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# Mathematics, Volume 7, Issue 8 (August 2019) – 108 articles

Cover Story (view full-size image): The problem of optimal box positioning is a problem of finding a position of a box with given edge lengths that maximizes the number of enclosed points of the given set (the figure illustrates a 3-dimensional example). In this paper the problem was proved to be NP-hard. View this paper.
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12 pages, 247 KiB
Article
On Stability of Iterative Sequences with Error
by Salwa Salman Abed and Noor Saddam Taresh
Mathematics 2019, 7(8), 765; https://doi.org/10.3390/math7080765 - 20 Aug 2019
Cited by 6 | Viewed by 2324
Abstract
Iterative methods were employed to obtain solutions of linear and non-linear systems of equations, solutions of differential equations, and roots of equations. In this paper, it was proved that s-iteration with error and Picard–Mann iteration with error converge strongly to the unique fixed [...] Read more.
Iterative methods were employed to obtain solutions of linear and non-linear systems of equations, solutions of differential equations, and roots of equations. In this paper, it was proved that s-iteration with error and Picard–Mann iteration with error converge strongly to the unique fixed point of Lipschitzian strongly pseudo-contractive mapping. This convergence was almost $F$ -stable and $F$ -stable. Applications of these results have been given to the operator equations $F x = f$ and $x + F x = f ,$ where $F$ is a strongly accretive and accretive mappings of $X$ into itself. Full article
22 pages, 916 KiB
Article
Linguistic Picture Fuzzy Dombi Aggregation Operators and Their Application in Multiple Attribute Group Decision Making Problem
by Muhammad Qiyas, Saleem Abdullah, Shahzaib Ashraf and Lazim Abdullah
Mathematics 2019, 7(8), 764; https://doi.org/10.3390/math7080764 - 20 Aug 2019
Cited by 26 | Viewed by 3145
Abstract
The aims of this study are to propose the linguistic picture fuzzy Dombi (LPFD) aggregation operators and decision-making approach to deal with uncertainties in the form of linguistic picture fuzzy sets. LPFD operators have more flexibility due to the general fuzzy set. Utilizing [...] Read more.
The aims of this study are to propose the linguistic picture fuzzy Dombi (LPFD) aggregation operators and decision-making approach to deal with uncertainties in the form of linguistic picture fuzzy sets. LPFD operators have more flexibility due to the general fuzzy set. Utilizing the Dombi operational rule, the series of Dombi aggregation operators were proposed, namely linguistic picture fuzzy Dombi arithmetic/geometric, ordered arithmetic/ordered geometric and Hybrid arithmetic/Hybrid geometric aggregation operators. The distinguished feature of these proposed operators is studied. At that point, we have used these Dombi operators to design a model to deal with multiple attribute decision-making (MADM) issues under linguistic picture fuzzy information. Finally, an illustrative example to evaluate the emerging technology enterprises is provided to demonstrate the effectiveness of the proposed approach, together with a sensitivity analysis and comparison analysis, proving that its results are feasible and credible. Full article
(This article belongs to the Special Issue Operations Research Using Fuzzy Sets Theory)
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16 pages, 1203 KiB
Article
Special Class of Second-Order Non-Differentiable Symmetric Duality Problems with (G,αf)-Pseudobonvexity Assumptions
by Ramu Dubey, Lakshmi Narayan Mishra and Rifaqat Ali
Mathematics 2019, 7(8), 763; https://doi.org/10.3390/math7080763 - 20 Aug 2019
Cited by 14 | Viewed by 2784
Abstract
In this paper, we introduce the various types of generalized invexities, i.e., $α f$ -invex/ $α f$ -pseudoinvex and $( G , α f )$ -bonvex/ $( G , α f )$ -pseudobonvex functions. Furthermore, we construct nontrivial numerical examples of [...] Read more.
In this paper, we introduce the various types of generalized invexities, i.e., $α f$ -invex/ $α f$ -pseudoinvex and $( G , α f )$ -bonvex/ $( G , α f )$ -pseudobonvex functions. Furthermore, we construct nontrivial numerical examples of $( G , α f )$ -bonvexity/ $( G , α f )$ -pseudobonvexity, which is neither $α f$ -bonvex/ $α f$ -pseudobonvex nor $α f$ -invex/ $α f$ -pseudoinvex with the same $η$ . Further, we formulate a pair of second-order non-differentiable symmetric dual models and prove the duality relations under $α f$ -invex/ $α f$ -pseudoinvex and $( G , α f )$ -bonvex/ $( G , α f )$ -pseudobonvex assumptions. Finally, we construct a nontrivial numerical example justifying the weak duality result presented in the paper. Full article
(This article belongs to the Special Issue Multivariate Approximation for solving ODE and PDE)
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13 pages, 261 KiB
Article
Cohen Macaulay Bipartite Graphs and Regular Element on the Powers of Bipartite Edge Ideals
by Arindam Banerjee and Vivek Mukundan
Mathematics 2019, 7(8), 762; https://doi.org/10.3390/math7080762 - 20 Aug 2019
Viewed by 2586
Abstract
In this article, we discuss new characterizations of Cohen-Macaulay bipartite edge ideals. For arbitrary bipartite edge ideals $I ( G )$ , we also discuss methods to recognize regular elements on $I ( G ) s$ for all $s ≥ 1$ in terms [...] Read more.
