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Symmetry, Volume 10, Issue 10 (October 2018)

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Open AccessArticle Feedforward Neural Networks with a Hidden Layer Regularization Method
Symmetry 2018, 10(10), 525; https://doi.org/10.3390/sym10100525
Received: 3 September 2018 / Revised: 13 October 2018 / Accepted: 14 October 2018 / Published: 19 October 2018
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Abstract
In this paper, we propose a group Lasso regularization term as a hidden layer regularization method for feedforward neural networks. Adding a group Lasso regularization term into the standard error function as a hidden layer regularization term is a fruitful approach to eliminate
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In this paper, we propose a group Lasso regularization term as a hidden layer regularization method for feedforward neural networks. Adding a group Lasso regularization term into the standard error function as a hidden layer regularization term is a fruitful approach to eliminate the redundant or unnecessary hidden layer neurons from the feedforward neural network structure. As a comparison, a popular Lasso regularization method is introduced into standard error function of the network. Our novel hidden layer regularization method can force a group of outgoing weights to become smaller during the training process and can eventually be removed after the training process. This means it can simplify the neural network structure and it minimizes the computational cost. Numerical simulations are provided by using K-fold cross-validation method with K = 5 to avoid overtraining and to select the best learning parameters. The numerical results show that our proposed hidden layer regularization method prunes more redundant hidden layer neurons consistently for each benchmark dataset without loss of accuracy. In contrast, the existing Lasso regularization method prunes only the redundant weights of the network, but it cannot prune any redundant hidden layer neurons. Full article
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Open AccessArticle Transition from the Wave Equation to Either the Heat or the Transport Equations through Fractional Differential Expressions
Symmetry 2018, 10(10), 524; https://doi.org/10.3390/sym10100524
Received: 29 September 2018 / Revised: 6 October 2018 / Accepted: 12 October 2018 / Published: 19 October 2018
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Abstract
We present a model that intermediates among the wave, heat, and transport equations. The approach considers the propagation of initial disturbances in a one-dimensional medium that can vibrate. The medium is nonlinear in such a form that nonlocal differential expressions are required to
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We present a model that intermediates among the wave, heat, and transport equations. The approach considers the propagation of initial disturbances in a one-dimensional medium that can vibrate. The medium is nonlinear in such a form that nonlocal differential expressions are required to describe the time evolution of solutions. Nonlocality was modeled with a space-time fractional differential equation of order 1 α 2 in time, and order 1 β 2 in space. We adopted the notion of Caputo for the time derivative and the Riesz pseudo-differential operator for the space derivative. The corresponding Cauchy problem was solved for zero initial velocity and initial disturbance, represented by either the Dirac delta or the Gaussian distributions. Well-known results for the conventional partial differential equations of wave propagation, diffusion, and (modified) transport processes were recovered as particular cases. In addition, regular solutions were found for the partial differential equation that arises from α = 2 and β = 1 . Unlike the above conventional cases, the latter equation permits the presence of nodes in its solutions. Full article
(This article belongs to the Special Issue Symmetry Breaking in Quantum Phenomena)
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Open AccessArticle Study on an Automatic Parking Method Based on the Sliding Mode Variable Structure and Fuzzy Logical Control
Symmetry 2018, 10(10), 523; https://doi.org/10.3390/sym10100523
Received: 29 September 2018 / Revised: 13 October 2018 / Accepted: 16 October 2018 / Published: 19 October 2018
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Abstract
This paper discusses an automatic parking control method based on the combination of the sliding mode variable structure control (SMVSC) and fuzzy logical control. SMVSC is applied to drive the vehicle from a random initial position and pose, to the designated parking position
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This paper discusses an automatic parking control method based on the combination of the sliding mode variable structure control (SMVSC) and fuzzy logical control. SMVSC is applied to drive the vehicle from a random initial position and pose, to the designated parking position and pose. Then, the vehicle is driven from the designated parking position to the target parking slot using the method of fuzzy logical control, whose rules are limited to the range of the effective initial position. To combine SMVSC with the fuzzy logical control, the experimental results demonstrate that effective parking can be guaranteed, even if the initial position is out of the effective parking area of the fuzzy logical control. Full article
(This article belongs to the Special Issue Fuzzy Techniques for Decision Making 2018)
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Open AccessArticle On the Existence of the Solutions of a Fredholm Integral Equation with a Modified Argument in Hölder Spaces
Symmetry 2018, 10(10), 522; https://doi.org/10.3390/sym10100522
Received: 29 September 2018 / Revised: 14 October 2018 / Accepted: 16 October 2018 / Published: 19 October 2018
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Abstract
This article concerns the entity of solutions of a quadratic integral equation of the Fredholm type with an altered argument, x(t)=p(t)+x(t)01k(t,τ)
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This article concerns the entity of solutions of a quadratic integral equation of the Fredholm type with an altered argument, x ( t ) = p ( t ) + x ( t ) 0 1 k ( t , τ ) ( T x ) ( τ ) d τ , where p , k are given functions, T is the given operator satisfying conditions specified later and x is an unknown function. Through the classical Schauder fixed point theorem and a new conclusion about the relative compactness in Hölder spaces, we obtain the existence of solutions under certain assumptions. Our work is more general than the previous works in the Conclusion section. At the end, we introduce several tangible examples where our entity result can be adopted. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
