Open AccessArticle
Factors Affecting the Perception of 3D Facial Symmetry from 2D Projections
Symmetry 2017, 9(10), 243; doi:10.3390/sym9100243 (registering DOI) -
Abstract
Facial symmetry is believed to have an evolutionary significance and so its detection should be robust in natural settings. Previous studies of facial symmetry detection have used front views of faces where the decision could be made on 2D image properties rather than
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Facial symmetry is believed to have an evolutionary significance and so its detection should be robust in natural settings. Previous studies of facial symmetry detection have used front views of faces where the decision could be made on 2D image properties rather than 3D facial properties. These studies also employed comparative judgements, which could be influenced by attractiveness rather than symmetry. Two experiments explored the ability to detect typical levels of 3D facial asymmetry (contrasted with wholly symmetrical faces) from 2D projections of faces. Experiment 1 showed that asymmetry detection was impaired by inversion but even more impaired by 90 degrees rotation demonstrating the importance of the vertical reflection. Asymmetry detection was also reduced by yaw rotation of the head but still above-chance at 30 degrees rotation. Experiment 2 explored the effect of asymmetrical lighting and yaw rotation up to 45 degrees. Detection of asymmetry was affected by asymmetrical lighting and yaw rotation in a non-additive manner. The results are discussed in terms of the special role that faces and vertical symmetry play in visual perception. Full article
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Open AccessArticle
Self-Adaptive Pre-Processing Methodology for Big Data Stream Mining in Internet of Things Environmental Sensor Monitoring
Symmetry 2017, 9(10), 244; doi:10.3390/sym9100244 (registering DOI) -
Abstract
Over the years, advanced IT technologies have facilitated the emergence of new ways of generating and gathering data rapidly, continuously, and largely and are associated with a new research and application branch, namely, data stream mining (DSM). Among those multiple scenarios of DSM,
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Over the years, advanced IT technologies have facilitated the emergence of new ways of generating and gathering data rapidly, continuously, and largely and are associated with a new research and application branch, namely, data stream mining (DSM). Among those multiple scenarios of DSM, the Internet of Things (IoT) plays a significant role, with a typical meaning of a tough and challenging computational case of big data. In this paper, we describe a self-adaptive approach to the pre-processing step of data stream classification. The proposed algorithm allows different divisions with both variable numbers and lengths of sub-windows under a whole sliding window on an input stream, and clustering-based particle swarm optimization (CPSO) is adopted as the main metaheuristic search method to guarantee that its stream segmentations are effective and adaptive to itself. In order to create a more abundant search space, statistical feature extraction (SFX) is applied after variable partitions of the entire sliding window. We validate and test the effort of our algorithm with other temporal methods according to several IoT environmental sensor monitoring datasets. The experiments yield encouraging outcomes, supporting the reality that picking significant appropriate variant sub-window segmentations heuristically with an incorporated clustering technique merit would allow these to perform better than others. Full article
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Open AccessArticle
Lattice-Based Revocable Certificateless Signature
Symmetry 2017, 9(10), 242; doi:10.3390/sym9100242 (registering DOI) -
Abstract
Certificateless signatures (CLS) are noticeable because they may resolve the key escrow problem in ID-based signatures and break away the management problem regarding certificate in conventional signatures. However, the security of the mostly previous CLS schemes relies on the difficulty of solving discrete
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Certificateless signatures (CLS) are noticeable because they may resolve the key escrow problem in ID-based signatures and break away the management problem regarding certificate in conventional signatures. However, the security of the mostly previous CLS schemes relies on the difficulty of solving discrete logarithm or large integer factorization problems. These two problems would be solved by quantum computers in the future so that the signature schemes based on them will also become insecure. For post-quantum cryptography, lattice-based cryptography is significant due to its efficiency and security. However, no study on addressing the revocation problem in the existing lattice-based CLS schemes is presented. In this paper, we focus on the revocation issue and present the first revocable CLS (RCLS) scheme over lattices. Based on the short integer solution (SIS) assumption over lattices, the proposed lattice-based RCLS scheme is shown to be existential unforgeability against adaptive chosen message attacks. By performance analysis and comparisons, the proposed lattice-based RCLS scheme is better than the previously proposed lattice-based CLS scheme, in terms of private key size, signature length and the revocation mechanism. Full article
Open AccessArticle
