Mathematics

10 pages, 275 KiB

Open AccessArticle

The Convergence of Gallego’s Iterative Method for Distribution-Free Inventory Models

by Ting-Chen Hu, Kuo-Chen Hung and Kuo-Lung Yang

Mathematics 2019, 7(5), 484; https://doi.org/10.3390/math7050484 - 27 May 2019

Cited by 2 | Viewed by 2258

For inventory models with unknown distribution demand, during shortages, researchers used the first and the second moments to derive an upper bound for the worst case, that is the min-max distribution-free procedure for inventory models. They applied an iterative method to generate a [...] Read more.

For inventory models with unknown distribution demand, during shortages, researchers used the first and the second moments to derive an upper bound for the worst case, that is the min-max distribution-free procedure for inventory models. They applied an iterative method to generate a sequence to obtain the optimal order quantity. A researcher developed a three-sequence proof for the convergence of the order quantity sequence. We directly provide proof for the original order quantity sequence. Under our proof, we can construct an increasing sequence and a decreasing sequence that both converge to the optimal order quantity such that we can obtain the optimal solution within the predesigned threshold value. Full article

18 pages, 300 KiB

Open AccessArticle

Some Incomplete Hypergeometric Functions and Incomplete Riemann-Liouville Fractional Integral Operators

by Mehmet Ali Özarslan and Ceren Ustaoğlu

Mathematics 2019, 7(5), 483; https://doi.org/10.3390/math7050483 - 27 May 2019

Cited by 22 | Viewed by 2945

Abstract

Very recently, the incomplete Pochhammer ratios were defined in terms of the incomplete beta function

B_{y} (x, z)

. With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss, confluent hypergeometric, and Appell’s functions and investigate [...] Read more.

Very recently, the incomplete Pochhammer ratios were defined in terms of the incomplete beta function

B_{y} (x, z)

. With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss, confluent hypergeometric, and Appell’s functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas, and recurrence relations. Furthermore, incomplete Riemann-Liouville fractional integral operators are introduced. This definition helps us to obtain linear and bilinear generating relations for the new incomplete Gauss hypergeometric functions. Full article

(This article belongs to the Special Issue Fractional-Order Integral and Derivative Operators and Their Applications)

19 pages, 318 KiB

Open AccessArticle

(C , Ψ * , G ) Class of Contractions and Fixed Points in a Metric Space Endowed with a Graph

by Reny George, Ekta Tamrakar, Jelena Vujaković, Hemant Kumar Pathak and Selvavinayagam Velusamy

Mathematics 2019, 7(5), 482; https://doi.org/10.3390/math7050482 - 27 May 2019

Cited by 2 | Viewed by 2272

Abstract

In this paper, we introduce the

(C, Ψ^{*}, G)

class of contraction mappings using C-class functions and some improved control functions for a pair of set valued mappings as well as a pair of single-valued mappings, and [...] Read more.

In this paper, we introduce the

(C, Ψ^{*}, G)

class of contraction mappings using C-class functions and some improved control functions for a pair of set valued mappings as well as a pair of single-valued mappings, and prove common fixed point theorems for such mappings in a metric space endowed with a graph. Our results unify and generalize many important fixed point results existing in literature. As an application of our main result, we have derived fixed point theorems for a pair of

α

-admissible set valued mappings in a metric space. Full article

(This article belongs to the Special Issue Fixed Point Theory and Related Nonlinear Problems with Applications)

14 pages, 273 KiB

Open AccessArticle

Generalized (σ,ξ)-Contractions and Related Fixed Point Results in a P.M.S

by Vahid Parvaneh, Babak Mohammadi, Hassen Aydi and Aiman Mukheimer

Mathematics 2019, 7(5), 481; https://doi.org/10.3390/math7050481 - 27 May 2019

Viewed by 2190

Abstract

In this paper, we present the concept of

Θ - {(σ, ξ)}_{Ω}

-contraction mappings and we nominate some related fixed point results in ordered p-metric spaces. Our results extend several famous ones in the literature. Some examples and [...] Read more.

In this paper, we present the concept of

Θ - {(σ, ξ)}_{Ω}

-contraction mappings and we nominate some related fixed point results in ordered p-metric spaces. Our results extend several famous ones in the literature. Some examples and an application are given in order to validate our results. Full article

(This article belongs to the Special Issue Applications in Theoretical and Computational Fixed Point Problems)

28 pages, 335 KiB

Open AccessArticle

Coordinating Supply-Chain Management under Stochastic Fuzzy Environment and Lead-Time Reduction

by Asif Iqbal Malik and Biswajit Sarkar

Mathematics 2019, 7(5), 480; https://doi.org/10.3390/math7050480 - 27 May 2019

Cited by 27 | Viewed by 4267

Abstract

In this paper, a supply-chain (SC) coordination method based on the lead-time crashing is proposed for a seller–buyer system. By considering different transportation modes, we control the lead-time (LT) variability. For the first time, we have attempted to determine the impact of the [...] Read more.