In this article, we discuss new characterizations of Cohen-Macaulay bipartite edge ideals. For arbitrary bipartite edge ideals $I ( G )$ , we also discuss methods to recognize regular elements on $I ( G ) s$ for all $s ≥ 1$ in terms of the combinatorics of the graph G. Full article
(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)
22 pages, 5959 KiB
Article
Understanding the Evolution of Tree Size Diversity within the Multivariate Nonsymmetrical Diffusion Process and Information Measures
by Petras Rupšys
Mathematics 2019, 7(8), 761; https://doi.org/10.3390/math7080761 - 19 Aug 2019
Cited by 13 | Viewed by 2883
Abstract
This study focuses on the stochastic differential calculus of Itô, as an effective tool for the analysis of noise in forest growth and yield modeling. Idea of modeling state (tree size) variable in terms of univariate stochastic differential equation is exposed to a [...] Read more.
This study focuses on the stochastic differential calculus of Itô, as an effective tool for the analysis of noise in forest growth and yield modeling. Idea of modeling state (tree size) variable in terms of univariate stochastic differential equation is exposed to a multivariate stochastic differential equation. The new developed multivariate probability density function and its marginal univariate, bivariate and trivariate distributions, and conditional univariate, bivariate and trivariate probability density functions can be applied for the modeling of tree size variables and various stand attributes such as the mean diameter, height, crown base height, crown width, volume, basal area, slenderness ratio, increments, and much more. This study introduces generalized multivariate interaction information measures based on the differential entropy to capture multivariate dependencies between state variables. The present study experimentally confirms the effectiveness of using multivariate interaction information measures to reconstruct multivariate relationships of state variables using measurements obtained from a real-world data set. Full article
(This article belongs to the Special Issue Stochastic Differential Equations and Their Applications)
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14 pages, 490 KiB
Article
A New Approach for the Black–Scholes Model with Linear and Nonlinear Volatilities
by Seda Gulen, Catalin Popescu and Murat Sari
Mathematics 2019, 7(8), 760; https://doi.org/10.3390/math7080760 - 19 Aug 2019
Cited by 16 | Viewed by 7351
Abstract
Since financial engineering problems are of great importance in the academic community, effective methods are still needed to analyze these models. Therefore, this article focuses mainly on capturing the discrete behavior of linear and nonlinear Black–Scholes European option pricing models. To achieve this, [...] Read more.
Since financial engineering problems are of great importance in the academic community, effective methods are still needed to analyze these models. Therefore, this article focuses mainly on capturing the discrete behavior of linear and nonlinear Black–Scholes European option pricing models. To achieve this, this article presents a combined method; a sixth order finite difference (FD6) scheme in space and a third–order strong stability preserving Runge–Kutta (SSPRK3) over time. The computed results are compared with available literature and the exact solution. The computed results revealed that the current method seems to be quite strong both quantitatively and qualitatively with minimal computational effort. Therefore, this method appears to be a very reliable alternative and flexible to implement in solving the problem while preserving the physical properties of such realistic processes. Full article
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23 pages, 372 KiB
Article
Hybrid Control Scheme for Projective Lag Synchronization of Riemann–Liouville Sense Fractional Order Memristive BAM NeuralNetworks with Mixed Delays
by Grienggrai Rajchakit, Anbalagan Pratap, Ramachandran Raja, Jinde Cao, Jehad Alzabut and Chuangxia Huang
Mathematics 2019, 7(8), 759; https://doi.org/10.3390/math7080759 - 19 Aug 2019
Cited by 138 | Viewed by 4924
Abstract
This sequel is concerned with the analysis of projective lag synchronization of Riemann–Liouville sense fractional order memristive BAM neural networks (FOMBNNs) with mixed time delays via hybrid controller. Firstly, a new type of hybrid control scheme, which is the combination of open loop [...] Read more.
This sequel is concerned with the analysis of projective lag synchronization of Riemann–Liouville sense fractional order memristive BAM neural networks (FOMBNNs) with mixed time delays via hybrid controller. Firstly, a new type of hybrid control scheme, which is the combination of open loop control and adaptive state feedback control is designed to guarantee the global projective lag synchronization of the addressed FOMBNNs model. Secondly, by using a Lyapunov–Krasovskii functional and Barbalet’s lemma, a new brand of sufficient criterion is proposed to ensure the projective lag synchronization of the FOMBNNs model considered. Moreover, as special cases by using a hybrid control scheme, some sufficient conditions are derived to ensure the global projective synchronization, global complete synchronization and global anti-synchronization for the FOMBNNs model considered. Finally, numerical simulations are provided to check the accuracy and validity of our obtained synchronization results. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
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8 pages, 738 KiB
Article
Neutrosophic Quadruple Vector Spaces and Their Properties
by Vasantha Kandasamy W.B., Ilanthenral Kandasamy and Florentin Smarandache
Mathematics 2019, 7(8), 758; https://doi.org/10.3390/math7080758 - 19 Aug 2019
Cited by 16 | Viewed by 3009
Abstract
In this paper authors for the first time introduce the concept of Neutrosophic Quadruple (NQ) vector spaces and Neutrosophic Quadruple linear algebras and study their properties. Most of the properties of vector spaces are true in case of Neutrosophic Quadruple vector spaces. Two [...] Read more.
