Meutzner, Hartmut Stöcker, Theresa Lemser, Tobias Tauscher, Claudia Funke, Charaf Cherkouk and Dirk C. Meyer
Symmetry 2018, 10(6), 228; https://doi.org/10.3390/sym10060228 - 15 Jun 2018
Cited by 7 | Viewed by 5971
Abstract
The formation of crystals and symmetry on the atomic scale has persistently attracted scientists through the ages. The structure itself and its subtle dependence on boundary conditions is a reflection of three principles: atomic attraction, repulsion, and the limitations in 3D space. This [...] Read more.
The formation of crystals and symmetry on the atomic scale has persistently attracted scientists through the ages. The structure itself and its subtle dependence on boundary conditions is a reflection of three principles: atomic attraction, repulsion, and the limitations in 3D space. This involves a competition between simplicity and high symmetry on the one hand and necessary structural complexity on the other. This work presents a simple atomistic crystal growth model derived for equivalent atoms and a pair potential. It highlights fundamental concepts, most prominently provided by a maximum number of equilibrium distances in the atom’s local vicinity, to obtain high symmetric structural motifs, among them the Platonic Solids. In this respect, the harmonically balanced interaction during the atomistic nucleation process may be regarded as origin of symmetry. The minimization of total energy is generalized for 3D periodic structures constituting these motifs. In dependence on the pair potential’s short- and long-range characteristics the, by symmetry, rigid lattices relax isotropically within the potential well. The first few coordination shells with lattice-specific fixed distances do not necessarily determine which equilibrium symmetry prevails. A phase diagram calculated on the basis of these few assumptions summarizes stable regions of close-packed fcc and hcp, next to bcc symmetry for predominantly soft short-range and hard long-range interaction. This lattice symmetry, which is evident for alkali metals as well as transition metals of the vanadium and chromium group, cannot be obtained from classical Morse or Lennard-Jones type potentials, but needs the range flexibility within the pair potential. Full article
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17 pages, 856 KiB  
Article
A Bi-Level Programming Model for the Railway Express Cargo Service Network Design Problem
by Boliang Lin, Jianping Wu, Jiaxi Wang, Jingsong Duan and Yinan Zhao
Symmetry 2018, 10(6), 227; https://doi.org/10.3390/sym10060227 - 15 Jun 2018
Cited by 5 | Viewed by 3627
Abstract
Service network design is fundamentally crucial for railway express cargo transportation. The main challenge is to strike a balance between two conflicting objectives: low network setup costs and high expected operational incomes. Different configurations of these objectives will have different impacts on the [...] Read more.
Service network design is fundamentally crucial for railway express cargo transportation. The main challenge is to strike a balance between two conflicting objectives: low network setup costs and high expected operational incomes. Different configurations of these objectives will have different impacts on the quality of freight transportation services. In this paper, a bi-level programming model for the railway express cargo service network design problem is proposed. The upper-level model forms the optimal decisions in terms of the service characteristics, and the low-level model selects the service arcs for each commodity. The rail express cargo is strictly subject to the service commitment, the capacity restriction, flow balance constraints, and logical relationship constraints among the decisions variables. Moreover, linearization techniques are used to convert the lower-level model to a linear one so that it can be directly solved by a standard optimization solver. Finally, a real-world case study based on the Beijing–Guangzhou Railway Line is carried out to demonstrate the effectiveness and efficiency of the proposed solution approach. Full article
(This article belongs to the Special Issue Symmetry and Engineering Design)
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22 pages, 1454 KiB  
Article
A Hybrid Neutrosophic Group ANP-TOPSIS Framework for Supplier Selection Problems
by Mohamed Abdel-Basset, Mai Mohamed and Florentin Smarandache
Symmetry 2018, 10(6), 226; https://doi.org/10.3390/sym10060226 - 15 Jun 2018
Cited by 93 | Viewed by 6495
Abstract
One of the most significant competitive strategies for organizations is sustainable supply chain management (SSCM). The vital part in the administration of a sustainable supply chain is the sustainable supplier selection, which is a multi-criteria decision-making issue, including many conflicting criteria. The valuation [...] Read more.
One of the most significant competitive strategies for organizations is sustainable supply chain management (SSCM). The vital part in the administration of a sustainable supply chain is the sustainable supplier selection, which is a multi-criteria decision-making issue, including many conflicting criteria. The valuation and selection of sustainable suppliers are difficult problems due to vague, inconsistent and imprecise knowledge of decision makers. In the literature on supply chain management for measuring green performance, the requirement for methodological analysis of how sustainable variables affect each other, and how to consider vague, imprecise and inconsistent knowledge, is still unresolved. This research provides an incorporated multi-criteria decision-making procedure for sustainable supplier selection problems (SSSPs). An integrated framework is presented via interval-valued neutrosophic sets to deal with vague, imprecise and inconsistent information that exists usually in real world. The analytic network process (ANP) is employed to calculate weights of selected criteria by considering their interdependencies. For ranking alternatives and avoiding additional comparisons of analytic network processes, the technique for order preference by similarity to ideal solution (TOPSIS) is used. The proposed framework is turned to account for analyzing and selecting the optimal supplier. An actual case study of a dairy company in Egypt is examined within the proposed framework. Comparison with other existing methods is implemented to confirm the effectiveness and efficiency of the proposed approach. Full article
(This article belongs to the Special Issue Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets)
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12 pages, 271 KiB  
Article
Improved Symmetry Measures of Simplified Neutrosophic Sets and Their Decision-Making Method Based on a Sine Entropy Weight Model
by Wenhua Cui and Jun Ye
Symmetry 2018, 10(6), 225; https://doi.org/10.3390/sym10060225 - 15 Jun 2018
Cited by 19 | Viewed by 2474
Abstract
This work indicates the insufficiency of existing symmetry measures (SMs) between asymmetry measures of simplified neutrosophic sets (SNSs) and proposes the improved normalized SMs of SNSs, including the improved SMs and weighted SMs in single-valued and interval neutrosophic settings. Then, the sine entropy [...] Read more.