In this paper, a supply-chain (SC) coordination method based on the lead-time crashing is proposed for a seller–buyer system. By considering different transportation modes, we control the lead-time (LT) variability. For the first time, we have attempted to determine the impact of the reliable and unreliable seller in a continuous-review supply-chain model under the stochastic environment. The authors discussed two reliability cases for the seller. First, we consider the seller is unreliable and in the second case, the seller is reliable. In addition, the demand during the lead time is stochastic with the known mean and variance. The proposed approach tries to find an optimal solution that performs well without a specific probability distribution. Besides, a discrete investment is made to reduce the setup cost, which will indirectly help supply-chain members to increase the total profit of the system. In the proposed model, the seller motivates the buyer by reducing lead time to take part in coordinating decision-making for the system’s profit optimization. We derive the coordination conditions for both members, the seller and the buyer, under which they are convinced to take part in the cooperative decision-making plan. Therefore, lead-time crashing is the proposed incentive mechanism for collaborative supply-chain management. We use a fixed-charge step function to calculate the lead-time crashing cost for slow and fast shipping mode. We give two numerical examples to validate the proposed models and demonstrate the service-level enhancement under the collaborative supply-chain management in case of an unreliable seller. Concluding remarks and future extensions are discussed at the end. Full article

(This article belongs to the Section Engineering Mathematics)

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13 pages, 338 KiB

Open AccessArticle

Characterization of n-Vertex Graphs of Metric Dimension n − 3 by Metric Matrix

by Juan Wang, Lianying Miao and Yunlong Liu

Mathematics 2019, 7(5), 479; https://doi.org/10.3390/math7050479 - 27 May 2019

Cited by 7 | Viewed by 2384

Abstract

Let

G = (V (G), E (G))

be a connected graph. An ordered set

W \subset V (G)

is a resolving set for G if every vertex of G is uniquely determined by its [...] Read more.

Let

G = (V (G), E (G))

be a connected graph. An ordered set

W \subset V (G)

is a resolving set for G if every vertex of G is uniquely determined by its vector of distances to the vertices in W. The metric dimension of G is the minimum cardinality of a resolving set. In this paper, we characterize the graphs of metric dimension

n - 3

by constructing a special distance matrix, called metric matrix. The metric matrix makes it so a class of graph and its twin graph are bijective and the class of graph is obtained from its twin graph, so it provides a basis for the extension of graphs with respect to metric dimension. Further, the metric matrix gives a new idea of the characterization of extremal graphs based on metric dimension. Full article

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15 pages, 264 KiB

Open AccessArticle

A Mizoguchi–Takahashi Type Fixed Point Theorem in Complete Extended b-Metric Spaces

by Nayab Alamgir, Quanita Kiran, Hassen Aydi and Aiman Mukheimer

Mathematics 2019, 7(5), 478; https://doi.org/10.3390/math7050478 - 26 May 2019

Cited by 11 | Viewed by 3016

Abstract

In this paper, we prove a new fixed point theorem for a multi-valued mapping from a complete extended b-metric space U into the non empty closed and bounded subsets of U, which generalizes Nadler’s fixed point theorem. We also establish some [...] Read more.

In this paper, we prove a new fixed point theorem for a multi-valued mapping from a complete extended b-metric space U into the non empty closed and bounded subsets of U, which generalizes Nadler’s fixed point theorem. We also establish some fixed point results, which generalize our first result. Furthermore, we establish Mizoguchi–Takahashi’s type fixed point theorem for a multi-valued mapping from a complete extended b-metric space U into the non empty closed and bounded subsets of U that improves many existing results in the literature. Full article

(This article belongs to the Special Issue Applications in Theoretical and Computational Fixed Point Problems)

10 pages, 254 KiB

Open AccessArticle

On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process

by Alexander Zeifman, Yacov Satin, Ksenia Kiseleva and Victor Korolev

Mathematics 2019, 7(5), 477; https://doi.org/10.3390/math7050477 - 26 May 2019

Cited by 4 | Viewed by 2475

Abstract

We consider a multidimensional inhomogeneous birth-death process. In this paper, a general situation is studied in which the intensity of birth and death for each coordinate (“each type of particle”) depends on the state vector of the whole process. A one-dimensional projection of [...] Read more.

We consider a multidimensional inhomogeneous birth-death process. In this paper, a general situation is studied in which the intensity of birth and death for each coordinate (“each type of particle”) depends on the state vector of the whole process. A one-dimensional projection of this process on one of the coordinate axes is considered. In this case, a non-Markov process is obtained, in which the transitions to neighboring states are possible in small periods of time. For this one-dimensional process, by modifying the method previously developed by the authors of the note, estimates of the rate of convergence in weakly ergodic and null-ergodic cases are obtained. The simplest example of a two-dimensional process of this type is considered. Full article

(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)

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17 pages, 308 KiB

Open AccessArticle

On Some New Fixed Point Results in Complete Extended b-Metric Spaces

by Quanita Kiran, Nayab Alamgir, Nabil Mlaiki and Hassen Aydi

Mathematics 2019, 7(5), 476; https://doi.org/10.3390/math7050476 - 25 May 2019

Cited by 15 | Viewed by 3677

Abstract

In this paper, we specified a method that generalizes a number of fixed point results for single and multi-valued mappings in the structure of extended b-metric spaces. Our results extend several existing ones including the results of Aleksic et al. for single-valued [...] Read more.

