Mathematics, Volume 7, Issue 3 (March 2019) – 94 articles

The purpose of this paper is to establish fixed point results for a pair ${\alpha }_{\ast }$ -dominated multivalued mappings fulfilling generalized locally new ${\alpha }_{\ast }$ - $\psi$ -Ćirić type rational contractive conditions on a closed ball in complete dislocated metric spaces. Examples and applications are given to demonstrate the novelty of our results. Our results extend several comparable results in the existing literature. Full article

Steffensen-type methods with memory were originally designed to solve nonlinear equations without the use of additional functional evaluations per computing step. In this paper, a variant of Steffensen’s method is proposed which is derivative-free and with memory. In fact, using an acceleration technique via interpolation polynomials of appropriate degrees, the computational efficiency index of this scheme is improved. It is discussed that the new scheme is quite fast and has a high efficiency index. Finally, numerical investigations are brought forward to uphold the theoretical discussions. Full article

In this paper, we consider an insurance risk model with mixed premium income, in which both constant premium income and stochastic premium income are considered. We assume that the stochastic premium income process follows a compound Poisson process and the premium sizes are exponentially distributed. A new method for estimating the expected discounted penalty function by Fourier-cosine series expansion is proposed. We show that the estimation is easily computed, and it has a fast convergence rate. Some numerical examples are also provided to show the good properties of the estimation when the sample size is finite. Full article

Mass vaccination campaigns play major roles in the war against epidemics. Such prevention strategies cannot always reach their goals significantly without the help of media and awareness campaigns used to prevent contacts between susceptible and infected people. Feelings of fear, infodemics, and misconception could lead to some fluctuations of such policies. In addition to the vaccination strategy, the movement restriction approach is essential because of the factor of mobility or travel. However, anti-epidemic border measures may also be disturbed if some infected travelers manage to escape and infiltrate into a safer region. In this paper, we aim to study infection dynamics related to the spatial spread of an epidemic in interconnected regions in the presence of random perturbations caused by the three above-mentioned reasons. Therefore, we devise a stochastic multi-region epidemic model in which contacts between susceptible and infected populations, vaccination-based and movement restriction optimal control approaches are all assumed to be unpredictable, and then, we discuss the effectiveness of such policies. In order to reach our goal, we employ a stochastic maximum principle version for noised systems, state and prove the sufficient and necessary conditions of optimality, and finally provide the numerical results obtained using a stochastic progressive-regressive schemes method. Full article

In this paper, we study those fuzzy metrics M on X, in the George and Veeramani’s sense, such that ${\bigwedge }_{t>0}M(x,y,t)>0$ . The continuous extension $M^0$ of M to $X^2\times \left[0,+\infty \right[$ is called extended fuzzy metric. We prove that $M^0$ generates a metrizable topology on X, which can be described in a similar way to a classical metric. $M^0$ can be used for simplifying or improving questions concerning M; in particular, we expose the interest of this kind of fuzzy metrics to obtain generalizations of fixed point theorems given in fuzzy metric spaces. Full article

In this paper, we study an absolutely new problem, namely, the Cayley inclusion problem which involves the Cayley operator and a multi-valued mapping with XOR-operation. We have shown that the Cayley operator is a single-valued comparison and it is Lipschitz-type-continuous. A fixed point formulation of the Cayley inclusion problem is shown by using the concept of a resolvent operator as well as the Yosida approximation operator. Finally, an existence and convergence result is proved. An example is constructed for some of the concepts used in this work. Full article

In this paper, we consider a two-machine job-shop scheduling problem of minimizing total completion time subject to n jobs with two operations and equal processing times on each machine. This problem occurs e.g., as a single-track railway scheduling problem with three stations and constant travel times between any two adjacent stations. We present a polynomial dynamic programming algorithm of the complexity $O(n^5)$ and a heuristic procedure of the complexity $O(n^3)$ . This settles the complexity status of the problem under consideration which was open before and extends earlier work for the two-station single-track railway scheduling problem. We also present computational results of the comparison of both algorithms. For the 30,000 instances with up to 30 jobs considered, the average relative error of the heuristic is less than $1\%$ . In our tests, the practical running time of the dynamic programming algorithm was even bounded by $O(n^4)$ . Full article