In this paper authors for the first time introduce the concept of Neutrosophic Quadruple (NQ) vector spaces and Neutrosophic Quadruple linear algebras and study their properties. Most of the properties of vector spaces are true in case of Neutrosophic Quadruple vector spaces. Two vital observations are, all quadruple vector spaces are of dimension four, be it defined over the field of reals R or the field of complex numbers C or the finite field of characteristic p, $Z p$ ; p a prime. Secondly all of them are distinct and none of them satisfy the classical property of finite dimensional vector spaces. So this problem is proposed as a conjecture in the final section. Full article
(This article belongs to the Special Issue New Challenges in Neutrosophic Theory and Applications)
15 pages, 1456 KiB
Article
Non-Intrusive Inference Reduced Order Model for Fluids Using Deep Multistep Neural Network
by Xuping Xie, Guannan Zhang and Clayton G. Webster
Mathematics 2019, 7(8), 757; https://doi.org/10.3390/math7080757 - 19 Aug 2019
Cited by 23 | Viewed by 4331
Abstract
In this effort we propose a data-driven learning framework for reduced order modeling of fluid dynamics. Designing accurate and efficient reduced order models for nonlinear fluid dynamic problems is challenging for many practical engineering applications. Classical projection-based model reduction methods generate reduced systems [...] Read more.
In this effort we propose a data-driven learning framework for reduced order modeling of fluid dynamics. Designing accurate and efficient reduced order models for nonlinear fluid dynamic problems is challenging for many practical engineering applications. Classical projection-based model reduction methods generate reduced systems by projecting full-order differential operators into low-dimensional subspaces. However, these techniques usually lead to severe instabilities in the presence of highly nonlinear dynamics, which dramatically deteriorates the accuracy of the reduced-order models. In contrast, our new framework exploits linear multistep networks, based on implicit Adams–Moulton schemes, to construct the reduced system. The advantage is that the method optimally approximates the full order model in the low-dimensional space with a given supervised learning task. Moreover, our approach is non-intrusive, such that it can be applied to other complex nonlinear dynamical systems with sophisticated legacy codes. We demonstrate the performance of our method through the numerical simulation of a two-dimensional flow past a circular cylinder with Reynolds number Re = 100. The results reveal that the new data-driven model is significantly more accurate than standard projection-based approaches. Full article
(This article belongs to the Special Issue Machine Learning in Fluid Dynamics: Theory and Applications)
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10 pages, 222 KiB
Article
Multiple Solutions for Nonlocal Elliptic Systems Involving p(x)-Biharmonic Operator
by Qing Miao
Mathematics 2019, 7(8), 756; https://doi.org/10.3390/math7080756 - 19 Aug 2019
Cited by 13 | Viewed by 2407
Abstract
This paper analyzes the nonlocal elliptic system involving the p(x)-biharmonic operator. We give the corresponding variational structure of the problem, and then by means of Ricceri’s Variational theorem and the definition of general Lebesgue-Sobolev space, we obtain sufficient conditions for the infinite solutions [...] Read more.
This paper analyzes the nonlocal elliptic system involving the p(x)-biharmonic operator. We give the corresponding variational structure of the problem, and then by means of Ricceri’s Variational theorem and the definition of general Lebesgue-Sobolev space, we obtain sufficient conditions for the infinite solutions to this problem. Full article
16 pages, 5399 KiB
Article
Rock Classification from Field Image Patches Analyzed Using a Deep Convolutional Neural Network
by Xiangjin Ran, Linfu Xue, Yanyan Zhang, Zeyu Liu, Xuejia Sang and Jinxin He
Mathematics 2019, 7(8), 755; https://doi.org/10.3390/math7080755 - 18 Aug 2019
Cited by 78 | Viewed by 11149
Abstract
The automatic identification of rock type in the field would aid geological surveying, education, and automatic mapping. Deep learning is receiving significant research attention for pattern recognition and machine learning. Its application here has effectively identified rock types from images captured in the [...] Read more.