Fuzzy Logic-Based Model That Incorporates Personality Traits for Heterogeneous Pedestrians
Symmetry 2017, 9(10), 239; doi:10.3390/sym9100239 (registering DOI) -
Abstract
Most models designed to simulate pedestrian dynamical behavior are based on the assumption that human decision-making can be described using precise values. This study proposes a new pedestrian model that incorporates fuzzy logic theory into a multi-agent system to address cognitive behavior that
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Most models designed to simulate pedestrian dynamical behavior are based on the assumption that human decision-making can be described using precise values. This study proposes a new pedestrian model that incorporates fuzzy logic theory into a multi-agent system to address cognitive behavior that introduces uncertainty and imprecision during decision-making. We present a concept of decision preferences to represent the intrinsic control factors of decision-making. To realize the different decision preferences of heterogeneous pedestrians, the Five-Factor (OCEAN) personality model is introduced to model the psychological characteristics of individuals. Then, a fuzzy logic-based approach is adopted for mapping the relationships between the personality traits and the decision preferences. Finally, we have developed an application using our model to simulate pedestrian dynamical behavior in several normal or non-panic scenarios, including a single-exit room, a hallway with obstacles, and a narrowing passage. The effectiveness of the proposed model is validated with a user study. The results show that the proposed model can generate more reasonable and heterogeneous behavior in the simulation and indicate that individual personality has a noticeable effect on pedestrian dynamical behavior. Full article
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Open AccessArticle
Multi-Objective Optimization Algorithm Based on Sperm Fertilization Procedure (MOSFP)
Symmetry 2017, 9(10), 241; doi:10.3390/sym9100241 (registering DOI) -
Abstract
In this paper, we propose an extended multi-objective version of single objective optimization algorithm called sperm swarm optimization algorithm. The proposed multi-objective optimization algorithm based on sperm fertilization procedure (MOSFP) operates based on Pareto dominance and a crowding factor, that crowd and filter
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In this paper, we propose an extended multi-objective version of single objective optimization algorithm called sperm swarm optimization algorithm. The proposed multi-objective optimization algorithm based on sperm fertilization procedure (MOSFP) operates based on Pareto dominance and a crowding factor, that crowd and filter out the list of the best sperms (global best values). We divide the sperm swarm into three equal parts, after that, different types of turbulence (mutation) operators are applied on these parts, such as uniform mutation, non-uniform mutation, and without any mutation. Our algorithm is compared against three well-known algorithms in the field of optimization. These algorithms are NSGA-II, SPEA2, and OMOPSO. These algorithms are compared using a very popular benchmark function suites called Zitzler-Deb-Thiele (ZDT) and Walking-Fish-Group (WFG). We also adopt three quality metrics to compare the convergence, accuracy, and diversity of these algorithms, including, inverted generational distance (IGD), spread (SP), and epsilon (∈). The experimental results show that the performance of the proposed MOSFP is highly competitive, which outperformed OMOPSO in solving problems such as ZDT3, WFG5, and WFG8. In addition, the proposed MOSFP outperformed both of NSGA-II or SPEA2 algorithms in solving all the problems. Full article
Open AccessArticle
On Elastic Symmetry Identification for Polycrystalline Materials
Symmetry 2017, 9(10), 240; doi:10.3390/sym9100240 (registering DOI) -
Abstract
The products made by the forming of polycrystalline metals and alloys, which are in high demand in modern industries, have pronounced inhomogeneous distribution of grain orientations. The presence of specific orientation modes in such materials, i.e., crystallographic texture, is responsible for anisotropy of
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The products made by the forming of polycrystalline metals and alloys, which are in high demand in modern industries, have pronounced inhomogeneous distribution of grain orientations. The presence of specific orientation modes in such materials, i.e., crystallographic texture, is responsible for anisotropy of their physical and mechanical properties, e.g., elasticity. A type of anisotropy is usually unknown a priori, and possible ways of its determination is of considerable interest both from theoretical and practical viewpoints. In this work, emphasis is placed on the identification of elasticity classes of polycrystalline materials. By the newly introduced concept of “elasticity class” the union of congruent tensor subspaces of a special form is understood. In particular, it makes it possible to consider the so-called symmetry classification, which is widely spread in solid mechanics. The problem of identification of linear elasticity class for anisotropic material with elastic moduli given in an arbitrary orthonormal basis is formulated. To solve this problem, a general procedure based on constructing the hierarchy of approximations of elasticity tensor in different classes is formulated. This approach is then applied to analyze changes in the elastic symmetry of a representative volume element of polycrystalline copper during numerical experiments on severe plastic deformation. The microstructure evolution is described using a two-level crystal elasto-visco-plasticity model. The well-defined structures, which are indicative of the existence of essentially inhomogeneous distribution of crystallite orientations, were obtained in each experiment. However, the texture obtained in the quasi-axial upsetting experiment demonstrates the absence of significant macroscopic elastic anisotropy. Using the identification framework, it has been shown that the elasticity tensor corresponding to the resultant microstructure proves to be almost isotropic. Full article