Multicriteria correlation preference information (MCCPI) refers to a special type of 2-dimensional explicit information: the importance and interaction preferences regarding multiple dependent decision criteria. A few identification models have been established and implemented to transform the MCCPI into the most satisfactory 2-additive capacity. However, as one of the most commonly accepted particular type of capacity, 2-additive capacity only takes into account 2-order interactions and ignores the higher order interactions, which is not always reasonable in a real decision-making environment. In this paper, we generalize those identification models into ordinary capacity cases to freely represent the complicated situations of higher order interactions among multiple decision criteria. Furthermore, a MCCPI-based comprehensive decision aid algorithm is proposed to represent various kinds of dominance relationships of all decision alternatives as well as other useful decision aiding information. An illustrative example is adopted to show the proposed MCCPI-based capacity identification method and decision aid algorithm. Full article

The aim of this paper is to present a new semi-local convergence analysis for Newton’s method in a Banach space setting. The novelty of this paper is that by using more precise Lipschitz constants than in earlier studies and our new idea of restricted convergence domains, we extend the applicability of Newton’s method as follows: The convergence domain is extended; the error estimates are tighter and the information on the location of the solution is at least as precise as before. These advantages are obtained using the same information as before, since new Lipschitz constant are tighter and special cases of the ones used before. Numerical examples and applications are used to test favorable the theoretical results to earlier ones. Full article

Selecting optimal suppliers in fuzzy environments has become a major challenge for enterprises. Reputation plays an important role in the process of supplier selection because of its fuzziness, dynamicity, and transitivity. In this study, we first present a novel intuitionistic fuzzy sets (IFS)-hyperlink-induced topic search (HITS) method that combines the intuitionistic fuzzy set with the hyperlink-induced topic search (HITS) algorithm to extend the ability of processing fuzzy information in order to obtain post-propagated reputation values of suppliers. Then, we employ the dynamic intuitionistic fuzzy weighted average operator to gain dynamic reputation values and other evaluation attribute values. After that, intuitionistic fuzzy entropy weight method is adopted to acquire more accurate weights for each evaluation attribute. Finally, we employ the Vlsekriterijumska Optimizacija I Kompromisno Resenje method to acquire comprehensive evaluation values of candidate supplier to select optimal suppliers. Two groups of experiments for supplier selection are given to explain feasibility and practicality of the proposed method. Full article

Mathematical thinking (MT) has been one of the most important goals for mathematics education as it can support sustainable mathematics learning. Its role in school mathematics has recently been explicitly identified as one of “Four Basics” in the latest national curriculum standard for compulsory education, which is seen as one of the prominent features of Chinese mathematics education. This paper reviewed definitions, descriptions, and explanations from a historical perspective and aimed to provide a comprehensive and contemporary conceptualization for MT in a Chinese context, which can be applied as a comparison to MT in English works. To achieve this, document analysis was applied in this study. Major official documents, papers, and books were reviewed to see the process of MT conceptualization given by the policy makers and researchers. Results indicated that MT places more emphasis on the process of mathematical methods application in problem solving, such as the method of combination of symbolic and graphic mathematics. Mathematical thought is also recommended by Chinese researchers to help students think like mathematicians. Another major characteristic is that the classification of major types of MT is usually focused on that which can make the concept more understandable. Full article