In this paper, we specified a method that generalizes a number of fixed point results for single and multi-valued mappings in the structure of extended b-metric spaces. Our results extend several existing ones including the results of Aleksic et al. for single-valued mappings and the results of Nadler and Miculescu et al. for multi-valued mappings. Moreover, an example is given at the end to show the superiority of our results. Full article

(This article belongs to the Special Issue Fixed Point Theory and Dynamical Systems with Applications)

20 pages, 1519 KiB

Open AccessArticle

An Efficient Memetic Algorithm for the Minimum Load Coloring Problem

by Zhiqiang Zhang, Zhongwen Li, Xiaobing Qiao and Weijun Wang

Mathematics 2019, 7(5), 475; https://doi.org/10.3390/math7050475 - 25 May 2019

Cited by 4 | Viewed by 2556

Abstract

Given a graph G with n vertices and l edges, the load distribution of a coloring q: V → {red, blue} is defined as d_q = (r_q, b_q), in which r_q is the number of [...] Read more.

Given a graph G with n vertices and l edges, the load distribution of a coloring q: V → {red, blue} is defined as d_q = (r_q, b_q), in which r_q is the number of edges with at least one end-vertex colored red and b_q is the number of edges with at least one end-vertex colored blue. The minimum load coloring problem (MLCP) is to find a coloring q such that the maximum load, l_q = 1/l × max{r_q, b_q}, is minimized. This problem has been proved to be NP-complete. This paper proposes a memetic algorithm for MLCP based on an improved K-OPT local search and an evolutionary operation. Furthermore, a data splitting operation is executed to expand the data amount of global search, and a disturbance operation is employed to improve the search ability of the algorithm. Experiments are carried out on the benchmark DIMACS to compare the searching results from memetic algorithm and the proposed algorithms. The experimental results show that a greater number of best results for the graphs can be found by the memetic algorithm, which can improve the best known results of MLCP. Full article

(This article belongs to the Special Issue Evolutionary Computation)

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42 pages, 3716 KiB

Open AccessFeature PaperArticle

Change Point Detection for Airborne Particulate Matter (PM_2.5, PM₁₀) by Using the Bayesian Approach

by Muhammad Rizwan Khan and Biswajit Sarkar

Mathematics 2019, 7(5), 474; https://doi.org/10.3390/math7050474 - 24 May 2019

Cited by 5 | Viewed by 2901

Abstract

Airborne particulate matter (PM) is a key air pollutant that affects human health adversely. Exposure to high concentrations of such particles may cause premature death, heart disease, respiratory problems, or reduced lung function. Previous work on particulate matter (

P M_{2.5}

and [...] Read more.

Airborne particulate matter (PM) is a key air pollutant that affects human health adversely. Exposure to high concentrations of such particles may cause premature death, heart disease, respiratory problems, or reduced lung function. Previous work on particulate matter (

P M_{2.5}

and

P M_{10}

) was limited to specific areas. Therefore, more studies are required to investigate airborne particulate matter patterns due to their complex and varying properties, and their associated (

P M_{10}

and

P M_{2.5}

) concentrations and compositions to assess the numerical productivity of pollution control programs for air quality. Consequently, to control particulate matter pollution and to make effective plans for counter measurement, it is important to measure the efficiency and efficacy of policies applied by the Ministry of Environment. The primary purpose of this research is to construct a simulation model for the identification of a change point in particulate matter (

P M_{2.5}

and

P M_{10}

) concentration, and if it occurs in different areas of the world. The methodology is based on the Bayesian approach for the analysis of different data structures and a likelihood ratio test is used to a detect change point at unknown time (k). Real time data of particulate matter concentrations at different locations has been used for numerical verification. The model parameters before change point (

θ

) and parameters after change point (

λ

) have been critically analyzed so that the proficiency and success of environmental policies for particulate matter (

P M_{2.5}

and

P M_{10}

) concentrations can be evaluated. The main reason for using different areas is their considerably different features, i.e., environment, population densities, and transportation vehicle densities. Consequently, this study also provides insights about how well this suggested model could perform in different areas. Full article

(This article belongs to the Special Issue Application of Optimization in Production, Logistics, Inventory, Supply Chain Management and Block Chain)

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16 pages, 3377 KiB

Open AccessArticle

A Probabilistic Classification Procedure Based on Response Time Analysis Towards a Quick Pre-Diagnosis of Student’s Attention Deficit

by M. Hernaiz-Guijarro, J. C. Castro-Palacio, E. Navarro-Pardo, J. M. Isidro and P. Fernández-de-Córdoba

Mathematics 2019, 7(5), 473; https://doi.org/10.3390/math7050473 - 24 May 2019

Cited by 10 | Viewed by 3680

Abstract

A classification methodology based on an experimental study is proposed towards a fast pre-diagnosis of attention deficit. Our sample consisted of school-aged children between 8 and 12 years from Valencia, Spain. The study was based on the response time (RT) to visual stimuli [...] Read more.