Open AccessArticle
The Interval Cognitive Network Process for Multi-Attribute Decision-Making
Symmetry 2017, 9(10), 238; doi:10.3390/sym9100238 -
Abstract
Aiming at combining the good characteristics of a differential scale in representing human cognition and the favorable properties of interval judgments in expressing decision-makers’ uncertainty, this paper proposes the interval cognitive network process (I-CNP) to extend the primitive cognition network process (P-CNP) to
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Aiming at combining the good characteristics of a differential scale in representing human cognition and the favorable properties of interval judgments in expressing decision-makers’ uncertainty, this paper proposes the interval cognitive network process (I-CNP) to extend the primitive cognition network process (P-CNP) to handle interval judgments. The key points of I-CNP include a consistency definition for an interval pairwise opposite matrix (IPOM) and a method to derive interval utilities from an IPOM. This paper defines a feasible region-based consistency definition and a transitivity based consistency definition for an IPOM. Both of the two definitions are equivalent to the consistency definition for a crisp pairwise opposite matrix (POM) when an IPOM is reduced to a POM. Two methods that are able to derive interval utilities from both consistent and inconsistent IPOMs are developed based on the two definitions. Four numerical examples are used to illustrate the proposed methods and to compare I-CNP to interval analytic hierarchy process (IAHP). The results show that I-CNP reflects the decision-makers’ cognition better, and that the suggestions provided by I-CNP are more convincing. I-CNP provides new useful alternative tools for multi-attribute decision-making problems. Full article
Open AccessFeature PaperArticle
A Retinal Vessel Detection Approach Based on Shearlet Transform and Indeterminacy Filtering on Fundus Images
Symmetry 2017, 9(10), 235; doi:10.3390/sym9100235 -
Abstract
A fundus image is an effective tool for ophthalmologists studying eye diseases. Retinal vessel detection is a significant task in the identification of retinal disease regions. This study presents a retinal vessel detection approach using shearlet transform and indeterminacy filtering. The fundus image’s
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A fundus image is an effective tool for ophthalmologists studying eye diseases. Retinal vessel detection is a significant task in the identification of retinal disease regions. This study presents a retinal vessel detection approach using shearlet transform and indeterminacy filtering. The fundus image’s green channel is mapped in the neutrosophic domain via shearlet transform. The neutrosophic domain images are then filtered with an indeterminacy filter to reduce the indeterminacy information. A neural network classifier is employed to identify the pixels whose inputs are the features in neutrosophic images. The proposed approach is tested on two datasets, and a receiver operating characteristic curve and the area under the curve are employed to evaluate experimental results quantitatively. The area under the curve values are 0.9476 and 0.9469 for each dataset respectively, and 0.9439 for both datasets. The comparison with the other algorithms also illustrates that the proposed method yields the highest evaluation measurement value and demonstrates the efficiency and accuracy of the proposed method. Full article
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Open AccessArticle
SINR-Based MCS Level Adaptation in CSMA/CA Wireless Networks to Embrace IoT Devices
Symmetry 2017, 9(10), 236; doi:10.3390/sym9100236 -
Abstract
This paper presents an automatic modulation and coding scheme (MCS) level adaptation algorithm to embrace Internet of Things (IoT) devices by improving the area spectral efficiency of carrier-sense multiple access with collision avoidance (CSMA/CA) wireless networks. In the proposed algorithm, senders of CSMA/CA
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This paper presents an automatic modulation and coding scheme (MCS) level adaptation algorithm to embrace Internet of Things (IoT) devices by improving the area spectral efficiency of carrier-sense multiple access with collision avoidance (CSMA/CA) wireless networks. In the proposed algorithm, senders of CSMA/CA wireless networks use the signal to interference plus noise ratio of acknowledgment frames from their receivers to estimate channel statuses between the senders and the receivers. Using the estimated channel status of each receiver, senders control sending rates of traffic by adjusting MCS levels of packets destined for each receiver. We use Poisson point processes (PPPs) to model the locations of participating nodes (i.e., access points and wireless devices) in a given area. We evaluate the effectiveness of the proposed algorithm using an event-driven ns-2 simulator for various PPP densities of access points and wireless devices. Full article