Open AccessArticle Underdetermined Blind Source Separation Combining Tensor Decomposition and Nonnegative Matrix Factorization
Symmetry 2018, 10(10), 521; https://doi.org/10.3390/sym10100521
Received: 25 September 2018 / Revised: 14 October 2018 / Accepted: 16 October 2018 / Published: 18 October 2018
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Abstract
Underdetermined blind source separation (UBSS) is a hot topic in signal processing, which aims at recovering the source signals from a number of observed mixtures without knowing the mixing system. Recently, expectation-maximization algorithm shows a great potential in the UBSS. However, the final
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Underdetermined blind source separation (UBSS) is a hot topic in signal processing, which aims at recovering the source signals from a number of observed mixtures without knowing the mixing system. Recently, expectation-maximization algorithm shows a great potential in the UBSS. However, the final separation results depend strongly on the parameter initialization, leading to poor separation performance. In this paper, we propose an effective algorithm that combines tensor decomposition and nonnegative matrix factorization (NMF). In the proposed algorithm, we first employ tensor decomposition to estimate the mixing matrix, and NMF source model is used to estimate the source spectrogram factors. Then a series of iterations are derived to update the model parameters. At the same time, the spatial images of source signals are estimated with Wiener filters constructed from the learned parameters. Therefore, time-domain sources can be obtained through inverse short-time Fourier transform. Finally, plenty of experimental results demonstrate the effectiveness and advantages of our proposed algorithm over the compared algorithms. Full article
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Open AccessFeature PaperArticle Dark Matter as a Non-Relativistic Bose–Einstein Condensate with Massive Gravitons
Symmetry 2018, 10(10), 520; https://doi.org/10.3390/sym10100520
Received: 15 September 2018 / Revised: 13 October 2018 / Accepted: 15 October 2018 / Published: 17 October 2018
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Abstract
We confront a non-relativistic Bose–Einstein Condensate (BEC) model of light bosons interacting gravitationally either through a Newtonian or a Yukawa potential with the observed rotational curves of 12 dwarf galaxies. The baryonic component is modeled as an axisymmetric exponential disk and its characteristics
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We confront a non-relativistic Bose–Einstein Condensate (BEC) model of light bosons interacting gravitationally either through a Newtonian or a Yukawa potential with the observed rotational curves of 12 dwarf galaxies. The baryonic component is modeled as an axisymmetric exponential disk and its characteristics are derived from the surface luminosity profile of the galaxies. The purely baryonic fit is unsatisfactory, hence a dark matter component is clearly needed. The rotational curves of five galaxies could be explained with high confidence level by the BEC model. For these galaxies, we derive: (i) upper limits for the allowed graviton mass; and (ii) constraints on a velocity-type and a density-type quantity characterizing the BEC, both being expressed in terms of the BEC particle mass, scattering length and chemical potential. The upper limit for the graviton mass is of the order of 10 26 eV / c 2 , three orders of magnitude stronger than the limit derived from recent gravitational wave detections. Full article
(This article belongs to the Special Issue Cosmological Inflation, Dark Matter and Dark Energy)
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Open AccessArticle Synthetic Medical Images Using F&BGAN for Improved Lung Nodules Classification by Multi-Scale VGG16
Symmetry 2018, 10(10), 519; https://doi.org/10.3390/sym10100519
Received: 28 September 2018 / Revised: 15 October 2018 / Accepted: 16 October 2018 / Published: 17 October 2018
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Abstract
Lung cancer is one of the highest causes of cancer-related death in both men and women. Therefore, various diagnostic methods for lung nodules classification have been proposed to implement the early detection. Due to the limited amount and diversity of samples, these methods
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Lung cancer is one of the highest causes of cancer-related death in both men and women. Therefore, various diagnostic methods for lung nodules classification have been proposed to implement the early detection. Due to the limited amount and diversity of samples, these methods encounter some bottlenecks. In this paper, we intend to develop a method to enlarge the dataset and enhance the performance of pulmonary nodules classification. We propose a data augmentation method based on generative adversarial network (GAN), called Forward and Backward GAN (F&BGAN), which can generate high-quality synthetic medical images. F&BGAN has two stages, Forward GAN (FGAN) generates diverse images, and Backward GAN (BGAN) is used to improve the quality of images. Besides, a hierarchical learning framework, multi-scale VGG16 (M-VGG16) network, is proposed to extract discriminative features from alternating stacked layers. The methodology was evaluated on the Lung Image Database Consortium and Image Database Resource Initiative (LIDC-IDRI) dataset, with the best accuracy of 95.24%, sensitivity of 98.67%, specificity of 92.47% and area under ROC curve (AUROC) of 0.980. Experimental results demonstrate the feasibility of F&BGAN in generating medical images and the effectiveness of M-VGG16 in classifying malignant and benign nodules. Full article
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Open AccessFeature PaperReview Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths
Symmetry 2018, 10(10), 518; https://doi.org/10.3390/sym10100518
Received: 13 September 2018 / Revised: 11 October 2018 / Accepted: 12 October 2018 / Published: 16 October 2018
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Abstract
Many open quantum systems encountered in both natural and synthetic situations are embedded in classical-like baths. Often, the bath degrees of freedom may be represented in terms of canonically conjugate coordinates, but in some cases they may require a non-canonical or non-Hamiltonian representation.