This work indicates the insufficiency of existing symmetry measures (SMs) between asymmetry measures of simplified neutrosophic sets (SNSs) and proposes the improved normalized SMs of SNSs, including the improved SMs and weighted SMs in single-valued and interval neutrosophic settings. Then, the sine entropy measures of SNSs are presented to establish a sine entropy weight model for solving the criteria weights in decision-making. Based on the improved weighted SMs of SNSs and the sine entropy weight model, a multi-criteria decision-making (MCDM) method with unknown criteria weights (an improved MCDM method) is established in the SNS setting. In the MCDM process, corresponding to the criteria weights obtained by the sine entropy model, the ranking order of all alternatives and the best one are given by means of the improved weighted SMs between the ideal solution and each alternative. Lastly, the improved MCDM method is applied to an actual decision example in single-valued and interval neutrosophic settings to indicate the feasibility of the improved MCDM method. By comparative analysis with existing MCDM methods, the improved SMs and the sine entropy weight model not only provide a simpler and more effective method for MCDM problems with unknown criteria weights in the SNS setting, but can also overcome the insufficiency of the existing SMs and MCDM method. Full article
12 pages, 378 KiB  
Article
A Comparative Study of Logistic Models Using an Asymmetric Link: Modelling the Away Victories in Football
by José María Pérez–Sánchez, Emilio Gómez–Déniz and Nancy Dávila–Cárdenes
Symmetry 2018, 10(6), 224; https://doi.org/10.3390/sym10060224 - 15 Jun 2018
Cited by 4 | Viewed by 3400
Abstract
The target of this paper is to study the relevant factors affecting the victories away from home of football teams in order to fit the probability of winning an away match. The paper addressed the following research issues: (a) Is the identification of [...] Read more.
The target of this paper is to study the relevant factors affecting the victories away from home of football teams in order to fit the probability of winning an away match. The paper addressed the following research issues: (a) Is the identification of the significant variables underlying the results plausible? (b) Can information of these factors increase the probability of winning away from home and assist coaches in their decisions? Empirically, it is shown that there are more home victories and draws than away victories in the professional football leagues in Europe and this fact has to be taken into account. Thus, the classical logistic and Bayesian regression models do not seem to be adequate in this case and an asymmetric logistic regression model is therefore considered. This paper analyses 380 games played in the First Division of the Spanish Football League during the 2013–2014 season. Asymmetric logistic regression from a Bayesian point of view is chosen as the best model. This model detects new relevant factors undetected by standard logistic regressions. In view of the paper’s findings, various practical recommendations were made in order to improve decision-making in this field. The Asymmetric logit link is a helpful device that can assist coaches in their game strategies. Full article
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29 pages, 1640 KiB  
Article
A Novel Selection Model of Surgical Treatments for Early Gastric Cancer Patients Based on Heterogeneous Multicriteria Group Decision-Making
by Dan-Ping Li, Ji-Qun He, Peng-Fei Cheng, Jian-Qiang Wang and Hong-Yu Zhang
Symmetry 2018, 10(6), 223; https://doi.org/10.3390/sym10060223 - 14 Jun 2018
Cited by 17 | Viewed by 3396
Abstract
Gastric cancer results in malignant tumors with high morbidity and mortality, and seriously affects the health and life quality of patients. Early detection and appropriate treatment for early-stage gastric cancer patients are very helpful to reducing the recurrence rate and improving survival rates. [...] Read more.
Gastric cancer results in malignant tumors with high morbidity and mortality, and seriously affects the health and life quality of patients. Early detection and appropriate treatment for early-stage gastric cancer patients are very helpful to reducing the recurrence rate and improving survival rates. Hence, the selection of a suitable surgical treatment is an important part. At present, surgical treatment selection has been researched in numerous studies, but there is no study integrating fuzzy decision-making theory with quantitative analysis, considering the patient’s conditions with other relative conditions, and which can handle multisource heterogeneous information at the same time. Hence, this paper proposes a novel selection model of surgical treatments for early gastric cancer based on heterogeneous multiple-criteria group decision-making (MCGDM), which is helpful to selecting the most appropriate surgery in the case of asymmetric information between doctors and patients. Subjective and objective criteria are comprehensively taken into account in the index system of the selection model for early gastric cancer, which combines fuzzy theory with quantitative data analysis. Moreover, the evaluation information obtained from the patient’s conditions, the surgery, and the hospital’s medical status, etc., including crisp numbers, interval numbers, neutrosophic numbers, and probabilistic linguistic labels, is more complete and real, so the surgical treatment selection is accurate and reliable. Furthermore, the technique for order of preference by similarity to ideal solution (TOPSIS) method is employed to solve the prioritization of early gastric cancer surgical treatments. Finally, an empirical study of surgical treatment selection for early gastric cancer surgery is conducted, and the results of sensitivity analysis and comparative analysis suggest that the proposed selection model of surgical treatments for early gastric cancer patients is reliable and effective. Full article
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15 pages, 4767 KiB  
Article
Bayer Image Demosaicking Using Eight-Directional Weights Based on the Gradient of Color Difference
by Yizheng Liu, Chengyou Wang, Hongming Zhao, Jiayang Song and Shiyue Chen
Symmetry 2018, 10(6), 222; https://doi.org/10.3390/sym10060222 - 14 Jun 2018
Cited by 1 | Viewed by 4882
Abstract
In this paper, we propose a new demosaicking algorithm which uses eight-directional weights based on the gradient of color difference (EWGCD) for Bayer image demosaicking. To obtain the interpolation of green (G) pixels, the eight-directional G pixel values are first estimated in red [...] Read more.
