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Mathematics
Volume 7
Issue 11
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Mathematics, Volume 7, Issue 11 (November 2019) – 130 articles

Cover Story (view full-size image): The Wiener index is the oldest and most-studied topological index in chemistry. However, at present, there is no exact description of fullerenes with a maximum index value. In our paper, we took several steps in this direction. In particular, configurations of pentagons in caps are determined for some families of nanotubical fullerenes with the maximum Wiener index. View this paper.
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21 pages, 5709 KiB  
Article
Discrete Mutation Hopfield Neural Network in Propositional Satisfiability
by Mohd Shareduwan Mohd Kasihmuddin, Mohd. Asyraf Mansor, Md Faisal Md Basir and Saratha Sathasivam
Mathematics 2019, 7(11), 1133; https://doi.org/10.3390/math7111133 - 19 Nov 2019
Cited by 48 | Viewed by 4318
Abstract
The dynamic behaviours of an artificial neural network (ANN) system are strongly dependent on its network structure. Thus, the output of ANNs has long suffered from a lack of interpretability and variation. This has severely limited the practical usability of the logical rule [...] Read more.
The dynamic behaviours of an artificial neural network (ANN) system are strongly dependent on its network structure. Thus, the output of ANNs has long suffered from a lack of interpretability and variation. This has severely limited the practical usability of the logical rule in the ANN. The work presents an integrated representation of k-satisfiability (kSAT) in a mutation hopfield neural network (MHNN). Neuron states of the hopfield neural network converge to minimum energy, but the solution produced is confined to the limited number of solution spaces. The MHNN is incorporated with the global search capability of the estimation of distribution algorithms (EDAs), which typically explore various solution spaces. The main purpose is to estimate other possible neuron states that lead to global minimum energy through available output measurements. Furthermore, it is shown that the MHNN can retrieve various neuron states with the lowest minimum energy. Subsequent simulations performed on the MHNN reveal that the approach yields a result that surpasses the conventional hybrid HNN. Furthermore, this study provides a new paradigm in the field of neural networks by overcoming the overfitting issue. Full article
(This article belongs to the Special Issue Evolutionary Computation and Mathematical Programming)
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21 pages, 316 KiB  
Article
Space of Quasi-Periodic Limit Functions and Its Applications
by Rui Xie, Zhinan Xia and Junwei Liu
Mathematics 2019, 7(11), 1132; https://doi.org/10.3390/math7111132 - 19 Nov 2019
Cited by 4 | Viewed by 2607
Abstract
We introduce a class consisting of what we call quasi-periodic limit functions and then establish the relation between quasi-periodic limit functions and asymptotically quasi-periodic functions. At last, these quasi-periodic limit functions are applied to study the existence of asymptotically quasi-periodic solutions of abstract [...] Read more.
We introduce a class consisting of what we call quasi-periodic limit functions and then establish the relation between quasi-periodic limit functions and asymptotically quasi-periodic functions. At last, these quasi-periodic limit functions are applied to study the existence of asymptotically quasi-periodic solutions of abstract Cauchy problems. Full article
(This article belongs to the Special Issue Special Functions and Applications)
11 pages, 271 KiB  
Article
An Arc-Sine Law for Last Hitting Points in the Two-Parameter Wiener Space
by Jeong-Gyoo KIM
Mathematics 2019, 7(11), 1131; https://doi.org/10.3390/math7111131 - 19 Nov 2019
Cited by 1 | Viewed by 2015
Abstract
We develop the two-parameter version of an arc-sine law for a last hitting time. The existing arc-sine laws are about a stochastic process X t with one parameter t. If there is another varying key factor of an event described by a [...] Read more.
We develop the two-parameter version of an arc-sine law for a last hitting time. The existing arc-sine laws are about a stochastic process X t with one parameter t. If there is another varying key factor of an event described by a process, then we need to consider another parameter besides t. That is, we need a system of random variables with two parameters, say X s , t , which is far more complex than one-parameter processes. In this paper we challenge to develop such an idea, and provide the two-parameter version of an arc-sine law for a last hitting time. An arc-sine law for a two-parameter process is hardly found in literature. We use the properties of the two-parameter Wiener process for our development. Our result shows that the probability of last hitting points in the two-parameter Wiener space turns out to be arcsine-distributed. One can use our results to predict an event happened in a system of random variables with two parameters, which is not available among existing arc-sine laws for one parameter processes. Full article
(This article belongs to the Section Mathematics and Computer Science)
12 pages, 262 KiB  
Article
Common Fixed Point Results for Generalized Wardowski Type Contractive Multi-Valued Mappings
by Hüseyin Işık, Vahid Parvaneh, Babak Mohammadi and Ishak Altun
Mathematics 2019, 7(11), 1130; https://doi.org/10.3390/math7111130 - 18 Nov 2019
Cited by 14 | Viewed by 2300
Abstract
In this paper, we introduce generalized Wardowski type quasi-contractions called α - ( φ , Ω ) -contractions for a pair of multi-valued mappings and prove the existence of the common fixed point for such mappings. An illustrative example and an application are [...] Read more.