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Many open quantum systems encountered in both natural and synthetic situations are embedded in classical-like baths. Often, the bath degrees of freedom may be represented in terms of canonically conjugate coordinates, but in some cases they may require a non-canonical or non-Hamiltonian representation. Herein, we review an approach to the dynamics and statistical mechanics of quantum subsystems embedded in either non-canonical or non-Hamiltonian classical-like baths which is based on operator-valued quasi-probability functions. These functions typically evolve through the action of quasi-Lie brackets and their associated Quantum-Classical Liouville Equations, or through quasi-Lie brackets augmented by dissipative terms. Quasi-Lie brackets possess the unique feature that, while conserving the energy (which the Noether theorem links to time-translation symmetry), they violate the time-translation symmetry of their algebra. This fact can be heuristically understood in terms of the dynamics of the open quantum subsystem. We then describe an example in which a quantum subsystem is embedded in a bath of classical spins, which are described by non-canonical coordinates. In this case, it has been shown that an off-diagonal open-bath geometric phase enters into the propagation of the quantum-classical dynamics. Next, we discuss how non-Hamiltonian dynamics may be employed to generate the constant-temperature evolution of phase space degrees of freedom coupled to the quantum subsystem. Constant-temperature dynamics may be generated by either a classical Langevin stochastic process or a Nosé–Hoover deterministic thermostat. These two approaches are not equivalent but have different advantages and drawbacks. In all cases, the calculation of the operator-valued quasi-probability function allows one to compute time-dependent statistical averages of observables. This may be accomplished in practice using a hybrid Molecular Dynamics/Monte Carlo algorithms, which we outline herein. Full article
(This article belongs to the Special Issue New Trends in Quantum Electrodynamics)
Open AccessArticle Harnack Inequality and No-Arbitrage Analysis
Symmetry 2018, 10(10), 517; https://doi.org/10.3390/sym10100517
Received: 20 September 2018 / Revised: 20 September 2018 / Accepted: 15 October 2018 / Published: 16 October 2018
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Abstract
The present paper attains a Harnack inequality for the option pricing (or Kolmogorov) equation with gradient estimate arguments. We then perform a no-arbitrage analysis of a financial market. Full article
Open AccessArticle Inconsistent Data Cleaning Based on the Maximum Dependency Set and Attribute Correlation
Symmetry 2018, 10(10), 516; https://doi.org/10.3390/sym10100516
Received: 11 September 2018 / Revised: 12 October 2018 / Accepted: 12 October 2018 / Published: 16 October 2018
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Abstract
In banks, governments, and Internet companies, inconsistent data problems may often arise when various information systems are collecting, processing, and updating data due to human or equipment reasons. The emergence of inconsistent data makes it impossible to obtain correct information from the data
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In banks, governments, and Internet companies, inconsistent data problems may often arise when various information systems are collecting, processing, and updating data due to human or equipment reasons. The emergence of inconsistent data makes it impossible to obtain correct information from the data and reduces its availability. Such problems may be fatal in data-intensive enterprises, which causes huge economic losses. Moreover, it is very difficult to clean inconsistent data in databases, especially for data containing conditional functional dependencies with built-in predicates (CFDPs), because it tends to contain more candidate repair values. For the inconsistent data containing CFDPs to detect incomplete and repair difficult problems in databases, we propose a dependency lifting algorithm (DLA) based on the maximum dependency set (MDS) and a reparation algorithm (C-Repair) based on integrating the minimum cost and attribute correlation, respectively. In detection, we find recessive dependencies from the original dependency set to obtain the MDS and improve the original algorithm by dynamic domain adjustment, which extends the applicability to continuous attributes and improves the detection accuracy. In reparation, we first set up a priority queue (PQ) for elements to be repaired based on the minimum cost idea to select a candidate element; then, we treat the corresponding conflict-free instance ( I n v ) as the training set to learn the correlation among attributes and compute the weighted distance (WDis) between the tuple of the candidate element and other tuples in I n v according to the correlation; and, lastly, we perform reparation based on the WDis and re-compute the PQ after each reparation round to improve the efficiency, and use a label, flag, to mark the repaired elements to ensure the convergence at the same time. By setting up a contrast experiment, we compare the DLA with the CFDPs based algorithm, and the C-Repair with a cost-based, interpolation-based algorithm on a simulated instance and a real instance. From the experimental results, the DLA and C-Repair algorithms have better detection and repair ability at a higher time cost. Full article
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Open AccessArticle Fuzzy Normed Rings
Symmetry 2018, 10(10), 515; https://doi.org/10.3390/sym10100515
Received: 12 September 2018 / Revised: 6 October 2018 / Accepted: 8 October 2018 / Published: 16 October 2018
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Abstract