In this paper, we propose a new demosaicking algorithm which uses eight-directional weights based on the gradient of color difference (EWGCD) for Bayer image demosaicking. To obtain the interpolation of green (G) pixels, the eight-directional G pixel values are first estimated in red (R)/blue (B) pixels. This estimate is used to calculate the color difference in R/B pixels of the Bayer image in diagonal directions. However, in horizontal and vertical directions, the new estimated G pixels are defined to obtain the color difference. The eight-directional weights of estimated G pixels can be obtained by considering the gradient of the color difference and the gradient of the RGB pixels of the Bayer image. Therefore, the eight-directional weighted values and the first estimated G pixel values are combined to obtain the full G image. Compared with six similar algorithms using the same eighteen McMaster images, the results of the experiment demonstrate that the proposed algorithm has a better performance not only in the subjective visual measurement but also in the assessments of peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) index measurement. Full article
(This article belongs to the Special Issue Symmetry in Computing Theory and Application)
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35 pages, 2025 KiB  
Article
A Hybrid Fuzzy Analytic Network Process (FANP) and Data Envelopment Analysis (DEA) Approach for Supplier Evaluation and Selection in the Rice Supply Chain
by Chia–Nan Wang, Van Thanh Nguyen, Duy Hung Duong and Hanh Tuong Do
Symmetry 2018, 10(6), 221; https://doi.org/10.3390/sym10060221 - 14 Jun 2018
Cited by 33 | Viewed by 5252
Abstract
In the market economy, competition is typically due to the difficulty in selecting the most suitable supplier, one that is capable to help a business to develop a profit to the highest value threshold and capable to meet sustainable development features. In addition, [...] Read more.
In the market economy, competition is typically due to the difficulty in selecting the most suitable supplier, one that is capable to help a business to develop a profit to the highest value threshold and capable to meet sustainable development features. In addition, this research discusses a wide range of consequences from choosing an effective supplier, including reducing production cost, improving product quality, delivering the product on time, and responding flexibly to customer requirements. Therefore, the activities noted above are able to increase an enterprise’s competitiveness. It can be seen that selecting a supplier is complex in that decision-makers must have an understanding of the qualitative and quantitative features for assessing the symmetrical impact of the criteria to reach the most accurate result. In this research, the multi-criteria group decision-making (MCGDM) approach was proposed to solve supplier selection problems. The authors collected data from 25 potential suppliers, and the four main criteria within contain 15 sub-criteria to define the most effective supplier, which has viewed factors, including financial efficiency guarantee, quality of materials, ability to deliver on time, and the conditioned response to the environment to improve the efficiency of the industry supply chain. Initially, fuzzy analytic network process (ANP) is used to evaluate and rank these criteria, which are able to be utilized to clarify important criteria that directly affect the profitability of the business. Subsequently, data envelopment analysis (DEA) models, including the Charnes Cooper Rhodes model (CCR model), Banker Charnes Cooper model (BCC model), and slacks-based measure model (SBM model), were proposed to rank suppliers. The result of the model has proposed 7/25 suppliers, which have a condition response to the enterprises’ supply requirements. Full article
(This article belongs to the Special Issue Fuzzy Techniques for Decision Making 2018)
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14 pages, 1466 KiB  
Article
Spatiotemporal and Luminance Contrast Properties of Symmetry Perception
by Rebecca J. Sharman and Elena Gheorghiu
Symmetry 2018, 10(6), 220; https://doi.org/10.3390/sym10060220 - 14 Jun 2018
Cited by 9 | Viewed by 3835
Abstract
Recent studies have shown that limiting the lifetime of pattern elements improves symmetry detection, potentially by increasing the number of element locations. Here, we investigate how spatial relocation, luminance contrast modulation and lifetime duration of elements affect symmetry perception in dynamic stimuli. Stimuli [...] Read more.
Recent studies have shown that limiting the lifetime of pattern elements improves symmetry detection, potentially by increasing the number of element locations. Here, we investigate how spatial relocation, luminance contrast modulation and lifetime duration of elements affect symmetry perception in dynamic stimuli. Stimuli were dynamic dot-patterns containing varying amounts of symmetry about a vertical axis. Symmetrical matched-pairs were: (i) relocated to multiple successive, but random locations (i.e., multiple locations condition); (ii) relocated between the same two locations (i.e., two locations condition); (iii) not, relocated, but their luminance contrast was modulated at different temporal frequencies (i.e., one location condition), and (iv) not relocated, but a single pattern was presented at full contrast (i.e., static condition). In the dynamic conditions, we varied the elements’ lifetime duration and temporal frequency of contrast modulation. We measured symmetry detection thresholds using a two-interval forced choice procedure. Our results show improved performance for the multiple locations condition compared to two-location and static conditions, suggesting a cumulative process whereby weak symmetry information is integrated by spatiotemporal filters to increase overall symmetry signal strength. Performance also improved for the static, contrast modulated patterns, but this was explained by a reduction in perceived density. This suggests that different mechanisms mediate symmetry detection in dynamic stimuli and static contrast modulated patterns. Full article
(This article belongs to the Special Issue Symmetry-Related Activity in Mid-Level Vision)
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7 pages, 211 KiB  
Article
Symmetric Identities for Fubini Polynomials
by Taekyun Kim, Dae San Kim, Gwan-Woo Jang and Jongkyum Kwon
Symmetry 2018, 10(6), 219; https://doi.org/10.3390/sym10060219 - 14 Jun 2018
Cited by 21 | Viewed by 3468
Abstract
We represent the generating function of w-torsion Fubini polynomials by means of a fermionic p-adic integral on Zp. Then we investigate a quotient of such p-adic integrals on Zp, representing generating functions of three w-torsion [...] Read more.