In this paper, we introduce generalized Wardowski type quasi-contractions called α - ( φ , Ω ) -contractions for a pair of multi-valued mappings and prove the existence of the common fixed point for such mappings. An illustrative example and an application are given to show the usability of our results. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
25 pages, 1352 KiB  
Article
Long Term Memory Assistance for Evolutionary Algorithms
by Matej Črepinšek, Shih-Hsi Liu, Marjan Mernik and Miha Ravber
Mathematics 2019, 7(11), 1129; https://doi.org/10.3390/math7111129 - 18 Nov 2019
Cited by 19 | Viewed by 2821
Abstract
Short term memory that records the current population has been an inherent component of Evolutionary Algorithms (EAs). As hardware technologies advance currently, inexpensive memory with massive capacities could become a performance boost to EAs. This paper introduces a Long Term Memory Assistance (LTMA) [...] Read more.
Short term memory that records the current population has been an inherent component of Evolutionary Algorithms (EAs). As hardware technologies advance currently, inexpensive memory with massive capacities could become a performance boost to EAs. This paper introduces a Long Term Memory Assistance (LTMA) that records the entire search history of an evolutionary process. With LTMA, individuals already visited (i.e., duplicate solutions) do not need to be re-evaluated, and thus, resources originally designated to fitness evaluations could be reallocated to continue search space exploration or exploitation. Three sets of experiments were conducted to prove the superiority of LTMA. In the first experiment, it was shown that LTMA recorded at least 50 % more duplicate individuals than a short term memory. In the second experiment, ABC and jDElscop were applied to the CEC-2015 benchmark functions. By avoiding fitness re-evaluation, LTMA improved execution time of the most time consuming problems F 03 and F 05 between 7% and 28% and 7% and 16%, respectively. In the third experiment, a hard real-world problem for determining soil models’ parameters, LTMA improved execution time between 26% and 69%. Finally, LTMA was implemented under a generalized and extendable open source system, called EARS. Any EA researcher could apply LTMA to a variety of optimization problems and evolutionary algorithms, either existing or new ones, in a uniform way. Full article
(This article belongs to the Special Issue Evolutionary Computation and Mathematical Programming)
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14 pages, 2126 KiB  
Article
Analysis of Queueing System MMPP/M/K/K with Delayed Feedback
by Agassi Melikov, Sevinj Aliyeva and Janos Sztrik
Mathematics 2019, 7(11), 1128; https://doi.org/10.3390/math7111128 - 18 Nov 2019
Cited by 10 | Viewed by 4449
Abstract
The model of multi-channel queuing system with Markov modulated Poisson process (MMPP) flow and delayed feedback is considered. After the customer is served completely, they will decide either to join the retrial group again for another service (feedback) with some state-dependent probability or [...] Read more.
The model of multi-channel queuing system with Markov modulated Poisson process (MMPP) flow and delayed feedback is considered. After the customer is served completely, they will decide either to join the retrial group again for another service (feedback) with some state-dependent probability or to leave the system forever with complimentary probability. Feedback calls organize an orbit of repeated calls (r-calls). If upon arrival of an r-call all the channels of the system are busy, then it either leaves the system with some state-dependent probability or with a complementary probability returns to orbit. Methods to calculate the steady-state probabilities of the appropriate three-dimensional Markov chain as well as performance measures of investigated system are developed. Results of numerical experiments are demonstrated. Full article
(This article belongs to the Special Issue Queueing Methods in Reliability Theory and Information Technology Systems)
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11 pages, 238 KiB  
Article
Absence of Global Solutions for a Fractional in Time and Space Shallow-Water System
by Mohamed Jleli, Mokhtar Kirane and Bessem Samet
Mathematics 2019, 7(11), 1127; https://doi.org/10.3390/math7111127 - 18 Nov 2019
Cited by 2 | Viewed by 1993
Abstract
An initial boundary value problem for a fractional in time and space shallow-water system involving ψ -Caputo fractional derivatives of different orders is considered. Using the test function method, sufficient criteria for the absence of global in time solutions of the system are [...] Read more.