In this paper, the concept of fuzzy normed ring is introduced and some basic properties related to it are established. Our definition of normed rings on fuzzy sets leads to a new structure, which we call a fuzzy normed ring. We define fuzzy
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In this paper, the concept of fuzzy normed ring is introduced and some basic properties related to it are established. Our definition of normed rings on fuzzy sets leads to a new structure, which we call a fuzzy normed ring. We define fuzzy normed ring homomorphism, fuzzy normed subring, fuzzy normed ideal, fuzzy normed prime ideal, and fuzzy normed maximal ideal of a normed ring, respectively. We show some algebraic properties of normed ring theory on fuzzy sets, prove theorems, and give relevant examples. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry)
Open AccessArticle Extension of Eigenvalue Problems on Gauss Map of Ruled Surfaces
Symmetry 2018, 10(10), 514; https://doi.org/10.3390/sym10100514
Received: 20 September 2018 / Revised: 9 October 2018 / Accepted: 12 October 2018 / Published: 16 October 2018
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Abstract
A finite-type immersion or smooth map is a nice tool to classify submanifolds of Euclidean space, which comes from the eigenvalue problem of immersion. The notion of generalized 1-type is a natural generalization of 1-type in the usual sense and pointwise 1-type. We
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A finite-type immersion or smooth map is a nice tool to classify submanifolds of Euclidean space, which comes from the eigenvalue problem of immersion. The notion of generalized 1-type is a natural generalization of 1-type in the usual sense and pointwise 1-type. We classify ruled surfaces with a generalized 1-type Gauss map as part of a plane, a circular cylinder, a cylinder over a base curve of an infinite type, a helicoid, a right cone and a conical surface of G-type. Full article
Open AccessArticle The Quaternionic Commutator Bracket and Its Implications
Symmetry 2018, 10(10), 513; https://doi.org/10.3390/sym10100513
Received: 11 August 2018 / Revised: 30 September 2018 / Accepted: 9 October 2018 / Published: 16 October 2018
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Abstract
A quaternionic commutator bracket for position and momentum shows that the quaternionic wave function, viz. ψ˜=(icψ0,ψ), represents a state of a particle with orbital angular momentum, L=3,
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A quaternionic commutator bracket for position and momentum shows that the quaternionic wave function, viz. ψ ˜ = ( i c ψ 0 , ψ ) , represents a state of a particle with orbital angular momentum, L = 3 , resulting from the internal structure of the particle. This angular momentum can be attributed to spin of the particle. The vector ψ , points in an opposite direction of L . When a charged particle is placed in an electromagnetic field, the interaction energy reveals that the magnetic moments interact with the electric and magnetic fields giving rise to terms similar to Aharonov–Bohm and Aharonov–Casher effects. Full article
Open AccessArticle A New Approach to the Solution of the Fredholm Integral Equation via a Fixed Point on Extended b-Metric Spaces
Symmetry 2018, 10(10), 512; https://doi.org/10.3390/sym10100512
Received: 7 September 2018 / Revised: 27 September 2018 / Accepted: 13 October 2018 / Published: 16 October 2018
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Abstract
It is very well known that real-life applications of fixed point theory are restricted with the transformation of the problem in the form of f(x)=x. (1) The Knaster–Tarski fixed point theorem underlies various approaches of checking the
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It is very well known that real-life applications of fixed point theory are restricted with the transformation of the problem in the form of f ( x ) = x . (1) The Knaster–Tarski fixed point theorem underlies various approaches of checking the correctness of programs. (2) The Brouwer fixed point theorem is used to prove the existence of Nash equilibria in games. (3) Dlala et al. proposed a solution for magnetic field problems via the fixed point approach. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
Open AccessFeature PaperArticle Cut-and-Project Schemes for Pisot Family Substitution Tilings
Symmetry 2018, 10(10), 511; https://doi.org/10.3390/sym10100511
Received: 19 September 2018 / Revised: 9 October 2018 / Accepted: 11 October 2018 / Published: 16 October 2018
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Abstract
We consider Pisot family substitution tilings in Rd whose dynamical spectrum is pure point. There are two cut-and-project schemes (CPSs) which arise naturally: one from the Pisot family property and the other from the pure point spectrum. The first CPS has an
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We consider Pisot family substitution tilings in R d whose dynamical spectrum is pure point. There are two cut-and-project schemes (CPSs) which arise naturally: one from the Pisot family property and the other from the pure point spectrum. The first CPS has an internal space R m for some integer m N defined from the Pisot family property, and the second CPS has an internal space H that is an abstract space defined from the condition of the pure point spectrum. However, it is not known how these two CPSs are related. Here we provide a sufficient condition to make a connection between the two CPSs. For Pisot unimodular substitution tiling in R , the two CPSs turn out to be same due to the remark by Barge-Kwapisz. Full article
Open AccessArticle New Soft Set Based Class of Linear Algebraic Codes
Symmetry 2018, 10(10), 510; https://doi.org/10.3390/sym10100510
Received: 17 September 2018 / Revised: 12 October 2018 / Accepted: 15 October 2018 / Published: 16 October 2018
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Abstract