We represent the generating function of w-torsion Fubini polynomials by means of a fermionic p-adic integral on Zp. Then we investigate a quotient of such p-adic integrals on Zp, representing generating functions of three w-torsion Fubini polynomials and derive some new symmetric identities for the w-torsion Fubini and two variable w-torsion Fubini polynomials. Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with their Applications)
39 pages, 442 KiB  
Article
Gauss Map and Its Applications on Ruled Submanifolds in Minkowski Space
by Sun Mi Jung and Young Ho Kim
Symmetry 2018, 10(6), 218; https://doi.org/10.3390/sym10060218 - 13 Jun 2018
Cited by 3 | Viewed by 2111
Abstract
We study ruled submanifolds in Minkowski space in regard to the Gauss map satisfying some partial differential equation. As a generalization of usual cylinders, cones and null scrolls in a three-dimensional Minkowski space, a cylinder over a space curve, a product manifold of [...] Read more.
We study ruled submanifolds in Minkowski space in regard to the Gauss map satisfying some partial differential equation. As a generalization of usual cylinders, cones and null scrolls in a three-dimensional Minkowski space, a cylinder over a space curve, a product manifold of a right cone and a k-plane, a product manifold of a hyperbolic cone and a k-plane which look like kinds of cylinders over cones in 3-space, and the generalized B-scroll kind in Minkowski space are characterized with the partial differential equation regarding the Gauss map, where k is a positive integer. Full article
10 pages, 240 KiB  
Article
Decision-Making via Neutrosophic Support Soft Topological Spaces
by Parimala Mani, Karthika Muthusamy, Saeid Jafari, Florentin Smarandache and Udhayakumar Ramalingam
Symmetry 2018, 10(6), 217; https://doi.org/10.3390/sym10060217 - 13 Jun 2018
Cited by 18 | Viewed by 3067
Abstract
The concept of interval neutrosophic sets has been studied and the introduction of a new kind of set in topological spaces called the interval valued neutrosophic support soft set has been suggested. We study some of its basic properties. The main purpose of [...] Read more.
The concept of interval neutrosophic sets has been studied and the introduction of a new kind of set in topological spaces called the interval valued neutrosophic support soft set has been suggested. We study some of its basic properties. The main purpose of this paper is to give the optimum solution to decision-making in real life problems the using interval valued neutrosophic support soft set. Full article
(This article belongs to the Special Issue Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets)
17 pages, 2225 KiB  
Article
Fuzzy Association Rule Based Froth Surface Behavior Control in Zinc Froth Flotation
by Jin Zhang, Zhaohui Tang, Mingxi Ai and Weihua Gui
Symmetry 2018, 10(6), 216; https://doi.org/10.3390/sym10060216 - 13 Jun 2018
Cited by 9 | Viewed by 3375
Abstract
Froth flotation is a vital mineral concentration process. Froth surface behavior is the knowledge about flotation working condition. However, in computer vision aided froth surface behavior control, there are still two challenges that need to be tackled seriously. Against the difficulty in the [...] Read more.
Froth flotation is a vital mineral concentration process. Froth surface behavior is the knowledge about flotation working condition. However, in computer vision aided froth surface behavior control, there are still two challenges that need to be tackled seriously. Against the difficulty in the froth surface behavior representation, this paper proposes to combine the bubble size distribution (BSD) and froth velocity distribution. As far as we know, this is the first time that the froth velocity distribution is presented. Against the difficulty in the adaptive generation of the optimal froth surface behavior feature (optimal setpoint), this paper introduces the fuzzy apriori to mine the association rule between the current working condition and the optimal setpoint. Then, a fuzzy inference module is constructed to generate optimal setpoint for current working condition adaptively. Many validation experiments and comparison experiments demonstrate the superiority and robustness of the proposed methods. Full article
(This article belongs to the Special Issue Novel Machine Learning Approaches for Intelligent Big Data)
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11 pages, 639 KiB  
Article
Multiple Attribute Decision-Making Method Using Similarity Measures of Neutrosophic Cubic Sets
by Angyan Tu, Jun Ye and Bing Wang
Symmetry 2018, 10(6), 215; https://doi.org/10.3390/sym10060215 - 12 Jun 2018
Cited by 13 | Viewed by 2730
Abstract
In inconsistent and indeterminate settings, as a usual tool, the neutrosophic cubic set (NCS) containing single-valued neutrosophic numbers and interval neutrosophic numbers can be applied in decision-making to present its partial indeterminate and partial determinate information. However, a few researchers have studied neutrosophic [...] Read more.