An initial boundary value problem for a fractional in time and space shallow-water system involving ψ -Caputo fractional derivatives of different orders is considered. Using the test function method, sufficient criteria for the absence of global in time solutions of the system are obtained. Full article
(This article belongs to the Special Issue Fractional-Order Integral and Derivative Operators and Their Applications)
25 pages, 5842 KiB  
Article
Memory-Based Evolutionary Algorithms for Nonlinear and Stochastic Programming Problems
by Abdel-Rahman Hedar, Amira A. Allam and Wael Deabes
Mathematics 2019, 7(11), 1126; https://doi.org/10.3390/math7111126 - 17 Nov 2019
Cited by 4 | Viewed by 3040
Abstract
In this paper, we target the problems of finding a global minimum of nonlinear and stochastic programming problems. To solve this type of problem, we propose new approaches based on combining direct search methods with Evolution Strategies (ESs) and Scatter Search (SS) metaheuristics [...] Read more.
In this paper, we target the problems of finding a global minimum of nonlinear and stochastic programming problems. To solve this type of problem, we propose new approaches based on combining direct search methods with Evolution Strategies (ESs) and Scatter Search (SS) metaheuristics approaches. First, we suggest new designs of ESs and SS with a memory-based element called Gene Matrix (GM) to deal with those type of problems. These methods are called Directed Evolution Strategies (DES) and Directed Scatter Search (DSS), respectively, and they are able to search for a global minima. Moreover, a faster convergence can be achieved by accelerating the evolutionary search process using GM, and in the final stage we apply the Nelder-Mead algorithm to find the global minimum from the solutions found so far. Then, the variable-sample method is invoked in the DES and DSS to compose new stochastic programming techniques. Extensive numerical experiments have been applied on some well-known functions to test the performance of the proposed methods. Full article
(This article belongs to the Special Issue Evolutionary Computation and Mathematical Programming)
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17 pages, 314 KiB  
Article
Stability of the Apollonius Type Additive Functional Equation in Modular Spaces and Fuzzy Banach Spaces
by Sang Og Kim and John Michael Rassias
Mathematics 2019, 7(11), 1125; https://doi.org/10.3390/math7111125 - 17 Nov 2019
Cited by 8 | Viewed by 2580
Abstract
In this work, we investigate the generalized Hyers-Ulam stability of the Apollonius type additive functional equation in modular spaces with or without Δ 2 -conditions. We study the same problem in fuzzy Banach spaces and β -homogeneous Banach spaces. We show the hyperstability [...] Read more.
In this work, we investigate the generalized Hyers-Ulam stability of the Apollonius type additive functional equation in modular spaces with or without Δ 2 -conditions. We study the same problem in fuzzy Banach spaces and β -homogeneous Banach spaces. We show the hyperstability of the functional equation associated with the Jordan triple product in fuzzy Banach algebras. The obtained results can be applied to differential and integral equations with kernels of non-power types. Full article
(This article belongs to the Special Issue Functional Inequalities and Equations)
12 pages, 318 KiB  
Article
A New Explicit Four-Step Symmetric Method for Solving Schrödinger’s Equation
by Saleem Obaidat and Said Mesloub
Mathematics 2019, 7(11), 1124; https://doi.org/10.3390/math7111124 - 17 Nov 2019
Cited by 5 | Viewed by 2240
Abstract
In this article we have developed a new explicit four-step linear method of fourth algebraic order with vanished phase-lag and its first derivative. The efficiency of the method is tested by solving effectively the one-dimensional time independent Schrödinger’s equation. The error and stability [...] Read more.
In this article we have developed a new explicit four-step linear method of fourth algebraic order with vanished phase-lag and its first derivative. The efficiency of the method is tested by solving effectively the one-dimensional time independent Schrödinger’s equation. The error and stability analysis are studied. Also, the new method is compared with other methods in the literature. It is found that this method is more efficient than these methods. Full article
(This article belongs to the Special Issue Numerical Modeling and Analysis)
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18 pages, 4443 KiB  
Article
Numerical Performance Evaluation of Solar Photovoltaic Water Pumping System under Partial Shading Condition using Modern Optimization
by Jouda Arfaoui, Hegazy Rezk, Mujahed Al-Dhaifallah, Feki Elyes and Mami Abdelkader
Mathematics 2019, 7(11), 1123; https://doi.org/10.3390/math7111123 - 16 Nov 2019
Cited by 15 | Viewed by 3868
Abstract
Renewable energy is an attractive solution for water pumping systems particularly in isolated regions where the utility grid is unavailable. An attempt is made to improve the performance of solar photovoltaic water pumping system (SPVWPS) under partial shading condition. Under this condition, the [...] Read more.