This paper extends the univariate Theory of Connections, introduced in (Mortari, 2017), to the multivariate case on rectangular domains with detailed attention to the bivariate case. In particular, it generalizes the bivariate Coons surface, introduced by (Coons, 1984), by providing analytical expressions, called constrained expressions, representing all possible surfaces with assigned boundary constraints in terms of functions and arbitrary-order derivatives. In two dimensions, these expressions, which contain a freely chosen function, $g(x,y)$ , satisfy all constraints no matter what the $g(x,y)$ is. The boundary constraints considered in this article are Dirichlet, Neumann, and any combinations of them. Although the focus of this article is on two-dimensional spaces, the final section introduces the Multivariate Theory of Connections, validated by mathematical proof. This represents the multivariate extension of the Theory of Connections subject to arbitrary-order derivative constraints in rectangular domains. The main task of this paper is to provide an analytical procedure to obtain constrained expressions in any space that can be used to transform constrained problems into unconstrained problems. This theory is proposed mainly to better solve PDE and stochastic differential equations. Full article

Let k be an algebraically closed field of prime characteristic and let G be a semisimple, simply connected, linear algebraic group. It is an open problem to find the cohomology of line bundles on the flag variety $G/B$ , where B is a Borel subgroup of G. In this paper we consider this problem in the case of $G=S{L}_3\left(k\right)$ and compute the cohomology for the case when $⟨\lambda ,{\alpha }^{\vee }⟩=-p^{n}a-1$ , $(1\le a\le p,n>0)$ or $⟨\lambda ,{\alpha }^{\vee }⟩=-p^n-r$ , $(r\ge 2,n\ge 0)$ . We also give the corresponding results for the two dimensional modules $N_{\alpha }\left(\lambda \right)$ . These results will help us understand the representations of $S{L}_3\left(k\right)$ in the given cases. Full article

The two procedures traditionally followed for group decision making with the Analytical Hierarchical Process (AHP) are the Aggregation of Individual Judgments (AIJ) and the Aggregation of Individual Priorities (AIP). In both cases, the geometric mean is used to synthesise judgments and individual priorities into a collective position. Unfortunately, positional measures (means) are only representative if dispersion is reduced. It is therefore necessary to develop decision tools that allow: (i) the identification of groups of actors that present homogeneous and differentiated behaviours; and, (ii) the aggregation of the priorities of the near groups to reach collective positions with the greatest possible consensus. Following a Bayesian approach to AHP in a local context (a single criterion), this work proposes a methodology to solve these problems when the number of actors is not high. The method is based on Bayesian comparison and selection of model tools which identify the number and composition of the groups as well as their priorities. This information can be very useful to identify agreement paths among the decision makers that can culminate in a more representative decision-making process. The proposal is illustrated by a real-life case study. Full article

New solutions and techniques for developing country policies are used under real conditions. The present study aims to propose a new approach for evaluating and ranking the European countries by using the interrelation between two groups of criteria, associated with the Human Development Index (HDI) and the World Internal Security and Police Index (WISPI). HDI and its components rank countries by value and detail the values of the components of longevity, education and income per capita. WISPI focuses on the effective rendering of security services and the outcome of rendered services. The priority of criteria is determined in the descending order of their correlation values with other group criteria. The criteria weights are set simultaneously for both groups by applying the weight balancing method WEBIRA. The methodology based on minimising sum of squared differences of the weighted sums within groups is used. Finally, the generalised criteria measuring the level of the country are calculated using the SAW method. Cluster analysis of the countries was carried out and compared with MCDM results. The study revealed that WEBIRA ranking of countries is basically consistent with the results of cluster analysis. The proposed methodology can be applied to develop the management policy of the countries, as well as to their evaluation and ranking by using various indices, criteria and procedures. The results of this research can also be used to reveal national policy choices, to point out government policy priorities. Full article

For the May–Leonard asymmetric system, which is a quadratic system of the Lotka–Volterra type depending on six parameters, we first look for subfamilies admitting invariant algebraic surfaces of degree two. Then for some such subfamilies we construct first integrals of the Darboux type, identifying the systems with one first integral or with two independent first integrals. Full article