In inconsistent and indeterminate settings, as a usual tool, the neutrosophic cubic set (NCS) containing single-valued neutrosophic numbers and interval neutrosophic numbers can be applied in decision-making to present its partial indeterminate and partial determinate information. However, a few researchers have studied neutrosophic cubic decision-making problems, where the similarity measure of NCSs is one of the useful measure methods. For this work, we propose the Dice, cotangent, and Jaccard measures between NCSs, and indicate their properties. Then, under an NCS environment, the similarity measures-based decision-making method of multiple attributes is developed. In the decision-making process, all the alternatives are ranked by the similarity measure of each alternative and the ideal solution to obtain the best one. Finally, two practical examples are applied to indicate the feasibility and effectiveness of the developed method. Full article
(This article belongs to the Special Issue Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets)
27 pages, 3113 KiB  
Article
Intelligent Prognostics of Degradation Trajectories for Rotating Machinery Based on Asymmetric Penalty Sparse Decomposition Model
by Qing Li and Steven Y. Liang
Symmetry 2018, 10(6), 214; https://doi.org/10.3390/sym10060214 - 12 Jun 2018
Cited by 7 | Viewed by 2851
Abstract
The ability to accurately track the degradation trajectories of rotating machinery components is arguably one of the challenging problems in prognostics and health management (PHM). In this paper, an intelligent prediction approach based on asymmetric penalty sparse decomposition (APSD) algorithm combined with wavelet [...] Read more.
The ability to accurately track the degradation trajectories of rotating machinery components is arguably one of the challenging problems in prognostics and health management (PHM). In this paper, an intelligent prediction approach based on asymmetric penalty sparse decomposition (APSD) algorithm combined with wavelet neural network (WNN) and autoregressive moving average-recursive least squares algorithm (ARMA-RLS) is proposed for degradation prognostics of rotating machinery, taking the accelerated life test of rolling bearings as an example. Specifically, the health indicators time series (e.g., peak-to-peak value and Kurtosis) is firstly decomposed into low frequency component (LFC) and high frequency component (HFC) using the APSD algorithm; meanwhile, the resulting non-convex regularization problem can be efficiently solved using the majorization-minimization (MM) method. In particular, the HFC part corresponds to the stable change around the zero line of health indicators which most extensively occurs; in contrast, the LFC part is essentially related to the evolutionary trend of health indicators. Furthermore, the nonparametric-based method, i.e., WNN, and parametric-based method, i.e., ARMA-RLS, are respectively introduced to predict the LFC and HFC that focus on abrupt degradation regions (e.g., last 100 points). Lastly, the final predicted data could be correspondingly obtained by integrating the predicted LFC and predicted HFC. The proposed methodology is tested using degradation health indicator time series from four rolling bearings. The proposed approach performed favorably when compared to some state-of-the-art benchmarks such as WNN and largest Lyapunov (LLyap) methods. Full article
(This article belongs to the Special Issue Symmetry and Complexity)
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16 pages, 297 KiB  
Article
Medical Diagnosis Based on Single-Valued Neutrosophic Probabilistic Rough Multisets over Two Universes
by Chao Zhang, Deyu Li, Said Broumi and Arun Kumar Sangaiah
Symmetry 2018, 10(6), 213; https://doi.org/10.3390/sym10060213 - 12 Jun 2018
Cited by 28 | Viewed by 3103
Abstract
In real-world diagnostic procedures, due to the limitation of human cognitive competence, a medical expert may not conveniently use some crisp numbers to express the diagnostic information, and plenty of research has indicated that generalized fuzzy numbers play a significant role in describing [...] Read more.
In real-world diagnostic procedures, due to the limitation of human cognitive competence, a medical expert may not conveniently use some crisp numbers to express the diagnostic information, and plenty of research has indicated that generalized fuzzy numbers play a significant role in describing complex diagnostic information. To deal with medical diagnosis problems based on generalized fuzzy sets (FSs), the notion of single-valued neutrosophic multisets (SVNMs) is firstly used to express the diagnostic information in this article. Then the model of probabilistic rough sets (PRSs) over two universes is applied to analyze SVNMs, and the concepts of single-valued neutrosophic rough multisets (SVNRMs) over two universes and probabilistic rough single-valued neutrosophic multisets (PRSVNMs) over two universes are introduced. Based on SVNRMs over two universes and PRSVNMs over two universes, single-valued neutrosophic probabilistic rough multisets (SVNPRMs) over two universes are further established. Next, a three-way decisions model by virtue of SVNPRMs over two universes in the context of medical diagnosis is constructed. Finally, a practical case study along with a comparative study are carried out to reveal the accuracy and reliability of the constructed three-way decisions model. Full article
(This article belongs to the Special Issue Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets)
11 pages, 287 KiB  
Article
Enumeration of Strongly Regular Graphs on up to 50 Vertices Having S3 as an Automorphism Group
by Marija Maksimović
Symmetry 2018, 10(6), 212; https://doi.org/10.3390/sym10060212 - 11 Jun 2018
Cited by 2 | Viewed by 2526
Abstract
One of the main problems in the theory of strongly regular graphs (SRGs) is constructing and classifying SRGs with given parameters. Strongly regular graphs with parameters (37,18,8,9), [...] Read more.
One of the main problems in the theory of strongly regular graphs (SRGs) is constructing and classifying SRGs with given parameters. Strongly regular graphs with parameters (37,18,8,9), (41,20,9,10), (45,22,10,11), (49,24,11,12), (49,18,7,6) and (50,21,8,9) are the only strongly regular graphs on up to 50 vertices that still have to be classified. In this paper, we give the enumeration of SRGs with these parameters having S3 as an automorphism group. The construction of SRGs in this paper is a step in the classification of SRGs on up to 50 vertices. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry)
12 pages, 306 KiB  
Article
Dynamics on Binary Relations over Topological Spaces
by Chung-Chuan Chen, J. Alberto Conejero, Marko Kostić and Marina Murillo-Arcila
Symmetry 2018, 10(6), 211; https://doi.org/10.3390/sym10060211 - 11 Jun 2018
Cited by 7 | Viewed by 3066
Abstract
The existence of chaos and the quest of dense orbits have been recently considered for dynamical systems given by multivalued linear operators. We consider the notions of topological transitivity, topologically mixing property, hypercyclicity, periodic points, and Devaney chaos in the general case of [...] Read more.