Renewable energy is an attractive solution for water pumping systems particularly in isolated regions where the utility grid is unavailable. An attempt is made to improve the performance of solar photovoltaic water pumping system (SPVWPS) under partial shading condition. Under this condition, the power versus voltage curve has more than one maximum power point (MPP), which makes the tracking of global MPP not an easy task. Two MPP tracking (MPPT) strategies are proposed and compared for tracking MPP of SPVWPS under shading condition. The first method is based on the classical perturb and observe (P&O) and the other method is based on a Salp Swarm Algorithm (SSA). Based on extensive MATLAB simulation, it is found that the SSA method can provide higher photovoltaic (PV) generated power than the P&O method under shading condition. Consequently, the pump flowrate is increased. But, under normal distribution of solar radiation, both MPPT techniques can extract the maximum power but SSA is considered a time-consuming approach. Moreover, SSA is compared with particle swarm optimization (PSO) and genetic algorithm (GA). The obtained results ensure the superiority of SSA compared with PSO and GA. SSA has high successful rate of reaching true global MPP. Full article
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10 pages, 3201 KiB  
Article
Optimal Propagating Fronts Using Hamilton-Jacobi Equations
by Angelo Alessandri, Patrizia Bagnerini, Roberto Cianci and Mauro Gaggero
Mathematics 2019, 7(11), 1122; https://doi.org/10.3390/math7111122 - 16 Nov 2019
Cited by 1 | Viewed by 2296
Abstract
The optimal handling of level sets associated to the solution of Hamilton-Jacobi equations such as the normal flow equation is investigated. The goal is to find the normal velocity minimizing a suitable cost functional that accounts for a desired behavior of level sets [...] Read more.
The optimal handling of level sets associated to the solution of Hamilton-Jacobi equations such as the normal flow equation is investigated. The goal is to find the normal velocity minimizing a suitable cost functional that accounts for a desired behavior of level sets over time. Sufficient conditions of optimality are derived that require the solution of a system of nonlinear Hamilton-Jacobi equations. Since finding analytic solutions is difficult in general, the use of numerical methods to obtain approximate solutions is addressed by dealing with some case studies in two and three dimensions. Full article
(This article belongs to the Section Engineering Mathematics)
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12 pages, 277 KiB  
Article
A Nonparametric Approach to Bond Portfolio Immunization
by Victor Lapshin
Mathematics 2019, 7(11), 1121; https://doi.org/10.3390/math7111121 - 16 Nov 2019
Cited by 3 | Viewed by 2887
Abstract
We consider the problem of short term immunization of a bond-like obligation with respect to changes in interest rates using a portfolio of bonds. In the case that the zero-coupon yield curve belongs to a fixed low-dimensional manifold, the problem is widely known [...] Read more.
We consider the problem of short term immunization of a bond-like obligation with respect to changes in interest rates using a portfolio of bonds. In the case that the zero-coupon yield curve belongs to a fixed low-dimensional manifold, the problem is widely known as parametric immunization. Parametric immunization seeks to make the sensitivities of the hedged portfolio price with respect to all model parameters equal to zero. However, within a popular approach of nonparametric (smoothing spline) term structure estimation, parametric hedging is not applicable right away. We present a nonparametric approach to hedging a bond-like obligation allowing for a general form of the term structure estimator with possible smoothing. We show that our approach yields the standard duration based immunization in the limit when the amount of smoothing goes to infinity. We also recover the industry best practice approach of hedging based on key rate durations as another particular case. The hedging portfolio is straightforward to calculate using only basic linear algebra operations. Full article
(This article belongs to the Special Issue Advanced Methods in Mathematical Finance)
18 pages, 420 KiB  
Article
Sine-Cosine Algorithm to Enhance Simulated Annealing for Unrelated Parallel Machine Scheduling with Setup Times
by Hamza Jouhari, Deming Lei, Mohammed A. A. Al-qaness, Mohamed Abd Elaziz, Ahmed A. Ewees and Osama Farouk
Mathematics 2019, 7(11), 1120; https://doi.org/10.3390/math7111120 - 16 Nov 2019
Cited by 41 | Viewed by 3962
Abstract
This paper presents a hybrid method of Simulated Annealing (SA) algorithm and Sine Cosine Algorithm (SCA) to solve unrelated parallel machine scheduling problems (UPMSPs) with sequence-dependent and machine-dependent setup times. The proposed method, called SASCA, aims to improve the SA algorithm using the [...] Read more.
This paper presents a hybrid method of Simulated Annealing (SA) algorithm and Sine Cosine Algorithm (SCA) to solve unrelated parallel machine scheduling problems (UPMSPs) with sequence-dependent and machine-dependent setup times. The proposed method, called SASCA, aims to improve the SA algorithm using the SCA as a local search method. The SCA provides a good tool for the SA to avoid getting stuck in a focal point and improving the convergence to an efficient solution. SASCA algorithm is used to solve UPMSPs by minimizing makespan. To evaluate the performance of SASCA, a set of experiments were performed using 30 tests for 4 problems. Moreover, the performance of the proposed method was compared with other meta-heuristic algorithms. The comparison results showed the superiority of SASCA over other methods in terms of performance dimensions. Full article
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28 pages, 319 KiB  
Article
Weighted Fractional Iyengar Type Inequalities in the Caputo Direction
by George A. Anastassiou
Mathematics 2019, 7(11), 1119; https://doi.org/10.3390/math7111119 - 16 Nov 2019
Cited by 1 | Viewed by 1852
Abstract
Here we present weighted fractional Iyengar type inequalities with respect to L p norms, with 1 ≤ p ≤ ∞ . Our employed fractional calculus is of Caputo type defined with respect to another function. Our results provide quantitative estimates for the approximation [...] Read more.