A classification methodology based on an experimental study is proposed towards a fast pre-diagnosis of attention deficit. Our sample consisted of school-aged children between 8 and 12 years from Valencia, Spain. The study was based on the response time (RT) to visual stimuli in computerized tasks. The process of answering consecutive questions usually follows an ex-Gaussian distribution of the RTs. Specifically, we seek to propose a simple automatic classification scheme of children based on the most recent evidence of the relationship between RTs and ADHD. Specifically, the prevalence percentage and reported evidence for RTs in relation to ADHD or to attention deficit symptoms were taken as reference in our study. We explain step by step how to go from the computer-based experiments and through the data analysis. Our desired aim is to provide a methodology to determine quickly those children who behave differently from the mean child in terms of response times and thus are potential candidates to be diagnosed for ADHD or any another cognitive disorder related to attention deficit. This is highly desirable as there is an urgent need for objective instruments to diagnose attention deficit symptomatology. Most of the methodologies available nowadays lead to an overdiagnosis of ADHD and are not based on direct measurement but on interviews of people related to the child such as parents or teachers. Although the ultimate diagnosis must be made by a psychologist, the selection provided by a methodology like ours could allow them to focus on assessing a smaller number of candidates which would help save time and other resources. Full article

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19 pages, 525 KiB

Open AccessArticle

On Subtrees of Fan Graphs, Wheel Graphs, and “Partitions” of Wheel Graphs under Dynamic Evolution

by Yu Yang, An Wang, Hua Wang, Wei-Ting Zhao and Dao-Qiang Sun

Mathematics 2019, 7(5), 472; https://doi.org/10.3390/math7050472 - 24 May 2019

Cited by 3 | Viewed by 3567

Abstract

The number of subtrees, or simply the subtree number, is one of the most studied counting-based graph invariants that has applications in many interdisciplinary fields such as phylogenetic reconstruction. Motivated from the study of graph surgeries on evolutionary dynamics, we consider the subtree [...] Read more.

The number of subtrees, or simply the subtree number, is one of the most studied counting-based graph invariants that has applications in many interdisciplinary fields such as phylogenetic reconstruction. Motivated from the study of graph surgeries on evolutionary dynamics, we consider the subtree problems of fan graphs, wheel graphs, and the class of graphs obtained from “partitioning” wheel graphs under dynamic evolution. The enumeration of these subtree numbers is done through the so-called subtree generation functions of graphs. With the enumerative result, we briefly explore the extremal problems in the corresponding class of graphs. Some interesting observations on the behavior of the subtree number are also presented. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

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16 pages, 320 KiB

Open AccessArticle

On Separate Fractional Sum-Difference Equations with n-Point Fractional Sum-Difference Boundary Conditions via Arbitrary Different Fractional Orders

by Saowaluck Chasreechai and Thanin Sitthiwirattham

Mathematics 2019, 7(5), 471; https://doi.org/10.3390/math7050471 - 24 May 2019

Cited by 7 | Viewed by 1959

Abstract

In this article, we study the existence and uniqueness results for a separate nonlinear Caputo fractional sum-difference equation with fractional difference boundary conditions by using the Banach contraction principle and the Schauder’s fixed point theorem. Our problem contains two nonlinear functions involving fractional [...] Read more.

In this article, we study the existence and uniqueness results for a separate nonlinear Caputo fractional sum-difference equation with fractional difference boundary conditions by using the Banach contraction principle and the Schauder’s fixed point theorem. Our problem contains two nonlinear functions involving fractional difference and fractional sum. Moreover, our problem contains different orders in

n + 1

fractional differences and

m + 1

fractional sums. Finally, we present an illustrative example. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications)

18 pages, 1014 KiB

Open AccessArticle

New Concepts of Picture Fuzzy Graphs with Application

by Cen Zuo, Anita Pal and Arindam Dey

Mathematics 2019, 7(5), 470; https://doi.org/10.3390/math7050470 - 24 May 2019

Cited by 62 | Viewed by 6728

Abstract

The picture fuzzy set is an efficient mathematical model to deal with uncertain real life problems, in which a intuitionistic fuzzy set may fail to reveal satisfactory results. Picture fuzzy set is an extension of the classical fuzzy set and intuitionistic fuzzy set. [...] Read more.

The picture fuzzy set is an efficient mathematical model to deal with uncertain real life problems, in which a intuitionistic fuzzy set may fail to reveal satisfactory results. Picture fuzzy set is an extension of the classical fuzzy set and intuitionistic fuzzy set. It can work very efficiently in uncertain scenarios which involve more answers to these type: yes, no, abstain and refusal. In this paper, we introduce the idea of the picture fuzzy graph based on the picture fuzzy relation. Some types of picture fuzzy graph such as a regular picture fuzzy graph, strong picture fuzzy graph, complete picture fuzzy graph, and complement picture fuzzy graph are introduced and some properties are also described. The idea of an isomorphic picture fuzzy graph is also introduced in this paper. We also define six operations such as Cartesian product, composition, join, direct product, lexicographic and strong product on picture fuzzy graph. Finally, we describe the utility of the picture fuzzy graph and its application in a social network. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications)

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10 pages, 251 KiB

Open AccessArticle

Solving Bi-Matrix Games Based on Fuzzy Payoffs via Utilizing the Interval Value Function Method

by Kaisheng Liu and Yumei Xing

Mathematics 2019, 7(5), 469; https://doi.org/10.3390/math7050469 - 24 May 2019

Cited by 3 | Viewed by 2211

Abstract

In this article, we introduce a model of bi-matrix games based on crisp parametric payoffs via utilizing the method of interval value function. Then, we get that equilibrium solutions of bi-matrix games on the basis of fuzzy payoffs and equilibrium solutions of the [...] Read more.