The existence of chaos and the quest of dense orbits have been recently considered for dynamical systems given by multivalued linear operators. We consider the notions of topological transitivity, topologically mixing property, hypercyclicity, periodic points, and Devaney chaos in the general case of binary relations on topological spaces, and we analyze how they can be particularized when they are represented with graphs and digraphs. The relations of these notions with different types of connectivity and with the existence of Hamiltonian paths are also exposed. Special attention is given to the study of dynamics over tournaments. Finally, we also show how disjointness can be introduced in this setting. Full article
(This article belongs to the Special Issue Symmetry in Graph Theory)
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31 pages, 17518 KiB  
Article
An Improved Whale Optimization Algorithm Based on Different Searching Paths and Perceptual Disturbance
by Wei-zhen Sun, Jie-sheng Wang and Xian Wei
Symmetry 2018, 10(6), 210; https://doi.org/10.3390/sym10060210 - 11 Jun 2018
Cited by 51 | Viewed by 6487
Abstract
Whale optimization algorithm (WOA) is a swarm intelligence optimization algorithm inspired by humpback whale hunting behavior. WOA has many similarities with other swarm intelligence algorithms (PSO, GWO, etc.). WOA’s unique search mechanism enables it to have a strong global search capability while taking [...] Read more.
Whale optimization algorithm (WOA) is a swarm intelligence optimization algorithm inspired by humpback whale hunting behavior. WOA has many similarities with other swarm intelligence algorithms (PSO, GWO, etc.). WOA’s unique search mechanism enables it to have a strong global search capability while taking into account the strong global search capabilities. In this work, considering the the deficiency of WOA in local search mechanism, combined with the optimization methods of other group intelligent algorithms, perceptual perturbation mechanism is introduced, which makes the agent perform more detailed searches near the local extreme point. At the same time, since the WOA uses a logarithmic spiral curve, the agent cannot fully search all the spaces within its search range, even though the introduction of the perturbation mechanism may still lead to the algorithm falling into a local optimum. Therefore, the equal pitch Archimedes spiral curve is chosen to replace the classic logarithmic spiral curve. In order to fully verify the effect of the search path on the performance of the algorithm, several other spiral curves have been chosen for experimental comparison. By utilizing the 23 benchmark test functions, the simulation results show that WOA (PDWOA) with perceptual perturbation significantly outperforms the standard WOA. Then, based on the PDWOA, the effect of the search path on the performance of the algorithm has been verified. The simulation results show that the equal pitch of the Archimedean spiral curve is best. Full article
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11 pages, 290 KiB  
Article
Computing the Metric Dimension of Gear Graphs
by Shahid Imran, Muhammad Kamran Siddiqui, Muhammad Imran, Muhammad Hussain, Hafiz Muhammad Bilal, Imran Zulfiqar Cheema, Ali Tabraiz and Zeeshan Saleem
Symmetry 2018, 10(6), 209; https://doi.org/10.3390/sym10060209 - 08 Jun 2018
Cited by 15 | Viewed by 4000
Abstract
Let G = (V, E) be a connected graph and d(u, v) denote the distance between the vertices u and v in G. A set of vertices W resolves a graph G if every vertex [...] Read more.
Let G = (V, E) be a connected graph and d(u, v) denote the distance between the vertices u and v in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). Let J2n,m be a m-level gear graph obtained by m-level wheel graph W2n,m ≅ mC2n + k1 by alternatively deleting n spokes of each copy of C2n and J3n be a generalized gear graph obtained by alternately deleting 2n spokes of the wheel graph W3n. In this paper, the metric dimension of certain gear graphs J2n,m and J3n generated by wheel has been computed. Also this study extends the previous result given by Tomescu et al. in 2007. Full article
(This article belongs to the Special Issue Symmetry in Graph Theory)
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22 pages, 982 KiB  
Article
Multi-Granulation Rough Set for Incomplete Interval-Valued Decision Information Systems Based on Multi-Threshold Tolerance Relation
by Bingyan Lin and Weihua Xu
Symmetry 2018, 10(6), 208; https://doi.org/10.3390/sym10060208 - 08 Jun 2018
Cited by 12 | Viewed by 2640
Abstract
A relation is viewed as a granularity from a granular computing perspective. A classic rough set contains only one granularity. A multi-granulation rough set contains multiple granularities, which promotes the applications of classical rough set. Firstly, this paper uses the incomplete interval-valued decision [...] Read more.
A relation is viewed as a granularity from a granular computing perspective. A classic rough set contains only one granularity. A multi-granulation rough set contains multiple granularities, which promotes the applications of classical rough set. Firstly, this paper uses the incomplete interval-valued decision information system (IIVDIS) as research object and constructs two rough set models in the light of single granularity rough set model for applying the rough set theory to real life more widely, which are optimistic multi-granulation rough set (OMGRS) model and pessimistic multi-granulation rough set (PMGRS) model in the IIVDIS. Secondly, we design two algorithms to compute the roughness and the degree of dependence that are two tools for measuring uncertainty of rough set. Finally, several experiments are performed on six UCI data sets to verify the validity of the proposed theorems. Full article
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14 pages, 241 KiB  
Article
Nonoscillatory Solutions to Second-Order Neutral Difference Equations
by Małgorzata Migda and Janusz Migda
Symmetry 2018, 10(6), 207; https://doi.org/10.3390/sym10060207 - 08 Jun 2018
Cited by 5 | Viewed by 2145
Abstract
We study asymptotic behavior of nonoscillatory solutions to second-order neutral difference equation of the form: [...] Read more.