Here we present weighted fractional Iyengar type inequalities with respect to L p norms, with 1 ≤ p ≤ ∞ . Our employed fractional calculus is of Caputo type defined with respect to another function. Our results provide quantitative estimates for the approximation of the Lebesgue–Stieljes integral of a function, based on its values over a finite set of points including at the endpoints of its interval of definition. Our method relies on the right and left generalized fractional Taylor’s formulae. The iterated generalized fractional derivatives case is also studied. We give applications at the end. Full article
(This article belongs to the Special Issue Multivariate Approximation for solving ODE and PDE)
16 pages, 4256 KiB  
Article
Blended Root Finding Algorithm Outperforms Bisection and Regula Falsi Algorithms
by Chaman Lal Sabharwal
Mathematics 2019, 7(11), 1118; https://doi.org/10.3390/math7111118 - 16 Nov 2019
Cited by 16 | Viewed by 9359
Abstract
Finding the roots of an equation is a fundamental problem in various fields, including numerical computing, social and physical sciences. Numerical techniques are used when an analytic solution is not available. There is not a single algorithm that works best for every function. [...] Read more.
Finding the roots of an equation is a fundamental problem in various fields, including numerical computing, social and physical sciences. Numerical techniques are used when an analytic solution is not available. There is not a single algorithm that works best for every function. We designed and implemented a new algorithm that is a dynamic blend of the bisection and regula falsi algorithms. The implementation results validate that the new algorithm outperforms both bisection and regula falsi algorithms. It is also observed that the new algorithm outperforms the secant algorithm and the Newton–Raphson algorithm because the new algorithm requires fewer computational iterations and is guaranteed to find a root. The theoretical and empirical evidence shows that the average computational complexity of the new algorithm is considerably less than that of the classical algorithms. Full article
(This article belongs to the Special Issue Numerical Modeling and Analysis)
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11 pages, 288 KiB  
Article
A New Fixed Point Theorem and a New Generalized Hyers-Ulam-Rassias Stability in Incomplete Normed Spaces
by Maryam Ramezani, Ozgur Ege and Manuel De la Sen
Mathematics 2019, 7(11), 1117; https://doi.org/10.3390/math7111117 - 16 Nov 2019
Cited by 3 | Viewed by 2016
Abstract
In this study, our goal is to apply a new fixed point method to prove the Hyers-Ulam-Rassias stability of a quadratic functional equation in normed spaces which are not necessarily Banach spaces. The results of the present paper improve and extend some previous [...] Read more.
In this study, our goal is to apply a new fixed point method to prove the Hyers-Ulam-Rassias stability of a quadratic functional equation in normed spaces which are not necessarily Banach spaces. The results of the present paper improve and extend some previous results. Full article
(This article belongs to the Special Issue Applications in Theoretical and Computational Fixed Point Problems)
7 pages, 241 KiB  
Article
On ω-Limit Sets of Zadeh’s Extension of Nonautonomous Discrete Systems on an Interval
by Guangwang Su and Taixiang Sun
Mathematics 2019, 7(11), 1116; https://doi.org/10.3390/math7111116 - 15 Nov 2019
Viewed by 1819
Abstract
Let I = [ 0 , 1 ] and f n be a sequence of continuous self-maps on I which converge uniformly to a self-map f on I. Denote by F ( I ) the set of fuzzy numbers on I, [...] Read more.
Let I = [ 0 , 1 ] and f n be a sequence of continuous self-maps on I which converge uniformly to a self-map f on I. Denote by F ( I ) the set of fuzzy numbers on I, and denote by ( F ( I ) , f ^ ) and ( F ( I ) , f ^ n ) the Zadeh ′ s extensions of ( I , f ) and ( I , f n ) , respectively. In this paper, we study the ω -limit sets of ( F ( I ) , f ^ n ) and show that, if all periodic points of f are fixed points, then ω ( A , f ^ n ) ⊂ F ( f ^ ) for any A ∈ F ( I ) , where ω ( A , f ^ n ) is the ω -limit set of A under ( F ( I ) , f ^ n ) and F ( f ^ ) = { A ∈ F ( I ) : f ^ ( A ) = A } . Full article
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications)
12 pages, 923 KiB  
Article
Reliability Evaluation for a Stochastic Flow Network Based on Upper and Lower Boundary Vectors
by Ding-Hsiang Huang, Cheng-Fu Huang and Yi-Kuei Lin
Mathematics 2019, 7(11), 1115; https://doi.org/10.3390/math7111115 - 15 Nov 2019
Cited by 10 | Viewed by 2323
Abstract
For stochastic flow network (SFN), given all the lower (or upper) boundary points, the classic problem is to calculate the probability that the capacity vectors are greater than or equal to the lower boundary points (less than or equal to the upper boundary [...] Read more.