The automatic identification of rock type in the field would aid geological surveying, education, and automatic mapping. Deep learning is receiving significant research attention for pattern recognition and machine learning. Its application here has effectively identified rock types from images captured in the field. This paper proposes an accurate approach for identifying rock types in the field based on image analysis using deep convolutional neural networks. The proposed approach can identify six common rock types with an overall classification accuracy of 97.96%, thus outperforming other established deep-learning models and a linear model. The results show that the proposed approach based on deep learning represents an improvement in intelligent rock-type identification and solves several difficulties facing the automated identification of rock types in the field. Full article
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19 pages, 330 KiB
Article
Some Generalized Contraction Classes and Common Fixed Points in b-Metric Space Endowed with a Graph
by Reny George, Hossam A. Nabwey, Rajagopalan Ramaswamy and Stojan Radenović
Mathematics 2019, 7(8), 754; https://doi.org/10.3390/math7080754 - 18 Aug 2019
Cited by 9 | Viewed by 2710
Abstract
We have introduced the new notions of R-weakly graph preserving and R-weakly $α$ -admissible pair of multivalued mappings which includes the class of graph preserving mappings, weak graph preserving mappings as well as $α$ -admissible mappings of type S, [...] Read more.
We have introduced the new notions of R-weakly graph preserving and R-weakly $α$ -admissible pair of multivalued mappings which includes the class of graph preserving mappings, weak graph preserving mappings as well as $α$ -admissible mappings of type S, $α *$ -admissible mappings of type S and $α *$ - orbital admissible mappings of type S respectively. Some generalized contraction and rational contraction classes are also introduced for a pair of multivalued mappings and common fixed point theorems are proved in a b-metric space endowed with a graph. We have also applied our results to obtain common fixed point theorems for R-weakly $α$ -admissible pair of multivalued mappings in a b-metric space which are the proper extension and generalization of many known results. Proper examples are provided in support of our results. Our main results and its consequences improve, generalize and extend many known fixed point results existing in literature. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Nonlinear Problems with Applications)
17 pages, 7158 KiB
Article
Quasi-Isometric Mesh Parameterization Using Heat-Based Geodesics and Poisson Surface Fills
by Daniel Mejia-Parra, Jairo R. Sánchez, Jorge Posada, Oscar Ruiz-Salguero and Carlos Cadavid
Mathematics 2019, 7(8), 753; https://doi.org/10.3390/math7080753 - 17 Aug 2019
Cited by 1 | Viewed by 4748
Abstract
In the context of CAD, CAM, CAE, and reverse engineering, the problem of mesh parameterization is a central process. Mesh parameterization implies the computation of a bijective map $ϕ$ from the original mesh $M ∈ R 3$ to the planar domain [...] Read more.
In the context of CAD, CAM, CAE, and reverse engineering, the problem of mesh parameterization is a central process. Mesh parameterization implies the computation of a bijective map $ϕ$ from the original mesh $M ∈ R 3$ to the planar domain $ϕ ( M ) ∈ R 2$ . The mapping may preserve angles, areas, or distances. Distance-preserving parameterizations (i.e., isometries) are obviously attractive. However, geodesic-based isometries present limitations when the mesh has concave or disconnected boundary (i.e., holes). Recent advances in computing geodesic maps using the heat equation in 2-manifolds motivate us to revisit mesh parameterization with geodesic maps. We devise a Poisson surface underlying, extending, and filling the holes of the mesh M. We compute a near-isometric mapping for quasi-developable meshes by using geodesic maps based on heat propagation. Our method: (1) Precomputes a set of temperature maps (heat kernels) on the mesh; (2) estimates the geodesic distances along the piecewise linear surface by using the temperature maps; and (3) uses multidimensional scaling (MDS) to acquire the 2D coordinates that minimize the difference between geodesic distances on M and Euclidean distances on $R 2$ . This novel heat-geodesic parameterization is successfully tested with several concave and/or punctured surfaces, obtaining bijective low-distortion parameterizations. Failures are registered in nonsegmented, highly nondevelopable meshes (such as seam meshes). These cases are the goal of future endeavors. Full article
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19 pages, 8152 KiB
Article
Development of Public Key Cryptographic Algorithm Using Matrix Pattern for Tele-Ultrasound Applications
by Seung-Hyeok Shin, Won-Sok Yoo and Hojong Choi
Mathematics 2019, 7(8), 752; https://doi.org/10.3390/math7080752 - 17 Aug 2019
Cited by 19 | Viewed by 4377
Abstract
A novel public key cryptographic algorithm using a matrix pattern is developed to improve encrypting strength. Compared to the Rivest–Sharmir–Adleman (RSA) and Elliptic Curve Cryptography (ECC) algorithms, our proposed algorithm has superior encrypting strength due to several unknown quantities and one additional sub-equation [...] Read more.