This paper is concerned with the estimation of spreading speed of a nonmonotone equation, which involves time delay and nonlocal dispersal. Due to the time delay, this equation does not generate monotone semiflows when the positive initial value is given. By constructing an auxiliary monotone equation, we obtain the spreading speed when the initial value admits nonempty compact support. Moreover, by passing to a limit function, we confirm the existence of traveling wave solutions if the wave speed equals to the spreading speed, which states the minimal wave speed of traveling wave solutions and improves the known results. Full article

In multi-scale information systems, the information is often characterized at multi scales and multi levels. To facilitate the computational process of multi-scale information systems, we employ the matrix method to represent the multi-scale information systems and to select the optimal scale combination of multi-scale decision information systems in this study. To this end, we first describe some important concepts and properties of information systems using some relational matrices. The relational matrix is then introduced into multi-scale information systems, and used to describe some main concepts in systems, including the lower and upper approximate sets and the consistence of systems. Furthermore, from the view of the relation matrix, the scale significance is defined to describe the global optimal scale and the local optimal scale of multi-scale information systems. Finally, the relational matrix is used to compute the scale significance and to construct the optimal scale selection algorithms. The efficiency of these algorithms is examined by several practical examples and experiments. Full article

Artificial bee colony is a powerful optimization method, which has strong search abilities to solve many optimization problems. However, some studies proved that ABC has poor exploitation abilities in complex optimization problems. To overcome this issue, an improved ABC variant based on elite strategy and dimension learning (called ABC-ESDL) is proposed in this paper. The elite strategy selects better solutions to accelerate the search of ABC. The dimension learning uses the differences between two random dimensions to generate a large jump. In the experiments, a classical benchmark set and the 2013 IEEE Congress on Evolutionary (CEC 2013) benchmark set are tested. Computational results show the proposed ABC-ESDL achieves more accurate solutions than ABC and five other improved ABC variants. Full article

Most power transformer faults are caused by iron core and winding faults. At present, the method that is most widely used for transformer iron core and winding faults identification is the vibration analysis method. The vibration analysis method generally determines the degree of fault by analyzing the energy spectrum of the transformer vibration signal. However, the noise reduction step in this method is complicated and costly, and the effect of denoising needs to be further improved to make the fault identification results more accurate. In addition, it is difficult to perform an accurate determination of the early mild failure of the transformer due to the effect of noise on the results. This paper presents a novel mathematical statistics method based on the vibration signal to optimize the vibration analysis method for the short-circuit failure of the transformer winding. The proposed method was used for linear analysis of the transformer vibration signal with different degrees of short-circuit failure of the transformer winding. By comparing the slope value of the transformer vibration signal cumulative probability distribution curve and analyzing the energy spectrum of the signal, the degree of short-circuit failure of the transformer winding was identified quickly and accurately. This method also simplified the signal denoising process in transformer fault detection, improved the accuracy of fault detection, reduced the time of fault detection, and provided good predictability for early mild faults of the transformer, thereby reducing the hidden hazards of operating the power transformer. The proposed optimization procedure offers a new research idea in transformer fault identification. Full article

The semigroup properties of the Riemann–Liouville fractional integral have played a key role in dealing with the existence of solutions to differential equations of fractional order. Based on some results of some experts’, we know that the Riemann–Liouville variable order fractional integral does not have semigroup property, thus the transform between the variable order fractional integral and derivative is not clear. These judgments bring us extreme difficulties in considering the existence of solutions of variable order fractional differential equations. In this work, we will introduce the concept of approximate solution to an initial value problem for differential equations of variable order involving the derivative argument on half-axis. Then, by our discussion and analysis, we investigate the unique existence of approximate solution to this initial value problem for differential equation of variable order involving the derivative argument on half-axis. Finally, we give examples to illustrate our results. Full article

The aim of this work is to study generalizations of the notion of the mean. Kolmogorov proposed a generalization based on an improper integral with a decay rate for the tail probabilities. This weak or Kolmogorov mean relates to the weak law of large numbers in the same way that the ordinary mean relates to the strong law. We propose a further generalization, also based on an improper integral, called the doubly-weak mean, applicable to heavy-tailed distributions such as the Cauchy distribution and the other symmetric stable distributions. We also consider generalizations arising from Abel–Feynman-type mollifiers that damp the behavior at infinity and alternative formulations of the mean in terms of the cumulative distribution and the characteristic function. Full article