In this article, we introduce a model of bi-matrix games based on crisp parametric payoffs via utilizing the method of interval value function. Then, we get that equilibrium solutions of bi-matrix games on the basis of fuzzy payoffs and equilibrium solutions of the game model are of equal value. Furthermore, it is concluded that equilibrium solutions of the game can be converted to optimal solutions of discrete nonlinear optimization problems with parameters. Lastly, the proposed methodology is illustrated by an example. Full article

(This article belongs to the Section Mathematics and Computer Science)

28 pages, 1324 KiB

Open AccessArticle

Some q-Rung Picture Fuzzy Dombi Hamy Mean Operators with Their Application to Project Assessment

by Jiahuan He, Xindi Wang, Runtong Zhang and Li Li

Mathematics 2019, 7(5), 468; https://doi.org/10.3390/math7050468 - 24 May 2019

Cited by 32 | Viewed by 3360

Abstract

The recently proposed q-rung picture fuzzy set (q-RPFSs) can describe complex fuzzy and uncertain information effectively. The Hamy mean (HM) operator gets good performance in the process of information aggregation due to its ability to capturing the interrelationships among aggregated [...] Read more.

The recently proposed q-rung picture fuzzy set (q-RPFSs) can describe complex fuzzy and uncertain information effectively. The Hamy mean (HM) operator gets good performance in the process of information aggregation due to its ability to capturing the interrelationships among aggregated values. In this study, we extend HM to q-rung picture fuzzy environment, propose novel q-rung picture fuzzy aggregation operators, and demonstrate their application to multi-attribute group decision-making (MAGDM). First of all, on the basis of Dombi t-norm and t-conorm (DTT), we propose novel operational rules of q-rung picture fuzzy numbers (q-RPFNs). Second, we propose some new aggregation operators of q-RPFNs based on the newly-developed operations, i.e., the q-rung picture fuzzy Dombi Hamy mean (q-RPFDHM) operator, the q-rung picture fuzzy Dombi weighted Hamy mean (q-RPFDWHM) operator, the q-rung picture fuzzy Dombi dual Hamy mean (q-RPFDDHM) operator, and the q-rung picture fuzzy Dombi weighted dual Hamy mean (q-RPFDWDHM) operator. Properties of these operators are also discussed. Third, a new q-rung picture fuzzy MAGDM method is proposed with the help of the proposed operators. Finally, a best project selection example is provided to demonstrate the practicality and effectiveness of the new method. The superiorities of the proposed method are illustrated through comparative analysis. Full article

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11 pages, 250 KiB

Open AccessArticle

Hermite-Hadamard-Fejér Type Inequalities for Preinvex Functions Using Fractional Integrals

by Sikander Mehmood, Fiza Zafar and Nusrat Yasmin

Mathematics 2019, 7(5), 467; https://doi.org/10.3390/math7050467 - 24 May 2019

Cited by 9 | Viewed by 2710

Abstract

In this paper, we have established the Hermite–Hadamard–Fejér inequality for fractional integrals involving preinvex functions. The results presented here provide new extensions of those given in earlier works as the weighted estimates of the left and right hand side of the Hermite–Hadamard inequalities [...] Read more.

In this paper, we have established the Hermite–Hadamard–Fejér inequality for fractional integrals involving preinvex functions. The results presented here provide new extensions of those given in earlier works as the weighted estimates of the left and right hand side of the Hermite–Hadamard inequalities for fractional integrals involving preinvex functions doesn’t exist previously. Full article

19 pages, 326 KiB

Open AccessArticle

New Inertial Forward-Backward Mid-Point Methods for Sum of Infinitely Many Accretive Mappings, Variational Inequalities, and Applications

by Li Wei, Yingzi Shang and Ravi P. Agarwal

Mathematics 2019, 7(5), 466; https://doi.org/10.3390/math7050466 - 24 May 2019

Cited by 3 | Viewed by 1998

Abstract

Some new inertial forward-backward projection iterative algorithms are designed in a real Hilbert space. Under mild assumptions, some strong convergence theorems for common zero points of the sum of two kinds of infinitely many accretive mappings are proved. New projection sets are constructed [...] Read more.

Some new inertial forward-backward projection iterative algorithms are designed in a real Hilbert space. Under mild assumptions, some strong convergence theorems for common zero points of the sum of two kinds of infinitely many accretive mappings are proved. New projection sets are constructed which provide multiple choices of the iterative sequences. Some already existing iterative algorithms are demonstrated to be special cases of ours. Some inequalities of metric projection and real number sequences are widely used in the proof of the main results. The iterative algorithms have also been modified and extended from pure discussion on the sum of accretive mappings or pure study on variational inequalities to that for both, which complements the previous work. Moreover, the applications of the abstract results on nonlinear capillarity systems are exemplified. Full article

(This article belongs to the Special Issue Fixed Point Theory and Related Nonlinear Problems with Applications)

20 pages, 880 KiB

Open AccessArticle

Multi-Product Production System with the Reduced Failure Rate and the Optimum Energy Consumption under Variable Demand

by Shaktipada Bhuniya, Biswajit Sarkar and Sarla Pareek

Mathematics 2019, 7(5), 465; https://doi.org/10.3390/math7050465 - 24 May 2019

Cited by 32 | Viewed by 3675

Abstract

The advertising of any smart product is crucial in generating customer demand, along with reducing sale prices. Naturally, a decrease in price always increases the demand for any smart product. This study introduces a multi-product production process, taking into consideration the advertising- and [...] Read more.