A novel public key cryptographic algorithm using a matrix pattern is developed to improve encrypting strength. Compared to the Rivest–Sharmir–Adleman (RSA) and Elliptic Curve Cryptography (ECC) algorithms, our proposed algorithm has superior encrypting strength due to several unknown quantities and one additional sub-equation during the encrypting process. Our proposed algorithm also provides a faster encoding/decoding speed when the patient’s images for tele-ultrasound applications are transmitted/received, compared to the RSA and ECC encrypting algorithms, because it encodes/decodes the plain memory block by simple addition and multiplication operations of n terms. However, the RSA and ECC algorithms encode/decode each memory block using complex mathematical exponentiation and congruence. To implement encrypting algorithms for tele-ultrasound applications, a streaming server was constructed to transmit the images to the systems using ultrasound machines. Using the obtained ultrasound images from a breast phantom, we compared our developed algorithm, utilizing a matrix pattern, with the RSA and ECC algorithms. The elapsed average time for our proposed algorithm is much faster than that for the RSA and ECC algorithms. Full article
(This article belongs to the Special Issue Information Theory, Cryptography, Randomness and Statistical Modeling)
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14 pages, 263 KiB
Article
Quantum Integral Inequalities of Simpson-Type for Strongly Preinvex Functions
by Yongping Deng, Muhammad Uzair Awan and Shanhe Wu
Mathematics 2019, 7(8), 751; https://doi.org/10.3390/math7080751 - 16 Aug 2019
Cited by 27 | Viewed by 2920
Abstract
In this paper, we establish a new q-integral identity, the result is then used to derive two q-integral inequalities of Simpson-type involving strongly preinvex functions. Some special cases of the obtained results are also considered, it is shown that several new [...] Read more.
In this paper, we establish a new q-integral identity, the result is then used to derive two q-integral inequalities of Simpson-type involving strongly preinvex functions. Some special cases of the obtained results are also considered, it is shown that several new and previously known results can be derived via generalized strongly preinvex functions and quantum integrals. Full article
(This article belongs to the Special Issue Inequalities in Geometry and Applications)
7 pages, 232 KiB
Article
Some Polynomial Sequence Relations
by Chan-Liang Chung
Mathematics 2019, 7(8), 750; https://doi.org/10.3390/math7080750 - 16 Aug 2019
Cited by 4 | Viewed by 2927
Abstract
We give some polynomial sequence relations that are generalizations of the Sury-type identities. We provide two proofs, one based on an elementary identity and the other using the method of generating functions. Full article
14 pages, 279 KiB
Article
Split Variational Inclusion Problem and Fixed Point Problem for a Class of Multivalued Mappings in CAT(0) Spaces
by Mujahid Abbas, Yusuf Ibrahim, Abdul Rahim Khan and Manuel De la Sen
Mathematics 2019, 7(8), 749; https://doi.org/10.3390/math7080749 - 16 Aug 2019
Cited by 9 | Viewed by 2529
Abstract
The aim of this paper is to introduce a modified viscosity iterative method to approximate a solution of the split variational inclusion problem and fixed point problem for a uniformly continuous multivalued total asymptotically strictly pseudocontractive mapping in [...] Read more.
The aim of this paper is to introduce a modified viscosity iterative method to approximate a solution of the split variational inclusion problem and fixed point problem for a uniformly continuous multivalued total asymptotically strictly pseudocontractive mapping in $C A T ( 0 )$ spaces. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. Furthermore, we solve a split Hammerstein integral inclusion problem and fixed point problem as an application to validate our result. It seems that our main result in the split variational inclusion problem is new in the setting of $C A T ( 0 )$ spaces. Full article
23 pages, 1967 KiB
Article
Three-Dimensional Hydro-Magnetic Flow Arising in a Long Porous Slider and a Circular Porous Slider with Velocity Slip
by Naeem Faraz, Yasir Khan, Amna Anjum and Muhammad Kahshan
Mathematics 2019, 7(8), 748; https://doi.org/10.3390/math7080748 - 16 Aug 2019
Cited by 6 | Viewed by 2903
Abstract
The current research explores the injection of a viscous fluid through a moving flat plate with a transverse uniform magneto-hydrodynamic (MHD) flow field to reduce sliding drag. Two cases of velocity slip between the slider and the ground are studied: a long slider [...] Read more.
The current research explores the injection of a viscous fluid through a moving flat plate with a transverse uniform magneto-hydrodynamic (MHD) flow field to reduce sliding drag. Two cases of velocity slip between the slider and the ground are studied: a long slider and a circular slider. Solving the porous slider problem is applicable to fluid-cushioned porous sliders, which are useful in reducing the frictional resistance of moving bodies. By using a similarity transformation, three dimensional Navier–Stokes equations are converted into coupled nonlinear ordinary differential equations. The resulting nonlinear boundary value problem was solved analytically using the homotopy analysis method (HAM). The HAM provided a fast convergent series solution, showing that this method is efficient, accurate, and has many advantages over the other existing methods. Solutions were obtained for the different values of Reynolds numbers (R), velocity slip, and magnetic fields. It was found that surface slip and Reynolds number had substantial influence on the lift and drag of the long and the circular sliders. Moreover, the effects of the applied magnetic field on the velocity components, load-carrying capacity, and friction force are discussed in detail with the aid of graphs and tables. Full article
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15 pages, 310 KiB
Article
A New Gronwall–Bellman Inequality in Frame of Generalized Proportional Fractional Derivative
Mathematics 2019, 7(8), 747; https://doi.org/10.3390/math7080747 - 15 Aug 2019
Cited by 10 | Viewed by 3223
Abstract
New versions of a Gronwall–Bellman inequality in the frame of the generalized (Riemann–Liouville and Caputo) proportional fractional derivative are provided. Before proceeding to the main results, we define the generalized Riemann–Liouville and Caputo proportional fractional derivatives and integrals and expose some of their [...] Read more.