For stochastic flow network (SFN), given all the lower (or upper) boundary points, the classic problem is to calculate the probability that the capacity vectors are greater than or equal to the lower boundary points (less than or equal to the upper boundary points). However, in some practical cases, SFN reliability would be evaluated between the lower and upper boundary points at the same time. The evaluation of SFN reliability with upper and lower boundary points at the same time is the focus of this paper. Because of intricate relationships among upper and lower boundary points, a decomposition approach is developed to obtain several simplified subsets. SFN reliability is calculated according to these subsets by means of the inclusion-exclusion principle. Two heuristic options are then established in order to calculate SFN reliability in an efficient direction based on the lower and upper boundary points. Full article
(This article belongs to the Special Issue Statistics and Modeling in Reliability Engineering)
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22 pages, 7970 KiB  
Article
On Solving Modified Helmholtz Equation in Layered Materials Using the Multiple Source Meshfree Approach
by Cheng-Yu Ku, Jing-En Xiao, Weichung Yeih and Chih-Yu Liu
Mathematics 2019, 7(11), 1114; https://doi.org/10.3390/math7111114 - 15 Nov 2019
Cited by 2 | Viewed by 2539
Abstract
This paper presents a study for solving the modified Helmholtz equation in layered materials using the multiple source meshfree approach (MSMA). The key idea of the MSMA starts with the method of fundamental solutions (MFS) as well as the collocation Trefftz method (CTM). [...] Read more.
This paper presents a study for solving the modified Helmholtz equation in layered materials using the multiple source meshfree approach (MSMA). The key idea of the MSMA starts with the method of fundamental solutions (MFS) as well as the collocation Trefftz method (CTM). The multiple source collocation scheme in the MSMA stems from the MFS and the basis functions are formulated using the CTM. The solution of the modified Helmholtz equation is therefore approximated by the superposition theorem using particular nonsingular functions by means of multiple sources located within the domain. To deal with the two-dimensional modified Helmholtz equation in layered materials, the domain decomposition method was adopted. Numerical examples were carried out to validate the method. The results illustrate that the MSMA is relatively simple because it avoids a complicated procedure for finding the appropriate position of the sources. Additionally, the MSMA for solving the modified Helmholtz equation is advantageous because the source points can be collocated on or within the domain boundary and the results are not sensitive to the location of source points. Finally, compared with other methods, highly accurate solutions can be obtained using the proposed method. Full article
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6 pages, 234 KiB  
Article
The Topological Transversality Theorem for Multivalued Maps with Continuous Selections
by Donal O’Regan
Mathematics 2019, 7(11), 1113; https://doi.org/10.3390/math7111113 - 15 Nov 2019
Cited by 1 | Viewed by 1915
Abstract
This paper considers a topological transversality theorem for multivalued maps with continuous, compact selections. Basically, this says, if we have two maps F and G with continuous compact selections and F ≅ G, then one map being essential guarantees the essentiality of [...] Read more.
This paper considers a topological transversality theorem for multivalued maps with continuous, compact selections. Basically, this says, if we have two maps F and G with continuous compact selections and F ≅ G, then one map being essential guarantees the essentiality of the other map. Full article
(This article belongs to the Special Issue Nonlinear Functional Analysis and Its Applications)
17 pages, 2419 KiB  
Article
Using G-Functions to Investigate the Evolutionary Stability of Bacterial Quorum Sensing
by Anne Mund, Christina Kuttler and Judith Pérez-Velázquez
Mathematics 2019, 7(11), 1112; https://doi.org/10.3390/math7111112 - 15 Nov 2019
Cited by 1 | Viewed by 2261
Abstract
In ecology, G-functions can be employed to define a growth function G for a population b, which can then be universally applied to all individuals or groups b i within this population. We can further define a strategy v i for [...] Read more.