The advertising of any smart product is crucial in generating customer demand, along with reducing sale prices. Naturally, a decrease in price always increases the demand for any smart product. This study introduces a multi-product production process, taking into consideration the advertising- and price-dependent demands of products, where the failure rate of the production system is reduced under the optimum energy consumption. For long-run production systems, unusual energy consumption and machine failures occur frequently, which are reduced in this study. All costs related with the production system are included in the optimum energy costs. The unit production cost is dependent on the production rate of the machine and its failure rate. The aim of this study is to obtain the optimum profit with a reduced failure rate, under the optimum advertising costs and the optimum sale price. The total profit of the model becomes a complex, non-linear function, with respect to the decision variables. For this reason, the model is solved numerically by an iterative method. However, the global optimality is proved numerically, by using the Hessian matrix. The numerical results obtained show that for smart production, the maximum profit always occurs at the optimum values of the decision variables. Full article

(This article belongs to the Special Issue Application of Optimization in Production, Logistics, Inventory, Supply Chain Management and Block Chain)

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18 pages, 467 KiB

Open AccessArticle

A New Method to Optimize the Satisfaction Level of the Decision Maker in Fuzzy Geometric Programming Problems

by Armita Khorsandi, Bing-Yuan Cao and Hadi Nasseri

Mathematics 2019, 7(5), 464; https://doi.org/10.3390/math7050464 - 23 May 2019

Cited by 3 | Viewed by 2950

Abstract

Geometric programming problems are well-known in mathematical modeling. They are broadly used in diverse practical fields that are contemplated through an appropriate methodology. In this paper, a multi-parametric vector

α

is proposed for approaching the highest decision maker satisfaction. Hitherto, the simple parameter [...] Read more.

Geometric programming problems are well-known in mathematical modeling. They are broadly used in diverse practical fields that are contemplated through an appropriate methodology. In this paper, a multi-parametric vector

α

is proposed for approaching the highest decision maker satisfaction. Hitherto, the simple parameter

α

, which has a scalar role, has been considered in the problem. The parameter

α

is a vector whose range is within the region of the satisfaction area. Conventionally, it is assumed that the decision maker is sure about the parameters, but, in reality, it is mostly hesitant about them, so the parameters are presented in fuzzy numbers. In this method, the decision maker can attain different satisfaction levels in each constraint, and even full satisfaction can be reached in some constraints. The goal is to find the highest satisfaction degree to maintain an optimal solution. Moreover, the objective function is turned into a constraint, i.e., one more dimension is added to n-dimensional multi-parametric

α

. Thus, the fuzzy geometric programming problem under this multi-parametric vector

α \in {(0, 1]}^{n + 1}

gives a maximum satisfaction level to the decision maker. A numerical example is presented to illustrate the proposed method and the superiority of this multi-parametric

α

over the simple one. Full article

(This article belongs to the Special Issue Operations Research Using Fuzzy Sets Theory)

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13 pages, 297 KiB

Open AccessArticle

Unified Local Convergence for Newton’s Method and Uniqueness of the Solution of Equations under Generalized Conditions in a Banach Space

by Ioannis K. Argyros, Ángel Alberto Magreñán, Lara Orcos and Íñigo Sarría

Mathematics 2019, 7(5), 463; https://doi.org/10.3390/math7050463 - 23 May 2019

Cited by 3 | Viewed by 2131

Abstract

Under the hypotheses that a function and its Fréchet derivative satisfy some generalized Newton–Mysovskii conditions, precise estimates on the radii of the convergence balls of Newton’s method, and of the uniqueness ball for the solution of the equations, are given for Banach space-valued [...] Read more.

Under the hypotheses that a function and its Fréchet derivative satisfy some generalized Newton–Mysovskii conditions, precise estimates on the radii of the convergence balls of Newton’s method, and of the uniqueness ball for the solution of the equations, are given for Banach space-valued operators. Some of the existing results are improved with the advantages of larger convergence region, tighter error estimates on the distances involved, and at-least-as-precise information on the location of the solution. These advantages are obtained using the same functions and Lipschitz constants as in earlier studies. Numerical examples are used to test the theoretical results. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

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20 pages, 1542 KiB

Open AccessArticle

Total Least Squares Spline Approximation

by Frank Neitzel, Nikolaj Ezhov and Svetozar Petrovic

Mathematics 2019, 7(5), 462; https://doi.org/10.3390/math7050462 - 22 May 2019

Cited by 8 | Viewed by 5255

Abstract

Spline approximation, using both values

y_{i}

and

x_{i}

as observations, is of vital importance for engineering geodesy, e.g., for approximation of profiles measured with terrestrial laser scanners, because it enables the consideration of arbitrary dispersion matrices for the observations. In the [...] Read more.