Topological indices are numerical values associated with a graph (structure) that can predict many physical, chemical, and pharmacological properties of organic molecules and chemical compounds. The distance degree ( $DD$ ) index was introduced by Dobrynin and Kochetova in 1994 for characterizing alkanes by an integer. In this paper, we have determined expressions for a $DD$ index of some derived graphs in terms of the parameters of the parent graph. Specifically, we establish expressions for the $DD$ index of a line graph, subdivision graph, vertex-semitotal graph, edge-semitotal graph, total graph, and paraline graph. Full article

In this paper, we discuss the existence of solutions for a hybrid boundary value problem of Caputo fractional differential equations. The main tool used in our study is associated with the technique of measures of noncompactness. As an application, we give an example to illustrate our results. Full article

Very often items that are substitutable and complementary to each other are sent from suppliers to retailers for business. In this paper, for these types of items, fixed charge (FC) four-dimensional (4D) multi-item transportation problems (MITPs) are formulated with both space and budget constraints under crisp and rough environments. These items are damageable/breakable. The rates of damageability of the items depend on the quantity transported and the distance of travel i.e., path. A fixed charge is applied to each of the routes (independent of items). There are some depots/warehouses (origins) from which the items are transported to the sales counters (destinations) through different conveyances and routes. In proposed FC 4D-MITP models, per unit selling prices, per unit purchasing prices, per unit transportation expenditures, fixed charges, availabilities at the sources, demands at the destinations, conveyance capacities, total available space and budget are expressed by rough intervals, where the transported items are substitutable and complementary in nature. In this business, the demands for the items at the destinations are directly related to their substitutability and complementary natures and prices. The suggested rough model is converted into a deterministic one using lower and upper approximation intervals following Hamzehee et al. as well as Expected Value Techniques. The converted model is optimized through the Generalized Reduced Gradient (GRG) techniques using LINGO 14 software. Finally, numerical examples are presented to illustrate the preciseness of the proposed model. As particular cases, different models such as 2D, 3D FCMITPs for two substitute items, one item with its complement and two non substitute non complementary items are derived and results are presented. Full article

In this paper, we investigate the generalized Hyers-Ulam stability of the Pexider functional equation $f(x+y,z+w)=g(x,z)+h(y,w)$ . Full article

The study is on the existence of the solution for a coupled system of fractional differential equations with integral boundary conditions. The first result will address the existence and uniqueness of solutions for the proposed problem and it is based on the contraction mapping principle. Secondly, by using Leray–Schauder’s alternative we manage to prove the existence of solutions. Finally, the conclusion is confirmed and supported by examples. Full article

The wide usage of information technologies in production has led to the Fourth Industrial Revolution, which has enabled real data collection from production tools that are capable of communicating with each other through the Internet of Things (IoT). Real time data improves production control especially in dynamic production environments. This study proposes a decision support system (DSS) designed to increase the performance of dispatching rules in dynamic scheduling using real time data, hence an increase in the overall performance of the job-shop. The DSS can work with all dispatching rules. To analyze its effects, it is run with popular dispatching rules selected from the literature on a simulation model created in Arena^®. When the number of jobs waiting in the queue of any workstation in the job-shop falls to a critical value, the DSS can change the order of schedules in its preceding workstations to feed the workstation as soon as possible. For this purpose, it first determines the jobs in the preceding workstations to be sent to the current workstation, then finds the job with the highest priority number according to the active dispatching rule, and lastly puts this job in the first position in its queue. The DSS is tested under low, normal, and high demand rate scenarios with respect to six performance criteria. It is observed that the DSS improves the system performance by increasing workstation utilization and decreasing both the number of tardy jobs and the amount of waiting time regardless of the employed dispatching rule. Full article