New versions of a Gronwall–Bellman inequality in the frame of the generalized (Riemann–Liouville and Caputo) proportional fractional derivative are provided. Before proceeding to the main results, we define the generalized Riemann–Liouville and Caputo proportional fractional derivatives and integrals and expose some of their features. We prove our main result in light of some efficient comparison analyses. The Gronwall–Bellman inequality in the case of weighted function is also obtained. By the help of the new proposed inequalities, examples of Riemann–Liouville and Caputo proportional fractional initial value problems are presented to emphasize the solution dependence on the initial data and on the right-hand side. Full article
9 pages, 268 KiB
Article
New Polynomial Bounds for Jordan’s and Kober’s Inequalities Based on the Interpolation and Approximation Method
by Lina Zhang and Xuesi Ma
Mathematics 2019, 7(8), 746; https://doi.org/10.3390/math7080746 - 15 Aug 2019
Cited by 6 | Viewed by 2276
Abstract
In this paper, new refinements and improvements of Jordan’s and Kober’s inequalities are presented. We give new polynomial bounds for the $s i n c ( x )$ and $cos ( x )$ functions based on the interpolation and approximation method. The results [...] Read more.
In this paper, new refinements and improvements of Jordan’s and Kober’s inequalities are presented. We give new polynomial bounds for the $s i n c ( x )$ and $cos ( x )$ functions based on the interpolation and approximation method. The results show that our bounds are tighter than the previous methods. Full article
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25 pages, 622 KiB
Article
A Modified Fletcher–Reeves Conjugate Gradient Method for Monotone Nonlinear Equations with Some Applications
by Auwal Bala Abubakar, Poom Kumam, Hassan Mohammad, Aliyu Muhammed Awwal and Kanokwan Sitthithakerngkiet
Mathematics 2019, 7(8), 745; https://doi.org/10.3390/math7080745 - 15 Aug 2019
Cited by 38 | Viewed by 8085
Abstract
One of the fastest growing and efficient methods for solving the unconstrained minimization problem is the conjugate gradient method (CG). Recently, considerable efforts have been made to extend the CG method for solving monotone nonlinear equations. In this research article, we present a [...] Read more.
One of the fastest growing and efficient methods for solving the unconstrained minimization problem is the conjugate gradient method (CG). Recently, considerable efforts have been made to extend the CG method for solving monotone nonlinear equations. In this research article, we present a modification of the Fletcher–Reeves (FR) conjugate gradient projection method for constrained monotone nonlinear equations. The method possesses sufficient descent property and its global convergence was proved using some appropriate assumptions. Two sets of numerical experiments were carried out to show the good performance of the proposed method compared with some existing ones. The first experiment was for solving monotone constrained nonlinear equations using some benchmark test problem while the second experiment was applying the method in signal and image recovery problems arising from compressive sensing. Full article
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
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10 pages, 337 KiB
Article
Global Stability of Fractional Order Coupled Systems with Impulses via a Graphic Approach
by Bei Zhang, Yonghui Xia, Lijuan Zhu, Haidong Liu and Longfei Gu
Mathematics 2019, 7(8), 744; https://doi.org/10.3390/math7080744 - 15 Aug 2019
Cited by 7 | Viewed by 2925
Abstract
Based on the graph theory and stability theory of dynamical system, this paper studies the stability of the trivial solution of a coupled fractional-order system. Some sufficient conditions are obtained to guarantee the global stability of the trivial solution. Finally, a comparison between [...] Read more.
Based on the graph theory and stability theory of dynamical system, this paper studies the stability of the trivial solution of a coupled fractional-order system. Some sufficient conditions are obtained to guarantee the global stability of the trivial solution. Finally, a comparison between fractional-order system and integer-order system ends the paper. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
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12 pages, 8826 KiB
Article
Design of a New Chaotic System Based on Van Der Pol Oscillator and Its Encryption Application
by Jianbin He and Jianping Cai
Mathematics 2019, 7(8), 743; https://doi.org/10.3390/math7080743 - 13 Aug 2019
Cited by 12 | Viewed by 4283
Abstract
The Van der Pol oscillator is investigated by the parameter control method. This method only needs to control one parameter of the Van der Pol oscillator by a simple periodic function; then, the Van der Pol oscillator can behave chaotically from the stable [...] Read more.