Spline approximation, using both values

y_{i}

and

x_{i}

as observations, is of vital importance for engineering geodesy, e.g., for approximation of profiles measured with terrestrial laser scanners, because it enables the consideration of arbitrary dispersion matrices for the observations. In the special case of equally weighted and uncorrelated observations, the resulting error vectors are orthogonal to the graph of the spline function and hence can be utilized for deformation monitoring purposes. Based on a functional model that uses cubic polynomials and constraints for continuity, smoothness and continuous curvature, the case of spline approximation with both the values

y_{i}

and

x_{i}

as observations is considered. In this case, some of the columns of the functional matrix contain observations and are thus subject to random errors. In the literature on mathematics and statistics this case is known as an errors-in-variables (EIV) model for which a so-called “total least squares” (TLS) solution can be computed. If weights for the observations and additional constraints for the unknowns are introduced, a “constrained weighted total least squares” (CWTLS) problem is obtained. In this contribution, it is shown that the solution for this problem can be obtained from a rigorous solution of an iteratively linearized Gauss-Helmert (GH) model. The advantage of this model is that it does not impose any restrictions on the form of the functional relationship between the involved quantities. Furthermore, dispersion matrices can be introduced without limitations, even the consideration of singular ones is possible. Therefore, the iteratively linearized GH model can be regarded as a generalized approach for solving CWTLS problems. Using a numerical example it is demonstrated how the GH model can be applied to obtain a spline approximation with orthogonal error vectors. The error vectors are compared with those derived from two least squares (LS) approaches. Full article

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8 pages, 270 KiB

Open AccessArticle

Picard-Jungck Operator for a Pair of Mappings and Simulation Type Functions

by Sumit Chandok, Aiman Mukheimer, Azhar Hussain and Ljiljana Paunović

Mathematics 2019, 7(5), 461; https://doi.org/10.3390/math7050461 - 22 May 2019

Cited by 4 | Viewed by 2100

Abstract

In this manuscript, we propose a new class of Picard-Jungck operators for a pair of mappings on complete metric spaces by taking into account of the

C_{G}

-simulation function. Also, some new results for the existence of such operators for a pair [...] Read more.

In this manuscript, we propose a new class of Picard-Jungck operators for a pair of mappings on complete metric spaces by taking into account of the

C_{G}

-simulation function. Also, some new results for the existence of such operators for a pair of self mappings in the setting of metric spaces are obtained. Some nontrivial examples are presented to show the usability of the results. Full article

(This article belongs to the Special Issue Applications in Theoretical and Computational Fixed Point Problems)

19 pages, 905 KiB

Open AccessArticle

An Optimized Network Representation Learning Algorithm Using Multi-Relational Data

by Zhonglin Ye, Haixing Zhao, Ke Zhang, Yu Zhu and Zhaoyang Wang

Mathematics 2019, 7(5), 460; https://doi.org/10.3390/math7050460 - 21 May 2019

Viewed by 2364

Abstract

Representation learning aims to encode the relationships of research objects into low-dimensional, compressible, and distributed representation vectors. The purpose of network representation learning is to learn the structural relationships between network vertices. Knowledge representation learning is oriented to model the entities and relationships [...] Read more.

Representation learning aims to encode the relationships of research objects into low-dimensional, compressible, and distributed representation vectors. The purpose of network representation learning is to learn the structural relationships between network vertices. Knowledge representation learning is oriented to model the entities and relationships in knowledge bases. In this paper, we first introduce the idea of knowledge representation learning into network representation learning, namely, we propose a new approach to model the vertex triplet relationships based on DeepWalk without TransE. Consequently, we propose an optimized network representation learning algorithm using multi-relational data, MRNR, which introduces the multi-relational data between vertices into the procedures of network representation learning. Importantly, we adopted a kind of higher order transformation strategy to optimize the learnt network representation vectors. The purpose of MRNR is that multi-relational data (triplets) can effectively guide and constrain the procedures of network representation learning. The experimental results demonstrate that the proposed MRNR can learn the discriminative network representations, which show better performance on network classification, visualization, and case study tasks compared to the proposed baseline algorithms in this paper. Full article

(This article belongs to the Section Network Science)

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11 pages, 586 KiB

Open AccessArticle

On a Variational Method for Stiff Differential Equations Arising from Chemistry Kinetics

by Sergio Amat, María José Legaz and Juan Ruiz-Álvarez

Mathematics 2019, 7(5), 459; https://doi.org/10.3390/math7050459 - 21 May 2019

Cited by 11 | Viewed by 2675

Abstract

For the approximation of stiff systems of ODEs arising from chemistry kinetics, implicit integrators emerge as good candidates. This paper proposes a variational approach for this type of systems. In addition to introducing the technique, we present its most basic properties and test [...] Read more.

For the approximation of stiff systems of ODEs arising from chemistry kinetics, implicit integrators emerge as good candidates. This paper proposes a variational approach for this type of systems. In addition to introducing the technique, we present its most basic properties and test its numerical performance through some experiments. The main advantage with respect to other implicit methods is that our approach has a global convergence. The other approaches need to ensure convergence of the iterative scheme used to approximate the associated nonlinear equations that appear for the implicitness. Notice that these iterative methods, for these nonlinear equations, have bounded basins of attraction. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

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13 pages, 260 KiB

Open AccessArticle

A New Subclass of Analytic Functions Defined by Using Salagean q-Differential Operator

by Muhammad Naeem, Saqib Hussain, Tahir Mahmood, Shahid Khan and Maslina Darus

Mathematics 2019, 7(5), 458; https://doi.org/10.3390/math7050458 - 21 May 2019

Cited by 19 | Viewed by 3296

Abstract

In our present investigation, we use the technique of convolution and quantum calculus to study the Salagean q-differential operator. By using this operator and the concept of the Janowski function, we define certain new classes of analytic functions. Some properties of these [...] Read more.