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Open AccessArticle
Internet of THings Area Coverage Analyzer (ITHACA) for Complex Topographical Scenarios
Symmetry 2017, 9(10), 237; doi:10.3390/sym9100237 -
Abstract
The number of connected devices is increasing worldwide. Not only in contexts like the Smart City, but also in rural areas, to provide advanced features like smart farming or smart logistics. Thus, wireless network technologies to efficiently allocate Internet of Things (IoT) and
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The number of connected devices is increasing worldwide. Not only in contexts like the Smart City, but also in rural areas, to provide advanced features like smart farming or smart logistics. Thus, wireless network technologies to efficiently allocate Internet of Things (IoT) and Machine to Machine (M2M) communications are necessary. Traditional cellular networks like Global System for Mobile communications (GSM) are widely used worldwide for IoT environments. Nevertheless, Low Power Wide Area Networks (LP-WAN) are becoming widespread as infrastructure for present and future IoT and M2M applications. Based also on a subscription service, the LP-WAN technology SIGFOXTM may compete with cellular networks in the M2M and IoT communications market, for instance in those projects where deploying the whole communications infrastructure is too complex or expensive. For decision makers to decide the most suitable technology for each specific application, signal coverage is within the key features. Unfortunately, besides simulated coverage maps, decision-makers do not have real coverage maps for SIGFOXTM, as they can be found for cellular networks. Thereby, we propose Internet of THings Area Coverage Analyzer (ITHACA), a signal analyzer prototype to provide automated signal coverage maps and analytics for LP-WAN. Experiments performed in the Gran Canaria Island, Spain (with both urban and complex topographic rural environments), returned a real SIGFOXTM service availability above 97% and above 11% more coverage with respect to the company-provided simulated maps. We expect that ITHACA may help decision makers to deploy the most suitable technologies for future IoT and M2M projects. Full article
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Open AccessFeature PaperArticle
Managing Non-Homogeneous Information and Experts’ Psychological Behavior in Group Emergency Decision Making
Symmetry 2017, 9(10), 234; doi:10.3390/sym9100234 -
Abstract
After an emergency event (EE) happens, emergency decision making (EDM) is a common and effective way to deal with the emergency situation, which plays an important role in mitigating its level of harm. In the real world, it is a big challenge for
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After an emergency event (EE) happens, emergency decision making (EDM) is a common and effective way to deal with the emergency situation, which plays an important role in mitigating its level of harm. In the real world, it is a big challenge for an individual emergency manager (EM) to make a proper and comprehensive decision for coping with an EE. Consequently, many practical EDM problems drive group emergency decision making (GEDM) problems whose main limitations are related to the lack of flexibility in knowledge elicitation, disagreements in the group and the consideration of experts’ psychological behavior in the decision process. Hence, this paper proposes a novel GEDM approach that allows more flexibility for preference elicitation under uncertainty, provides a consensus process to avoid disagreements and considers experts’ psychological behavior by using the fuzzy TODIM method based on prospect theory. Eventually, a group decision support system (GDSS) is developed to support the whole GEDM process defined in the proposed method demonstrating its novelty, validity and feasibility. Full article
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Open AccessArticle
How to Obtain Common Criteria Certification of Smart TV for Home IoT Security and Reliability
Symmetry 2017, 9(10), 233; doi:10.3390/sym9100233 -
Abstract
In the new era of IoT (Internet of Things), numerous gadgets and services include innovative IoT technologies that provide customers with convenience and improve their quality of life. Smart TVs are typical IoT devices that offer broadcasting services. However, they are susceptible to
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In the new era of IoT (Internet of Things), numerous gadgets and services include innovative IoT technologies that provide customers with convenience and improve their quality of life. Smart TVs are typical IoT devices that offer broadcasting services. However, they are susceptible to security intrusions via e-mail, media players, cameras, and internet connectivity. The frequency of hacking through malicious applications installed in Smart TV has rapidly increased. Therefore, appropriate countermeasures should be developed immediately. In April 2017, we (with LG electronics) received the ‘world-first’ Common Criteria EAL2 (Evaluation Assurance Level 2) certification for Smart TVs. As far as we know, no Smart TV has received a Common Criteria EAL2 security certification until now. This article describes our experience with the certification process and examines several security and reliability aspects of Smart TVs. Full article