The Van der Pol oscillator is investigated by the parameter control method. This method only needs to control one parameter of the Van der Pol oscillator by a simple periodic function; then, the Van der Pol oscillator can behave chaotically from the stable limit cycle. Based on the new Van der Pol oscillator with variable parameter (VdPVP), some dynamical characteristics are discussed by numerical simulations, such as the Lyapunov exponents and bifurcation diagrams. The numerical results show that there exists a positive Lyapunov exponent in the VdPVP. Therefore, an encryption algorithm is designed by the pseudo-random sequences generated from the VdPVP. This simple algorithm consists of chaos scrambling and chaos XOR (exclusive-or) operation, and the statistical analyses show that it has good security and encryption effectiveness. Finally, the feasibility and validity are verified by simulation experiments of image encryption. Full article
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21 pages, 350 KiB
Article
A Novel Approach to Generalized Intuitionistic Fuzzy Soft Sets and Its Application in Decision Support System
by Muhammad Jabir Khan, Poom Kumam, Peide Liu, Wiyada Kumam and Shahzaib Ashraf
Mathematics 2019, 7(8), 742; https://doi.org/10.3390/math7080742 - 13 Aug 2019
Cited by 59 | Viewed by 4592
Abstract
The basic idea underneath the generalized intuitionistic fuzzy soft set is very constructive in decision making, since it considers how to exploit an extra intuitionistic fuzzy input from the director to make up for any distortion in the information provided by the evaluation [...] Read more.
The basic idea underneath the generalized intuitionistic fuzzy soft set is very constructive in decision making, since it considers how to exploit an extra intuitionistic fuzzy input from the director to make up for any distortion in the information provided by the evaluation experts, which is redefined and clarified by F. Feng. In this paper, we introduced a method to solve decision making problems using an adjustable weighted soft discernibility matrix in a generalized intuitionistic fuzzy soft set. We define the threshold functions like mid-threshold, top-bottom-threshold, bottom-bottom-threshold, top-top-threshold, med-threshold function and their level soft sets of the generalized intuitionistic fuzzy soft set. After, we proposed two algorithms based on threshold functions, a weighted soft discernibility matrix and a generalized intuitionistic fuzzy soft set and also to show the supremacy of the given methods we illustrate a descriptive example using a weighted soft discernibility matrix in the generalized intuitionistic fuzzy soft set. Results indicate that the proposed method is more effective and generalized over all existing methods of the fuzzy soft set. Full article
(This article belongs to the Special Issue Operations Research Using Fuzzy Sets Theory)
34 pages, 18308 KiB
Article
Modeling and Control of IPMC Actuators Based on LSSVM-NARX Paradigm
by Liangsong Huang, Yu Hu, Yun Zhao and Yuxia Li
Mathematics 2019, 7(8), 741; https://doi.org/10.3390/math7080741 - 13 Aug 2019
Cited by 12 | Viewed by 3450
Abstract
Ionic polymer-metal composites are electrically driven intelligent composites that are readily exposed to bending deformations in the presence of external electric fields. Owing to their advantages, ionicpolymer-metal composites are promising candidates for actuators. However, ionicpolymer-metal composites exhibit strong nonlinear properties, especially hysteresis characteristics, [...] Read more.
Ionic polymer-metal composites are electrically driven intelligent composites that are readily exposed to bending deformations in the presence of external electric fields. Owing to their advantages, ionicpolymer-metal composites are promising candidates for actuators. However, ionicpolymer-metal composites exhibit strong nonlinear properties, especially hysteresis characteristics, resulting in severely reduced control accuracy. This study proposes an ionic polymer-metal composite platform and investigates its modeling and control. First, the hysteresis characteristics of the proposed Pt-electrode ionic polymer-metal composite are tested. Based on the hysteresis characteristics, ionic polymer-metal composites are modeled using the Prandtl-Ishlinskii model and the least squares support vector machine-nonlinear autoregressive model, respectively. Then, the ionic polymer-metal composite is driven by a random sinusoidal voltage, and the LSSVM-NARX model is established on the basis of the displacement data obtained. In addition, an artificial bee colony algorithm is proposed for accuracy optimization of the model parameters. Finally, an inverse controller based on the least squares support vector machine-nonlinear autoregressive model is proposed to compensate the hysteresis characteristics of the ionic polymer-metal composite. A hybrid PID feedback controller is developed by combining the inverse controller with PID feedback control, followed by simulation and testing of its actual position control on the ionic polymer-metal composite platform. The results show that the hybrid PID feedback control system can effectively eliminate the effects of the hysteresis characteristics on ionic polymer-metal composite control. Full article
(This article belongs to the Special Issue Mathematical Modeling: From Nonlinear Dynamics to Complex Systems)
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36 pages, 1428 KiB
Article
A Soft-Rough Set Based Approach for Handling Contextual Sparsity in Context-Aware Video Recommender Systems
by Syed Manzar Abbas, Khubaib Amjad Alam and Shahaboddin Shamshirband
Mathematics 2019, 7(8), 740; https://doi.org/10.3390/math7080740 - 12 Aug 2019
Cited by 14 | Viewed by 5854
Abstract
Context-aware video recommender systems (CAVRS) seek to improve recommendation performance by incorporating contextual features along with the conventional user-item ratings used by video recommender systems. In addition, the selection of influential and relevant contexts has a significant effect on the performance of CAVRS. [...] Read more.