In this paper, we design and develop a new class of linear algebraic codes defined as soft linear algebraic codes using soft sets. The advantage of using these codes is that they have the ability to transmit m-distinct messages to m-set of receivers
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In this paper, we design and develop a new class of linear algebraic codes defined as soft linear algebraic codes using soft sets. The advantage of using these codes is that they have the ability to transmit m-distinct messages to m-set of receivers simultaneously. The methods of generating and decoding these new classes of soft linear algebraic codes have been developed. The notion of soft canonical generator matrix, soft canonical parity check matrix, and soft syndrome are defined to aid in construction and decoding of these codes. Error detection and correction of these codes are developed and illustrated by an example. Full article
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Open AccessArticle Fibonacci and Lucas Numbers of the Form 2a + 3b + 5c + 7d
Symmetry 2018, 10(10), 509; https://doi.org/10.3390/sym10100509
Received: 22 September 2018 / Revised: 14 October 2018 / Accepted: 14 October 2018 / Published: 16 October 2018
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Abstract
In this paper, we find all Fibonacci and Lucas numbers written in the form 2a+3b+5c+7d, in non-negative integers a,b,c,d, with 0max{a
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In this paper, we find all Fibonacci and Lucas numbers written in the form 2 a + 3 b + 5 c + 7 d , in non-negative integers a , b , c , d , with 0 max { a , b , c } d . Full article
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with their Applications)
Open AccessArticle A Class of Nonlinear Boundary Value Problems for an Arbitrary Fractional-Order Differential Equation with the Riemann-Stieltjes Functional Integral and Infinite-Point Boundary Conditions
Symmetry 2018, 10(10), 508; https://doi.org/10.3390/sym10100508
Received: 7 September 2018 / Revised: 12 October 2018 / Accepted: 14 October 2018 / Published: 16 October 2018
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Abstract
In this paper, we investigate the existence of an absolute continuous solution to a class of first-order nonlinear differential equation with integral boundary conditions (BCs) or with infinite-point BCs. The Liouville-Caputo fractional derivative is involved in the nonlinear function. We first consider the
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In this paper, we investigate the existence of an absolute continuous solution to a class of first-order nonlinear differential equation with integral boundary conditions (BCs) or with infinite-point BCs. The Liouville-Caputo fractional derivative is involved in the nonlinear function. We first consider the existence of a solution for the first-order nonlinear differential equation with m-point nonlocal BCs. The existence of solutions of our problems is investigated by applying the properties of the Riemann sum for continuous functions. Several examples are given in order to illustrate our results. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
Open AccessArticle A Low Rank Channel Estimation Scheme in Massive Multiple-Input Multiple-Output
Symmetry 2018, 10(10), 507; https://doi.org/10.3390/sym10100507
Received: 21 September 2018 / Revised: 7 October 2018 / Accepted: 12 October 2018 / Published: 16 October 2018
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Abstract
Aiming at the problem of computational complexity of channel estimation, this paper proposes a low-complexity block matching pursuit (BMP) algorithm based on antenna grouping and block sparsity for frequency division duplex (FDD) massive Multiple-input Multiple-output orthogonal frequency division multiplexing (OFDM) systems. The system
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Aiming at the problem of computational complexity of channel estimation, this paper proposes a low-complexity block matching pursuit (BMP) algorithm based on antenna grouping and block sparsity for frequency division duplex (FDD) massive Multiple-input Multiple-output orthogonal frequency division multiplexing (OFDM) systems. The system coherence time may be exceeded as a result of time consumption when adopting an orthogonal pilot symbol in the time domain. To solve this problem, an antenna grouping transmission scheme is proposed to reduce the total channel estimation time by sacrificing the observed data length. The simulation results show that the proposed BMP algorithm has good anti-noise performance, and it can accurately determine the non-zero position of the sparse vector and adaptively determine the sparsity of the channel, which effectively translates to improved channel estimation performance and better overall system performance than the existing algorithms. Full article
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Open AccessArticle Nekhoroshev Stability for the Dirichlet Toda Lattice
Symmetry 2018, 10(10), 506; https://doi.org/10.3390/sym10100506
Received: 3 September 2018 / Revised: 8 October 2018 / Accepted: 11 October 2018 / Published: 16 October 2018
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Abstract
In this work, we prove a Nekhoroshev-type stability theorem for the Toda lattice with Dirichlet boundary conditions, i.e., with fixed ends. The Toda lattice is a member of the family of Fermi-Pasta-Ulam (FPU) chains, and in view of the unexpected recurrence phenomena numerically