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Open AccessArticle
Multiple Signal Classification Algorithm Based Electric Dipole Source Localization Method in an Underwater Environment
Symmetry 2017, 9(10), 231; doi:10.3390/sym9100231 -
Abstract
A novel localization method based on multiple signal classification (MUSIC) algorithm is proposed for positioning an electric dipole source in a confined underwater environment by using electric dipole-receiving antenna array. In this method, the boundary element method (BEM) is introduced to analyze the
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A novel localization method based on multiple signal classification (MUSIC) algorithm is proposed for positioning an electric dipole source in a confined underwater environment by using electric dipole-receiving antenna array. In this method, the boundary element method (BEM) is introduced to analyze the boundary of the confined region by use of a matrix equation. The voltage of each dipole pair is used as spatial-temporal localization data, and it does not need to obtain the field component in each direction compared with the conventional fields based localization method, which can be easily implemented in practical engineering applications. Then, a global-multiple region-conjugate gradient (CG) hybrid search method is used to reduce the computation burden and to improve the operation speed. Two localization simulation models and a physical experiment are conducted. Both the simulation results and physical experiment result provide accurate positioning performance, with the help to verify the effectiveness of the proposed localization method in underwater environments. Full article
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Open AccessArticle
Open Gromov-Witten Invariants from the Augmentation Polynomial
Symmetry 2017, 9(10), 232; doi:10.3390/sym9100232 -
Abstract
A conjecture of Aganagic and Vafa relates the open Gromov-Witten theory of X=OP1(1,1) to the augmentation polynomial of Legendrian contact homology. We describe how to use this conjecture to compute genus zero,
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A conjecture of Aganagic and Vafa relates the open Gromov-Witten theory of X=OP1(1,1) to the augmentation polynomial of Legendrian contact homology. We describe how to use this conjecture to compute genus zero, one boundary component open Gromov-Witten invariants for Lagrangian submanifolds LKX obtained from the conormal bundles of knots KS3. This computation is then performed for two non-toric examples (the figure-eight and three-twist knots). For (r,s) torus knots, the open Gromov-Witten invariants can also be computed using Atiyah-Bott localization. Using this result for the unknot and the (3,2) torus knot, we show that the augmentation polynomial can be derived from these open Gromov-Witten invariants. Full article
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Open AccessArticle
Gravity Theories with Background Fields and Spacetime Symmetry Breaking
Symmetry 2017, 9(10), 230; doi:10.3390/sym9100230 -
Abstract
An overview is given of effective gravitational field theories with fixed background fields that break spacetime symmetry. The behavior of the background fields and the types of excitations that can occur depend on whether the symmetry breaking is explicit or spontaneous. For example,
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An overview is given of effective gravitational field theories with fixed background fields that break spacetime symmetry. The behavior of the background fields and the types of excitations that can occur depend on whether the symmetry breaking is explicit or spontaneous. For example, when the breaking is spontaneous, the background field is dynamical and massless Nambu–Goldstone and massive Higgs excitations can appear. However, if the breaking is explicit, the background is nondynamical, and in this case additional metric or vierbein excitations occur due to the loss of local symmetry, or these excitations can be replaced by dynamical scalar fields using a Stückelberg approach. The interpretation of Noether identities that must hold in each case differs, depending on the type of symmetry breaking, and this affects the nature of the consistency conditions that must hold. The Noether identities also shed light on why the Stückelberg approach works, and how it is able to restore the broken spacetime symmetry in a theory with explicit breaking. Full article
Open AccessArticle
A General Zero Attraction Proportionate Normalized Maximum Correntropy Criterion Algorithm for Sparse System Identification
Symmetry 2017, 9(10), 229; doi:10.3390/sym9100229 -
Abstract
A general zero attraction (GZA) proportionate normalized maximum correntropy criterion (GZA-PNMCC) algorithm is devised and presented on the basis of the proportionate-type adaptive filter techniques and zero attracting theory to highly improve the sparse system estimation behavior of the classical MCC algorithm within