We study asymptotic behavior of nonoscillatory solutions to second-order neutral difference equation of the form: Δ ( r n Δ ( x n + p n x n − τ ) ) = a n f ( n , x n ) + b n . The obtained results are based on the discrete Bihari type lemma and a Stolz type lemma. Full article
(This article belongs to the Special Issue Difference Equations, Symmetric, Close to Symmetric and Cyclic Systems of Difference Equations)
13 pages, 779 KiB  
Article
Laplacian Spectra for Categorical Product Networks and Its Applications
by Shin Min Kang, Muhammad Kamran Siddiqui, Najma Abdul Rehman, Muhammad Imran and Mehwish Hussain Muhammad
Symmetry 2018, 10(6), 206; https://doi.org/10.3390/sym10060206 - 07 Jun 2018
Cited by 11 | Viewed by 3100
Abstract
The Kirchhoff index, global mean-first passage time, average path length and number of spanning trees are of great importance in the field of networking. The “Kirchhoff index” is known as a structure descriptor index. The “global mean-first passage time” is known as a [...] Read more.
The Kirchhoff index, global mean-first passage time, average path length and number of spanning trees are of great importance in the field of networking. The “Kirchhoff index” is known as a structure descriptor index. The “global mean-first passage time” is known as a measure for nodes that are quickly reachable from the whole network. The “average path length” is a measure of the efficiency of information or mass transport on a network, and the “number of spanning trees” is used to minimize the cost of power networks, wiring connections, etc. In this paper, we have selected a complex network based on a categorical product and have used the spectrum approach to find the Kirchhoff index, global mean-first passage time, average path length and number of spanning trees. We find the expressions for the product and sum of reciprocals of all nonzero eigenvalues of a categorical product network with the help of the eigenvalues of the path and cycles. Full article
(This article belongs to the Special Issue Symmetry and Complexity)
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18 pages, 678 KiB  
Article
The Recalculation of the Weights of Criteria in MCDM Methods Using the Bayes Approach
by Irina Vinogradova, Valentinas Podvezko and Edmundas Kazimieras Zavadskas
Symmetry 2018, 10(6), 205; https://doi.org/10.3390/sym10060205 - 07 Jun 2018
Cited by 91 | Viewed by 4677
Abstract
The application of multiple criteria decision-making methods (MCDM) is aimed at choosing the best alternative out of the number of available versions in the absence of the apparently dominant alternative. One of the two major components of multiple criteria decision-making methods is represented [...] Read more.
The application of multiple criteria decision-making methods (MCDM) is aimed at choosing the best alternative out of the number of available versions in the absence of the apparently dominant alternative. One of the two major components of multiple criteria decision-making methods is represented by the weights of the criteria describing the considered process. The weights of the criteria quantitatively express their significance and influence on the evaluation result. The criterion weights can be subjective, i.e., based on the estimates assigned by the experts, and the so-called objective, i.e., those which assess the structure of the data array at the time of evaluation. Several groups of experts, representing the opinions of various interested parties may take part in the evaluation of criteria. The evaluation data on the criterion weights also depend on the mathematical methods used for calculations and the estimation scales. In determining the objective weights, several methods, assessing various properties or characteristics of the data array’s structure, are usually employed. Therefore, the use of the procedures, improving the accuracy of the evaluation of the weights’ values and the integration of the obtained data into a single value, is often required. The present paper offers a new approach to more accurate evaluation of the criteria weights obtained by using various methods based on the idea of the Bayes hypothesis. The performed investigation shows that the suggested method is symmetrical and does not depend on the fact whether a priori or posterior values of the weights are recalculated. This result is the theoretical basis for practical use of the method of combining the weights obtained by various approaches as the geometric mean of various estimates. The ideas suggested by the authors have been repeatedly used in the investigation for combining the objective weights, for recalculating the criteria weights after obtaining the estimates of other groups of experts and for combining the subjective and the objective weights. The recalculated values of the weights of the criteria are used in the work for evaluating the quality of the distant courses taught to the students. Full article
(This article belongs to the Special Issue Solution Models based on Symmetric and Asymmetric Information)
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12 pages, 381 KiB  
Article
Intrinsic Metrics on Sierpinski-Like Triangles and Their Geometric Properties
by Mustafa Saltan
Symmetry 2018, 10(6), 204; https://doi.org/10.3390/sym10060204 - 07 Jun 2018
Cited by 10 | Viewed by 3973
Abstract
The classical Sierpinski Gasket defined on the equilateral triangle is a typical example of fractals. Sierpinski-like triangles can also be constructed on isosceles or scalene triangles. An explicit formula for the intrinsic metric on the classical Sierpinski Gasket via code representation of its [...] Read more.