In our present investigation, we use the technique of convolution and quantum calculus to study the Salagean q-differential operator. By using this operator and the concept of the Janowski function, we define certain new classes of analytic functions. Some properties of these classes are discussed, and numerous sharp results such as coefficient estimates, distortion theorem, radii of star-likeness, convexity, close-to-convexity, extreme points, and integral mean inequalities of functions belonging to these classes are obtained and studied. Full article

(This article belongs to the Special Issue Complex Analysis and Its Applications)

16 pages, 356 KiB

Open AccessArticle

On the Performance of Variable Selection and Classification via Rank-Based Classifier

by Md Showaib Rahman Sarker, Michael Pokojovy and Sangjin Kim

Mathematics 2019, 7(5), 457; https://doi.org/10.3390/math7050457 - 21 May 2019

Cited by 2 | Viewed by 3116

Abstract

In high-dimensional gene expression data analysis, the accuracy and reliability of cancer classification and selection of important genes play a very crucial role. To identify these important genes and predict future outcomes (tumor vs. non-tumor), various methods have been proposed in the literature. [...] Read more.

In high-dimensional gene expression data analysis, the accuracy and reliability of cancer classification and selection of important genes play a very crucial role. To identify these important genes and predict future outcomes (tumor vs. non-tumor), various methods have been proposed in the literature. But only few of them take into account correlation patterns and grouping effects among the genes. In this article, we propose a rank-based modification of the popular penalized logistic regression procedure based on a combination of

ℓ_{1}

and

ℓ_{2}

penalties capable of handling possible correlation among genes in different groups. While the

ℓ_{1}

penalty maintains sparsity, the

ℓ_{2}

penalty induces smoothness based on the information from the Laplacian matrix, which represents the correlation pattern among genes. We combined logistic regression with the BH-FDR (Benjamini and Hochberg false discovery rate) screening procedure and a newly developed rank-based selection method to come up with an optimal model retaining the important genes. Through simulation studies and real-world application to high-dimensional colon cancer gene expression data, we demonstrated that the proposed rank-based method outperforms such currently popular methods as lasso, adaptive lasso and elastic net when applied both to gene selection and classification. Full article

(This article belongs to the Special Issue Uncertainty Quantification Techniques in Statistics)

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11 pages, 808 KiB

Open AccessArticle

Approximation-Free Output-Feedback Non-Backstepping Controller for Uncertain SISO Nonautonomous Nonlinear Pure-Feedback Systems

by Jang-Hyun Park, Tae-Sik Park and Seong-Hwan Kim

Mathematics 2019, 7(5), 456; https://doi.org/10.3390/math7050456 - 21 May 2019

Cited by 11 | Viewed by 2848

Abstract

A novel differentiator-based approximation-free output-feedback controller for uncertain nonautonomous nonlinear pure-feedback systems is proposed. Using high-order sliding mode observer, which is a finite-time exact differentiator, the time-derivatives of the signal generated using tracking error and filtered input are directly estimated. As a result, [...] Read more.

A novel differentiator-based approximation-free output-feedback controller for uncertain nonautonomous nonlinear pure-feedback systems is proposed. Using high-order sliding mode observer, which is a finite-time exact differentiator, the time-derivatives of the signal generated using tracking error and filtered input are directly estimated. As a result, the proposed non-backstepping control law and stability analysis are drastically simple. The tracking error vector is guaranteed to be exponentially stable in finite time regardless of the nonautonomous property in the considered system. It does not require neural networks or fuzzy logic systems, which are typically adopted to capture unstructured uncertainties intrinsic in the controlled system. As far as the authors know, there are no research results on the output-feedback controller for the uncertain nonautonomous pure-feedback nonlinear systems. The results of the simulation show clearly the performance and compactness of the control scheme proposed. Full article

(This article belongs to the Section Engineering Mathematics)

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6 pages, 227 KiB

Open AccessArticle

Some Remarks on a Variational Method for Stiff Differential Equations

by Sergio Amat, María José Legaz and Pablo Pedregal

Mathematics 2019, 7(5), 455; https://doi.org/10.3390/math7050455 - 20 May 2019

Cited by 1 | Viewed by 2015

Abstract

We have recently proposed a variational framework for the approximation of systems of differential equations. We associated, in a natural way, with the original problem, a certain error functional. The discretization is based on standard descent schemes, and we can use a variable-step [...] Read more.

We have recently proposed a variational framework for the approximation of systems of differential equations. We associated, in a natural way, with the original problem, a certain error functional. The discretization is based on standard descent schemes, and we can use a variable-step implementation. The minimization problem has a unique solution, and the approach has a global convergence. The use of our error-functional strategy was considered by other authors, but using a completely different way to derive the discretization. Their technique was based on the use of an integral form of the Euler equation for a related optimal control problem, combined with an adapted version of the shooting method, and the cyclic coordinate descent method. In this note, we illustrate and compare our strategy to theirs from a numerical point of view. Full article

(This article belongs to the Special Issue Computational Methods in Applied Analysis and Mathematical Modeling)

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Mathematics, Volume 7, Issue 5 (May 2019) – 106 articles

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