In ecology, G-functions can be employed to define a growth function G for a population b, which can then be universally applied to all individuals or groups b i within this population. We can further define a strategy v i for every group b i . Examples for strategies include diverse behaviour such as number of offspring, habitat choice, and time of nesting for birds. In this work, we employ G-functions to investigate the evolutionary stability of the bacterial cooperation process known as quorum sensing. We employ the G-function ansatz to model both the population dynamics and the resulting evolutionary pressure in order to find evolutionary stable states. This results in a semi-linear parabolic system of equations, where cost and benefit are taken into account separately. Depending on different biological assumptions, we analyse a variety of typical model functions. These translate into different long-term scenarios for different functional responses, ranging from single-strategy states to coexistence. As a special feature, we distinguish between the production of public goods, available for all subpopulations, and private goods, from which only the producers can benefit. Full article
(This article belongs to the Special Issue Partial Differential Equations in Ecology: 80 Years and Counting)
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13 pages, 854 KiB  
Article
Removing Twins in Graphs to Break Symmetries
by Antonio González and María Luz Puertas
Mathematics 2019, 7(11), 1111; https://doi.org/10.3390/math7111111 - 15 Nov 2019
Cited by 7 | Viewed by 4107
Abstract
Determining vertex subsets are known tools to provide information about automorphism groups of graphs and, consequently about symmetries of graphs. In this paper, we provide both lower and upper bounds of the minimum size of such vertex subsets, called the determining number of [...] Read more.
Determining vertex subsets are known tools to provide information about automorphism groups of graphs and, consequently about symmetries of graphs. In this paper, we provide both lower and upper bounds of the minimum size of such vertex subsets, called the determining number of the graph. These bounds, which are performed for arbitrary graphs, allow us to compute the determining number in two different graph families such are cographs and unit interval graphs. Full article
(This article belongs to the Special Issue Distances and Domination in Graphs)
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12 pages, 1698 KiB  
Article
Total and Double Total Domination Number on Hexagonal Grid
by Antoaneta Klobučar and Ana Klobučar
Mathematics 2019, 7(11), 1110; https://doi.org/10.3390/math7111110 - 15 Nov 2019
Cited by 6 | Viewed by 2970
Abstract
In this paper, we determine the upper and lower bound for the total domination number and exact values and the upper bound for the double-total domination number on hexagonal grid H m , n with m hexagons in a row and n hexagons [...] Read more.
In this paper, we determine the upper and lower bound for the total domination number and exact values and the upper bound for the double-total domination number on hexagonal grid H m , n with m hexagons in a row and n hexagons in a column. Further, we explore the ratio between the total domination number and the number of vertices of H m , n when m and n tend to infinity. Full article
(This article belongs to the Special Issue Graph Theory at Work in Carbon Chemistry)
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9 pages, 240 KiB  
Article
A Comparison of Methods for Determining the Time Step When Propagating with the Lanczos Algorithm
by N. Mohankumar and Tucker Carrington
Mathematics 2019, 7(11), 1109; https://doi.org/10.3390/math7111109 - 15 Nov 2019
Cited by 2 | Viewed by 2171
Abstract
To use the short iterative Lanczos algorithm to solve the time-dependent Schroedinger equation, one must choose, for a given Lanczos space size, a time step. We compare the derivation of the well-known Lubich and Hochbruck time step from SIAM J. Numer. Anal. 34 [...] Read more.
To use the short iterative Lanczos algorithm to solve the time-dependent Schroedinger equation, one must choose, for a given Lanczos space size, a time step. We compare the derivation of the well-known Lubich and Hochbruck time step from SIAM J. Numer. Anal. 34 (1997) 1911 with the a priori time step we proposed in Mohankumar and Carrington (MC) Comput. Phys. Commun., 181 (2010) 1859 and demonstrate that the MC time step is somewhat larger, i.e., that the MC error bound is tighter. In addition, we use the MC approach to derive an error bound and time step for imaginary time propagation. The error bound we derive is much tighter than the error bound of Stewart and Leyk. Full article
(This article belongs to the Special Issue Computational Methods in Analysis and Applications 2020)
15 pages, 290 KiB  
Article
Nonlinear Impulsive Multi-Order Caputo-Type Generalized Fractional Differential Equations with Infinite Delay
by Bashir Ahmad, Madeaha Alghanmi, Ahmed Alsaedi and Ravi P. Agarwal
Mathematics 2019, 7(11), 1108; https://doi.org/10.3390/math7111108 - 15 Nov 2019
Cited by 6 | Viewed by 2136
Abstract
We establish sufficient conditions for the existence of solutions for a nonlinear impulsive multi-order Caputo-type generalized fractional differential equation with infinite delay and nonlocal generalized integro-initial value conditions. The existence result is proved by means of Krasnoselskii’s fixed point theorem, while the contraction [...] Read more.