The classical Sierpinski Gasket defined on the equilateral triangle is a typical example of fractals. Sierpinski-like triangles can also be constructed on isosceles or scalene triangles. An explicit formula for the intrinsic metric on the classical Sierpinski Gasket via code representation of its points is given. The aim of this paper is to generalize this formula to the Sierpinski-like triangles. We also investigate geometrical properties of these triangles with respect to the intrinsic metric. Moreover, we describe certain properties of the classical Sierpinski gasket which are not shared by all of its analogues. Full article
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14 pages, 294 KiB  
Article
Some Results on Neutrosophic Triplet Group and Their Applications
by Tèmítópé Gbóláhàn Jaíyéolá and Florentin Smarandache
Symmetry 2018, 10(6), 202; https://doi.org/10.3390/sym10060202 - 06 Jun 2018
Cited by 15 | Viewed by 3442
Abstract
This article is based on new developments on a neutrosophic triplet group (NTG) and applications earlier introduced in 2016 by Smarandache and Ali. NTG sprang up from neutrosophic triplet set X: a collection of triplets [...] Read more.
This article is based on new developments on a neutrosophic triplet group (NTG) and applications earlier introduced in 2016 by Smarandache and Ali. NTG sprang up from neutrosophic triplet set X: a collection of triplets ( b , n e u t ( b ) , a n t i ( b ) ) for an b ∈ X that obeys certain axioms (existence of neutral(s) and opposite(s)). Some results that are true in classical groups were investigated in NTG and were shown to be either universally true in NTG or true in some peculiar types of NTG. Distinguishing features between an NTG and some other algebraic structures such as: generalized group (GG), quasigroup, loop and group were investigated. Some neutrosophic triplet subgroups (NTSGs) of a neutrosophic triplet group were studied. In particular, for any arbitrarily fixed a ∈ X , the subsets X a = { b ∈ X : n e u t ( b ) = n e u t ( a ) } and ker f a = { b ∈ X | f ( b ) = n e u t ( f ( a ) ) } of X, where f : X → Y is a neutrosophic triplet group homomorphism, were shown to be NTSG and normal NTSG, respectively. Both X a and ker f a were shown to be a-normal NTSGs and found to partition X. Consequently, a Lagrange-like formula was found for a finite NTG X ; | X | = ∑ a ∈ X [ X a : ker f a ] | ker f a | based on the fact that | ker f a | | | X a | . The first isomorphism theorem X / ker f ≅ Im f was established for NTGs. Using an arbitrary non-abelian NTG X and its NTSG X a , a Bol structure was constructed. Applications of the neutrosophic triplet set, and our results on NTG in relation to management and sports, are highlighted and discussed. Full article
(This article belongs to the Special Issue Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets)
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22 pages, 1700 KiB  
Article
A Study on Neutrosophic Cubic Graphs with Real Life Applications in Industries
by Muhammad Gulistan, Naveed Yaqoob, Zunaira Rashid, Florentin Smarandache and Hafiz Abdul Wahab
Symmetry 2018, 10(6), 203; https://doi.org/10.3390/sym10060203 - 05 Jun 2018
Cited by 22 | Viewed by 3978
Abstract
Neutrosophic cubic sets are the more generalized tool by which one can handle imprecise information in a more effective way as compared to fuzzy sets and all other versions of fuzzy sets. Neutrosophic cubic sets have the more flexibility, precision and compatibility to [...] Read more.
Neutrosophic cubic sets are the more generalized tool by which one can handle imprecise information in a more effective way as compared to fuzzy sets and all other versions of fuzzy sets. Neutrosophic cubic sets have the more flexibility, precision and compatibility to the system as compared to previous existing fuzzy models. On the other hand the graphs represent a problem physically in the form of diagrams, matrices etc. which is very easy to understand and handle. So the authors applied the Neutrosophic cubic sets to graph theory in order to develop a more general approach where they can model imprecise information through graphs. We develop this model by introducing the idea of neutrosophic cubic graphs and introduce many fundamental binary operations like cartesian product, composition, union, join of neutrosophic cubic graphs, degree and order of neutrosophic cubic graphs and some results related with neutrosophic cubic graphs. One of very important futures of two neutrosophic cubic sets is the R-union that R-union of two neutrosophic cubic sets is again a neutrosophic cubic set, but here in our case we observe that R-union of two neutrosophic cubic graphs need not be a neutrosophic cubic graph. Since the purpose of this new model is to capture the uncertainty, so we provide applications in industries to test the applicability of our defined model based on present time and future prediction which is the main advantage of neutrosophic cubic sets. Full article
(This article belongs to the Special Issue Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets)
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15 pages, 9185 KiB  
Article
Topological Characterization of the Symmetrical Structure of Bismuth Tri-Iodide
by Muhammad Imran, Muhammad Arfan Ali, Sarfraz Ahmad, Muhammad Kamran Siddiqui and Abdul Qudair Baig
Symmetry 2018, 10(6), 201; https://doi.org/10.3390/sym10060201 - 04 Jun 2018
Cited by 25 | Viewed by 3782
Abstract
The bismuth tri-iodide ( B i I 3 ) is an inorganic compound. It is the result of the response of bismuth and iodine, which has inspired enthusiasm for subjective inorganic investigation. The topological indices are the numerical invariants of the molecular graph [...] Read more.
The bismuth tri-iodide ( B i I 3 ) is an inorganic compound. It is the result of the response of bismuth and iodine, which has inspired enthusiasm for subjective inorganic investigation. The topological indices are the numerical invariants of the molecular graph that portray its topology and are normally graph invariants. In 1975, Randic presented, in a bond-added substance, a topological index as a descriptor for portraying subatomic branching. In this paper, we investigate the precious stone structure of bismuth tri-iodide chain and sheet. Moreover, exact formulas of degree-based added-substance topological indices principally the first, second, and hyper Zagreb indices, the general Randic index, the geometric-arithmetic index, the fourth atom-bond connectivity index, and the fifth geometric arithmetic index of the subatomic graph of bismuth tri-iodide for both chain and sheet structures are determined. Full article
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