Context-aware video recommender systems (CAVRS) seek to improve recommendation performance by incorporating contextual features along with the conventional user-item ratings used by video recommender systems. In addition, the selection of influential and relevant contexts has a significant effect on the performance of CAVRS. However, it is not guaranteed that, under the same contextual scenario, all the items are evaluated by users for providing dense contextual ratings. This problem cause contextual sparsity in CAVRS because the influence of each contextual factor in traditional CAVRS assumes the weights of contexts homogeneously for each of the recommendations. Hence, the selection of influencing contexts with minimal conflicts is identified as a potential research challenge. This study aims at resolving the contextual sparsity problem to leverage user interactions at varying contexts with an item in CAVRS. This problem may be investigated by considering a formal approximation of contextual attributes. For the purpose of improving the accuracy of recommendation process, we have proposed a novel contextual information selection process using Soft-Rough Sets. The proposed model will select a minimal set of influencing contexts using a weights assign process by Soft-Rough sets. Moreover, the proposed algorithm has been extensively evaluated using “LDOS-CoMoDa” dataset, and the outcome signifies the accuracy of our approach in handling contextual sparsity by exploiting relevant contextual factors. The proposed model outperforms existing solutions by identifying relevant contexts efficiently based on certainty, strength, and relevancy for effective recommendations. Full article
(This article belongs to the Special Issue Fuzziness and Mathematical Logic )
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40 pages, 4688 KiB
Article
Fuzzy Programming Approaches for Modeling a Customer-Centred Freight Routing Problem in the Road-Rail Intermodal Hub-and-Spoke Network with Fuzzy Soft Time Windows and Multiple Sources of Time Uncertainty
by Yan Sun and Xinya Li
Mathematics 2019, 7(8), 739; https://doi.org/10.3390/math7080739 - 12 Aug 2019
Cited by 28 | Viewed by 5987
Abstract
In this study, we systematically investigate a road-rail intermodal routing problem the optimization of which is oriented on the customer demands on transportation economy, timeliness and reliability. The road-rail intermodal transportation system is modelled as a hub-and-spoke network that contains time-flexible container truck [...] Read more.
(This article belongs to the Special Issue Operations Research Using Fuzzy Sets Theory)
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10 pages, 284 KiB
Article
On Ball Numbers
by Wolf-Dieter Richter
Mathematics 2019, 7(8), 738; https://doi.org/10.3390/math7080738 - 12 Aug 2019
Cited by 3 | Viewed by 2419
Abstract
We first shortly review, in part throwing a new light on, basics of ball numbers for balls having a positively homogeneous Minkowski functional and turn over then to a new particular class of ball numbers of balls having a Minkowski functional being homogeneous [...] Read more.
We first shortly review, in part throwing a new light on, basics of ball numbers for balls having a positively homogeneous Minkowski functional and turn over then to a new particular class of ball numbers of balls having a Minkowski functional being homogeneous with respect to multiplication with a specific diagonal matrix. Applications to crystal breeding, temperature expansion and normalizing density generating functions in big data analysis are indicated and a challenging problem from the inhomogeneity program is stated. Full article
18 pages, 285 KiB
Article
A New Approach for Exponential Stability Criteria of New Certain Nonlinear Neutral Differential Equations with Mixed Time-Varying Delays
by Janejira Tranthi, Thongchai Botmart, Wajaree Weera and Piyapong Niamsup
Mathematics 2019, 7(8), 737; https://doi.org/10.3390/math7080737 - 12 Aug 2019
Cited by 4 | Viewed by 2500
Abstract
This work is concerned with the delay-dependent criteria for exponential stability analysis of neutral differential equation with a more generally interval-distributed and discrete time-varying delays. By using a novel Lyapunov–Krasovkii functional, descriptor model transformation, utilization of the Newton–Leibniz formula, and the zero equation, [...] Read more.
This work is concerned with the delay-dependent criteria for exponential stability analysis of neutral differential equation with a more generally interval-distributed and discrete time-varying delays. By using a novel Lyapunov–Krasovkii functional, descriptor model transformation, utilization of the Newton–Leibniz formula, and the zero equation, the criteria for exponential stability are in the form of linear matrix inequalities (LMIs). Finally, we present the effectiveness of the theoretical results in numerical examples to show less conservative conditions than the others in the literature. Full article
11 pages, 1181 KiB
Article
Differential Equations Arising from the Generating Function of the (r, β)-Bell Polynomials and Distribution of Zeros of Equations
by Kyung-Won Hwang, Cheon Seoung Ryoo and Nam Soon Jung
Mathematics 2019, 7(8), 736; https://doi.org/10.3390/math7080736 - 12 Aug 2019
Cited by 5 | Viewed by 3008
Abstract
In this paper, we study differential equations arising from the generating function of the $( r , β )$ -Bell polynomials. We give explicit identities for the $( r , β )$ -Bell polynomials. Finally, we find the zeros of the [...] Read more.
In this paper, we study differential equations arising from the generating function of the $( r , β )$ -Bell polynomials. We give explicit identities for the $( r , β )$ -Bell polynomials. Finally, we find the zeros of the $( r , β )$ -Bell equations with numerical experiments. Full article