We establish sufficient conditions for the existence of solutions for a nonlinear impulsive multi-order Caputo-type generalized fractional differential equation with infinite delay and nonlocal generalized integro-initial value conditions. The existence result is proved by means of Krasnoselskii’s fixed point theorem, while the contraction mapping principle is employed to obtain the uniqueness of solutions for the problem at hand. The paper concludes with illustrative examples. Full article
(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)
15 pages, 788 KiB  
Article
Fundamental Questions and New Counterexamples for b-Metric Spaces and Fatou Property
by Ning Lu, Fei He and Wei-Shih Du
Mathematics 2019, 7(11), 1107; https://doi.org/10.3390/math7111107 - 14 Nov 2019
Cited by 7 | Viewed by 3287
Abstract
In this paper, we give new examples to show that the continuity actually strictly stronger than the Fatou property in b-metric spaces. We establish a new fixed point theorem for new essential and fundamental sufficient conditions such that a Ćirić type contraction [...] Read more.
In this paper, we give new examples to show that the continuity actually strictly stronger than the Fatou property in b-metric spaces. We establish a new fixed point theorem for new essential and fundamental sufficient conditions such that a Ćirić type contraction with contraction constant λ ∈ [ 1 s , 1 ) in a complete b-metric space with s > 1 have a unique fixed point. Many new examples illustrating our results are also given. Our new results extend and improve many recent results and they are completely original and quite different from the well known results on the topic in the literature. Full article
(This article belongs to the Special Issue Recent Advances in Fixed Point and Best Proximity Point Problems and Their Applications)
11 pages, 905 KiB  
Article
Singularities of Non-Developable Surfaces in Three-Dimensional Euclidean Space
by Jie Huang and Donghe Pei
Mathematics 2019, 7(11), 1106; https://doi.org/10.3390/math7111106 - 14 Nov 2019
Cited by 8 | Viewed by 3312
Abstract
We study the singularity on principal normal and binormal surfaces generated by smooth curves with singular points in the Euclidean 3-space. We discover the existence of singular points on such binormal surfaces and study these singularities by the method of singularity theory. By [...] Read more.
We study the singularity on principal normal and binormal surfaces generated by smooth curves with singular points in the Euclidean 3-space. We discover the existence of singular points on such binormal surfaces and study these singularities by the method of singularity theory. By using structure functions, we can characterize the ruled surface generated by special curves. Full article
(This article belongs to the Special Issue Advances in singularities throughout Mathematics: Algebra, Topology, Geometry, Analysis, Number Theory, Combinatorics and Cryptography)
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16 pages, 293 KiB  
Article
Finding Determinant Forms of Certain Hybrid Sheffer Sequences
by Monairah Alansari, Mumtaz Riyasat, Subuhi Khan and Kaleem Raza Kazmi
Mathematics 2019, 7(11), 1105; https://doi.org/10.3390/math7111105 - 14 Nov 2019
Cited by 1 | Viewed by 1662
Abstract
In this article, the integral transform is used to introduce a new family of extended hybrid Sheffer sequences via generating functions and operational rules. The determinant forms and other properties of these sequences are established using a matrix approach. The corresponding results for [...] Read more.
In this article, the integral transform is used to introduce a new family of extended hybrid Sheffer sequences via generating functions and operational rules. The determinant forms and other properties of these sequences are established using a matrix approach. The corresponding results for the extended hybrid Appell sequences are also obtained. Certain examples in terms of the members of the extended hybrid Sheffer and Appell sequences are framed. By employing operational rules, the identities involving the Lah, Stirling and Pascal matrices are derived for the aforementioned sequences. Full article
(This article belongs to the Section Engineering Mathematics)
42 pages, 565 KiB  
Article
Dynamic Restructuring Framework for Scheduling with Release Times and Due-Dates
by Nodari Vakhania
Mathematics 2019, 7(11), 1104; https://doi.org/10.3390/math7111104 - 14 Nov 2019
Cited by 5 | Viewed by 2513
Abstract
Scheduling jobs with release and due dates on a single machine is a classical strongly NP-hard combination optimization problem. It has not only immediate real-life applications but also it is effectively used for the solution of more complex multiprocessor and shop scheduling problems. [...] Read more.
Scheduling jobs with release and due dates on a single machine is a classical strongly NP-hard combination optimization problem. It has not only immediate real-life applications but also it is effectively used for the solution of more complex multiprocessor and shop scheduling problems. Here, we propose a general method that can be applied to the scheduling problems with job release times and due-dates. Based on this method, we carry out a detailed study of the single-machine scheduling problem, disclosing its useful structural properties. These properties give us more insight into the complex nature of the problem and its bottleneck feature that makes it intractable. This method also helps us to expose explicit conditions when the problem can be solved in polynomial time. In particular, we establish the complexity status of the special case of the problem in which job processing times are mutually divisible by constructing a polynomial-time algorithm that solves this setting. Apparently, this setting is a maximal polynomially solvable special case of the single-machine scheduling problem with non-arbitrary job processing times. Full article
(This article belongs to the Special Issue Advances and Novel Approaches in Discrete Optimization)
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