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A general zero attraction (GZA) proportionate normalized maximum correntropy criterion (GZA-PNMCC) algorithm is devised and presented on the basis of the proportionate-type adaptive filter techniques and zero attracting theory to highly improve the sparse system estimation behavior of the classical MCC algorithm within the framework of the sparse system identifications. The newly-developed GZA-PNMCC algorithm is carried out by introducing a parameter adjusting function into the cost function of the typical proportionate normalized maximum correntropy criterion (PNMCC) to create a zero attraction term. The developed optimization framework unifies the derivation of the zero attraction-based PNMCC algorithms. The developed GZA-PNMCC algorithm further exploits the impulsive response sparsity in comparison with the proportionate-type-based NMCC algorithm due to the GZA zero attraction. The superior performance of the GZA-PNMCC algorithm for estimating a sparse system in a non-Gaussian noise environment is proven by simulations. Full article
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Open AccessArticle
Bounded Solutions to Nonhomogeneous Linear Second-Order Difference Equations
Symmetry 2017, 9(10), 227; doi:10.3390/sym9100227 -
Abstract
By using some solvability methods and the contraction mapping principle are investigated bounded, as well as periodic solutions to some classes of nonhomogeneous linear second-order difference equations on domains N0, ZN2 and Z. The case when the
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By using some solvability methods and the contraction mapping principle are investigated bounded, as well as periodic solutions to some classes of nonhomogeneous linear second-order difference equations on domains N0, ZN2 and Z. The case when the coefficients of the equation are constant and the zeros of the characteristic polynomial associated to the corresponding homogeneous equation do not belong to the unit circle is described in detail. Full article
Open AccessArticle
Decomposition and Intersection of Two Fuzzy Numbers for Fuzzy Preference Relations
Symmetry 2017, 9(10), 228; doi:10.3390/sym9100228 -
Abstract
In fuzzy decision problems, the ordering of fuzzy numbers is the basic problem. The fuzzy preference relation is the reasonable representation of preference relations by a fuzzy membership function. This paper studies Nakamura’s and Kołodziejczyk’s preference relations. Eight cases, each representing different levels
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In fuzzy decision problems, the ordering of fuzzy numbers is the basic problem. The fuzzy preference relation is the reasonable representation of preference relations by a fuzzy membership function. This paper studies Nakamura’s and Kołodziejczyk’s preference relations. Eight cases, each representing different levels of overlap between two triangular fuzzy numbers are considered. We analyze the ranking behaviors of all possible combinations of the decomposition and intersection of two fuzzy numbers through eight extensive test cases. The results indicate that decomposition and intersection can affect the fuzzy preference relations, and thereby the final ranking of fuzzy numbers. Full article
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Open AccessFeature PaperArticle
Skein Invariants of Links and Their State Sum Models
Symmetry 2017, 9(10), 226; doi:10.3390/sym9100226 -
Abstract
We present the new skein invariants of classical links, H[H], K[K] and D[D], based on the invariants of links, H, K and D, denoting the regular isotopy version of the
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We present the new skein invariants of classical links, H[H], K[K] and D[D], based on the invariants of links, H, K and D, denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. The invariants are obtained by abstracting the skein relation of the corresponding invariant and making a new skein algorithm comprising two computational levels: first producing unlinked knotted components, then evaluating the resulting knots. The invariants in this paper, were revealed through the skein theoretic definition of the invariants Θd related to the Yokonuma–Hecke algebras and their 3-variable generalization Θ, which generalizes the Homflypt polynomial. H[H] is the regular isotopy counterpart of Θ. The invariants K[K] and D[D] are new generalizations of the Kauffman and the Dubrovnik polynomials. We sketch skein theoretic proofs of the well-definedness and topological properties of these invariants. The invariants of this paper are reformulated into summations of the generating invariants (H, K, D) on sublinks of the given link L, obtained by partitioning L into collections of sublinks. The first such reformulation was achieved by W.B.R. Lickorish for the invariant Θ and we generalize it to the Kauffman and Dubrovnik polynomial cases. State sum models are formulated for all the invariants. These state summation models are based on our skein template algorithm which formalizes the skein theoretic process as an analogue of a statistical mechanics partition function. Relationships with statistical mechanics models are articulated. Finally, we discuss physical situations where a multi-leveled course of action is taken naturally. Full article
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Open AccessArticle
Asymmetric Equivalences in Fuzzy Logic
Symmetry 2017, 9(10), 224; doi:10.3390/sym9100224 -
Abstract
We introduce a new class of operations called asymmetric equivalences. Several properties of asymmetric equivalence operations have been investigated. Based on the asymmetric equivalence, quasi-metric spaces are constructed on [0, 1]. Finally, we discuss symmetrization of asymmetric equivalences. Full article
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