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In this work, we prove a Nekhoroshev-type stability theorem for the Toda lattice with Dirichlet boundary conditions, i.e., with fixed ends. The Toda lattice is a member of the family of Fermi-Pasta-Ulam (FPU) chains, and in view of the unexpected recurrence phenomena numerically observed in these chains, it has been a long-standing research aim to apply the theory of perturbed integrable systems to these chains, in particular to the Toda lattice which has been shown to be a completely integrable system. The Dirichlet Toda lattice can be treated mathematically by using symmetries of the periodic Toda lattice. Precisely, by treating the phase space of the former system as an invariant subset of the latter one, namely as the fixed point set of an important symmetry of the periodic lattice, the results already obtained for the periodic lattice can be used to obtain analogous results for the Dirichlet lattice. In this way, we transfer our stability results for the periodic lattice to the Dirichlet lattice. The Nekhoroshev theorem is a perturbation theory result which does not have the probabilistic character of related theorems, and the lattice with fixed ends is more important for applications than the periodic one. Full article
Open AccessArticle Pythagorean Fuzzy Hamy Mean Operators in Multiple Attribute Group Decision Making and Their Application to Supplier Selection
Symmetry 2018, 10(10), 505; https://doi.org/10.3390/sym10100505
Received: 26 September 2018 / Revised: 5 October 2018 / Accepted: 10 October 2018 / Published: 15 October 2018
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Abstract
In this paper, we extend the Hamy mean (HM) operator and dual Hamy mean (DHM) operator with Pythagorean fuzzy numbers (PFNs) to propose Pythagorean fuzzy Hamy mean (PFHM) operator, weighted Pythagorean fuzzy Hamy mean (WPFHM) operator, Pythagorean fuzzy dual Hamy mean (PFDHM) operator,
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In this paper, we extend the Hamy mean (HM) operator and dual Hamy mean (DHM) operator with Pythagorean fuzzy numbers (PFNs) to propose Pythagorean fuzzy Hamy mean (PFHM) operator, weighted Pythagorean fuzzy Hamy mean (WPFHM) operator, Pythagorean fuzzy dual Hamy mean (PFDHM) operator, weighted Pythagorean fuzzy dual Hamy mean (WPFDHM) operator. Then the multiple attribute group decision making (MAGDM) methods are proposed with these operators. In the end, we utilize an applicable example for supplier selection to prove the proposed methods. Full article
(This article belongs to the Special Issue Multi-Criteria Decision Aid methods in fuzzy decision problems)
Open AccessArticle Invariant Graph Partition Comparison Measures
Symmetry 2018, 10(10), 504; https://doi.org/10.3390/sym10100504
Received: 10 September 2018 / Revised: 10 October 2018 / Accepted: 11 October 2018 / Published: 15 October 2018
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Abstract
Symmetric graphs have non-trivial automorphism groups. This article starts with the proof that all partition comparison measures we have found in the literature fail on symmetric graphs, because they are not invariant with regard to the graph automorphisms. By the construction of a
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Symmetric graphs have non-trivial automorphism groups. This article starts with the proof that all partition comparison measures we have found in the literature fail on symmetric graphs, because they are not invariant with regard to the graph automorphisms. By the construction of a pseudometric space of equivalence classes of permutations and with Hausdorff’s and von Neumann’s methods of constructing invariant measures on the space of equivalence classes, we design three different families of invariant measures, and we present two types of invariance proofs. Last, but not least, we provide algorithms for computing invariant partition comparison measures as pseudometrics on the partition space. When combining an invariant partition comparison measure with its classical counterpart, the decomposition of the measure into a structural difference and a difference contributed by the group automorphism is derived. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry)
Open AccessArticle Solution of Fractional Differential Equation Systems and Computation of Matrix Mittag–Leffler Functions
Symmetry 2018, 10(10), 503; https://doi.org/10.3390/sym10100503
Received: 15 September 2018 / Revised: 3 October 2018 / Accepted: 11 October 2018 / Published: 15 October 2018
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Abstract
In this paper, solutions for systems of linear fractional differential equations are considered. For the commensurate order case, solutions in terms of matrix Mittag–Leffler functions were derived by the Picard iterative process. For the incommensurate order case, the system was converted to a
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In this paper, solutions for systems of linear fractional differential equations are considered. For the commensurate order case, solutions in terms of matrix Mittag–Leffler functions were derived by the Picard iterative process. For the incommensurate order case, the system was converted to a commensurate order case by newly introducing unknown functions. Computation of matrix Mittag–Leffler functions was considered using the methods of the Jordan canonical matrix and minimal polynomial or eigenpolynomial, respectively. Finally, numerical examples were solved using the proposed methods. Full article
Open AccessArticle Visual Simulation of Detailed Turbulent Water by Preserving the Thin Sheets of Fluid
Symmetry 2018, 10(10), 502; https://doi.org/10.3390/sym10100502
Received: 19 September 2018 / Revised: 29 September 2018 / Accepted: 2 October 2018 / Published: 15 October 2018
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Abstract
When we perform particle-based water simulation, water particles are often increased dramatically because of particle splitting around breaking holes to maintain the thin fluid sheets. Because most of the existing approaches do not consider the volume of the water particles, the water particles
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When we perform particle-based water simulation, water particles are often increased dramatically because of particle splitting around breaking holes to maintain the thin fluid sheets. Because most of the existing approaches do not consider the volume of the water particles, the water particles must have a very low mass to satisfy the law of the conservation of mass. This phenomenon smears the motion of the water, which would otherwise result in splashing, thereby resulting in artifacts such as numerical dissipation. Thus, we propose a new fluid-implicit, particle-based framework for maintaining and representing the thin sheets and turbulent flows of water. After splitting the water particles, the proposed method uses the ghost density and ghost mass to redistribute the difference in mass based on the volume of the water particles. Next, small-scale turbulent flows are formed in local regions and transferred in a smooth manner to the global flow field. Our results show us the turbulence details as well as the thin sheets of water, thereby obtaining an aesthetically pleasing improvement compared with existing methods. Full article
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Open AccessArticle Third-Order Hankel Determinant for Certain Class of Analytic Functions Related with Exponential Function
Symmetry 2018, 10(10), 501; https://doi.org/10.3390/sym10100501
Received: 30 July 2018 / Revised: 7 October 2018 / Accepted: 9 October 2018 / Published: 15 October 2018
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Abstract
Let Sl* denote the class of analytic functions f in the open unit disk D={z:|z|<1} normalized by f(0)=f(0)1=0
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Let S l * denote the class of analytic functions f in the open unit disk D = { z : | z | < 1 } normalized by f ( 0 ) = f ( 0 ) 1 = 0 , which is subordinate to exponential function, z f ( z ) f ( z ) e z ( z D ) . In this paper, we aim to investigate the third-order Hankel determinant H 3 ( 1 ) for this function class S l * associated with exponential function and obtain the upper bound of the determinant H 3 ( 1 ) . Meanwhile, we give two examples to illustrate the results obtained. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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Open AccessArticle Fault Diagnosis of Mine Shaft Guide Rails Using Vibration Signal Analysis Based on Dynamic Time Warping
Symmetry 2018, 10(10), 500; https://doi.org/10.3390/sym10100500
Received: 21 September 2018 / Revised: 10 October 2018 / Accepted: 11 October 2018 / Published: 15 October 2018
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Abstract
Guide rails are amongst the most important components of mine hoist systems, and faults in them must be detected as early as possible to avoid fatal breakdowns in mine production. Presently, guide rail inspection is performed visually in most situations. In this paper,
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Guide rails are amongst the most important components of mine hoist systems, and faults in them must be detected as early as possible to avoid fatal breakdowns in mine production. Presently, guide rail inspection is performed visually in most situations. In this paper, we examine a more efficient approach based on multi-time scale and dynamic time warping (DTW) to diagnose guide rail faults including embossment, bumps, and clearance. Firstly, vibration signals collected from operational conveyance under different fault categories are analyzed and the corresponding characteristic waveforms (CWs) are extracted. Embossment faults are identified with priority according to visible disparities in CW patterns on a large time scale. Then, templates for bump and clearance faults are established through processing the CWs on a small time scale. Subsequently, the distances of DTW (DDTWs) between test samples and the selected templates are calculated. Finally, the remaining fault conditions are classified according to the DDTW results since the same fault category has the smallest distance. Experiments are conducted on a guide rail fault simulator to demonstrate the reliability of the proposed method. The resultant diagnosis accuracies are 100%, 90.40%, and 84.53%, respectively, for embossment, bump, and clearance faults, which indicates that the proposed approach is feasible and effective for diagnosing guide rail faults under variable operating conditions and different fault severities. Full article
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Open AccessArticle Heat Induction by Viscous Dissipation Subjected to Symmetric and Asymmetric Boundary Conditions on a Small Oscillating Flow in a Microchannel
Symmetry 2018, 10(10), 499; https://doi.org/10.3390/sym10100499
Received: 29 August 2018 / Revised: 24 September 2018 / Accepted: 10 October 2018 / Published: 15 October 2018
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Abstract
The heat induced by viscous dissipation in a microchannel fluid, due to a small oscillating motion of the lower plate, is investigated for the first time. The methodology is by applying the momentum and energy equations and solving them for three cases of
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The heat induced by viscous dissipation in a microchannel fluid, due to a small oscillating motion of the lower plate, is investigated for the first time. The methodology is by applying the momentum and energy equations and solving them for three cases of standard thermal boundary conditions. The first two cases involve symmetric boundary conditions of constant surface temperature on both plates and both plates insulated, respectively. The third case has the asymmetric conditions that the lower plate is insulated while the upper plate is maintained at constant temperature. Results reveal that, although the fluid velocity is only depending on the oscillation rate of the plate, the temperature field for all three cases show that the induced heating is dependent on the oscillation rate of the plate, but strongly dependent on the parameters Brinkman number and Prandtl number. All three cases prove that the increasing oscillation rate or Brinkman number and decreasing Prandtl number, when it is less than unity, will significantly increase the temperature field. The present model is applied to the synovial fluid motion in artificial hip implant and results in heat induced by viscous dissipation for the second case shows remarkably close agreement with the experimental literature. Full article
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Open AccessArticle Localization Free Energy Efficient and Cooperative Routing Protocols for Underwater Wireless Sensor Networks
Symmetry 2018, 10(10), 498; https://doi.org/10.3390/sym10100498
Received: 27 September 2018 / Revised: 9 October 2018 / Accepted: 10 October 2018 / Published: 15 October 2018
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Abstract
Mitigation of channel unfavorable circumstances during data routing in underwater wireless sensor networks (UWSNs) has utmost significance. It guarantees saving packet corruption along unfavorable channels so that vital data is not lost or become meaningless. This paper proposes two routing protocols for UWSNs:
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Mitigation of channel unfavorable circumstances during data routing in underwater wireless sensor networks (UWSNs) has utmost significance. It guarantees saving packet corruption along unfavorable channels so that vital data is not lost or become meaningless. This paper proposes two routing protocols for UWSNs: localization free energy efficient routing (LFEER) and its improved version, localization free energy efficient cooperative routing (Co-LFEER). The LFEER makes decision of choosing a relay based on its maximum residual energy, number of hops and the bit error rate of the link over which packets are transmitted. These metrics are chosen to save packets from corruption to the maximum limit and maintain stable paths (where nodes do not die soon). Since a single link is used in the LFEER for packets forwarding, the link may become worse with changing circumstances of the channel. To deal with this issue, cooperative routing is added to the LFFER to construct the Co-LFEER protocol, in which some copies of packets are received by destination to decide about packets quality. Converse to some prevalent protocols, both LFEER and Co-LFEER are independent of knowing the sensor nodes’ positions, which increases computational complexity and wasteful utilization of resources. Based on extensive simulations, the proposed schemes are better than Co-DBR in reducing energy utilization and advancing packets to the desired destination. Full article
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Open AccessArticle An Extended VIKOR Method for Multiple Criteria Group Decision Making with Triangular Fuzzy Neutrosophic Numbers
Symmetry 2018, 10(10), 497; https://doi.org/10.3390/sym10100497
Received: 1 August 2018 / Revised: 8 October 2018 / Accepted: 8 October 2018 / Published: 15 October 2018
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Abstract
In this article, we combine the original VIKOR model with a triangular fuzzy neutrosophic set to propose the triangular fuzzy neutrosophic VIKOR method. In the extended method, we use the triangular fuzzy neutrosophic numbers (TFNNs) to present the criteria values in multiple criteria
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In this article, we combine the original VIKOR model with a triangular fuzzy neutrosophic set to propose the triangular fuzzy neutrosophic VIKOR method. In the extended method, we use the triangular fuzzy neutrosophic numbers (TFNNs) to present the criteria values in multiple criteria group decision making (MCGDM) problems. Firstly, we summarily introduce the fundamental concepts, operation formulas and distance calculating method of TFNNs. Then we review some aggregation operators of TFNNs. Thereafter, we extend the original VIKOR model to the triangular fuzzy neutrosophic environment and introduce the calculating steps of the TFNNs VIKOR method, our proposed method which is more reasonable and scientific for considering the conflicting criteria. Furthermore, a numerical example for potential evaluation of emerging technology commercialization is presented to illustrate the new method, and some comparisons are also conducted to further illustrate advantages of the new method. Full article
Open AccessArticle Acceptance Sampling Plans for Finite and Infinite Lot Size under Power Lindley Distribution
Symmetry 2018, 10(10), 496; https://doi.org/10.3390/sym10100496
Received: 20 September 2018 / Revised: 8 October 2018 / Accepted: 11 October 2018 / Published: 15 October 2018
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Abstract
In this paper, we have developed single and double acceptance sampling plans when the product life length follows the power Lindley distribution. The sampling plans have been developed by assuming infinite and finite lot sizes. We have obtained the operating characteristic curves for
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In this paper, we have developed single and double acceptance sampling plans when the product life length follows the power Lindley distribution. The sampling plans have been developed by assuming infinite and finite lot sizes. We have obtained the operating characteristic curves for the resultant sampling plans. The sampling plans have been obtained for various values of the parameters. It has been found that for a finite lot size, the sampling plans provide smaller values of the parameters to achieve the specified acceptance probabilities. Full article
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