Mathematics, Volume 8, Issue 8 (August 2020) – 188 articles

In 2008, I. Włoch introduced a new generalization of Pell numbers. She used special initial conditions so that this sequence describes the total number of special families of subsets of the set of n integers. In this paper, we prove some results about the roots of the characteristic polynomial of this sequence, but we will consider general initial conditions. Since there are currently several types of generalizations of the Pell sequence, it is very difficult for anyone to realize what type of sequence an author really means. Thus, we will call this sequence the generalized k-distance Tribonacci sequence ${(T_n^{(k)})}_{n\ge 0}$. Full article

The study of the mixed initial-boundary value problem, corresponding to the thermoelasticity of porous micromorphic materials under the influence of microtemperatures, represents the main objective of this article. Achieving qualitative results on the existence, uniqueness and continuous dependence on the initial data and loads, of the solution of the mixed problem, implies a new perspective of approaching these topics, imposed by the large number of unknowns, which increases the complexity of equations and conditions that characterize the thermoelastic porous micromorphic materials with microtemperatures. The use of the semigroup theory of operators represents the optimal solution for deducing these results, the theory being adaptable to the requirements of the demonstrations, the mixed problem turning into a problem of Cauchy type, with regards to an equation of evolution on a Hilbert space, chosen appropriately. Full article

In this paper, we review the Fourier-spectral method for some phase-field models: Allen–Cahn (AC), Cahn–Hilliard (CH), Swift–Hohenberg (SH), phase-field crystal (PFC), and molecular beam epitaxy (MBE) growth. These equations are very important parabolic partial differential equations and are applicable to many interesting scientific problems. The AC equation is a reaction-diffusion equation modeling anti-phase domain coarsening dynamics. The CH equation models phase segregation of binary mixtures. The SH equation is a popular model for generating patterns in spatially extended dissipative systems. A classical PFC model is originally derived to investigate the dynamics of atomic-scale crystal growth. An isotropic symmetry MBE growth model is originally devised as a method for directly growing high purity epitaxial thin film of molecular beams evaporating on a heated substrate. The Fourier-spectral method is highly accurate and simple to implement. We present a detailed description of the method and explain its connection to MATLAB usage so that the interested readers can use the Fourier-spectral method for their research needs without difficulties. Several standard computational tests are done to demonstrate the performance of the method. Furthermore, we provide the MATLAB codes implementation in the Appendix A. Full article

The unit return is determined as the return in the quotation currency (QCR) per the unit of base exchange medium (BEM). The main purpose is to examine the applicability of a trading system with a constant modulus of unit return (CMUR). The CMUR system supports speculative operations related to the exchange rate, given as the BEM quotation per the QCR. Premises for investment decisions are based on knowledge about the quotation dynamics described by its binary representation. This knowledge is described by a prediction table containing the conditional probability distributions of exchange rate increments. Any prediction table depends on observation range. Financial effectiveness of any CMUR system is assessed in the usual way by interest rate and risk index based on Shannon entropy. The main aim of our paper is to present algorithms which may be used for selecting effective CMUR systems. Required unit return modulus and observation range are control parameters applied for management of CMUR systems. Optimal values of these parameters are obtained by implementation of the proposed algorithm. All formal considerations are illustrated by an extensive case study linked to gold trading. Full article

Over the years, fire departments have been searching for methods to identify their operational disruptions and establish strategies that allow them to efficiently organize their resources. The present work develops a methodology for breakage calculation and another for predicting disruptions based on machine learning techniques. The main objective is to establish indicators to identify the failures due to the temporal state of the organization in the human and vehicular material. Likewise, by forecasting disruptions, to determine strategies for the deployment or acquisition of the necessary armament. This would allow improving operational resilience and increasing the efficiency of the firemen over time. The methodology was applied to the Departmental Fire and Rescue Doubs (SDIS25) in France. However, it is generic enough to be extended and adapted to other fire departments. Considering a historic of breakdowns of 2017 and 2018, the best predictions of public service breakdowns for the year 2019, presented a root mean squared error of 2.5602 and a mean absolute error of 2.0240 on average with the XGBoost technique. Full article

The Non-Uniform Rational B-spline (NURBS) surface not only has the characteristics of the rational Bézier surface, but also has changeable knot vectors and weights, which can express the quadric surface accurately. In this paper, we investigated new bounds of the first- and second-order partial derivatives of NURBS surfaces. A pilot study was performed using inequality theorems and degree reduction of B-spline basis functions. Theoretical analysis provides simple forms of the new bounds. Numerical examples are performed to illustrate that our method has sharper bounds than the existing ones. Full article

In this paper, we enlarge the class of $C^{*}$-algebra valued partial metric spaces as well as the class of $C^{*}$-algebra valued b-metric spaces by introducing the class of $C^{*}$-algebra valued partial b-metric spaces and utilize the same to prove our fixed point results. We furnish an example to highlight the utility of our main result. Finally, we apply our result in order to examine the existence and uniqueness of a solution for the system of Fredholm integral equations. Full article

In the consumerist world, there is an ever-increasing demand for consumption in urban life. Thus, the demand for shopping malls is growing. For a developer, site selection is an important issue as the optimal selection involves several complex factors and sub-factors for a successful investment venture. Thus, these tangible and intangible factors can be best solved by the Multi Criteria Decision Making (MCDM) models. In this study, optimal site selection has been done out of multiple alternative locations in and around the city of Kolkata, West Bengal, India. The Fuzzy Analytic Hierarchy Process (FAHP) and Fuzzy Technique for Order of Preference by Similarity to Ideal Solution (FTOPSIS) has been applied for shopping mall site selection. The AHP is used to obtain the crispified weight of factors. Imprecise linguistic terms used by the decision-maker are converted to Triangular Fuzzy Numbers (TFNs). This research used integrated sub-factors fuzzy weights using FAHP to FTOPSIS for ranking of the alternatives. Hardly any research is done with the use of sub-factors. In this study, seven factors and seventeen sub-factors are considered, the authors collected data from different locations with the help of municipal authorities and architects. This work further provides useful guidelines for shopping mall selection in different states and countries. Full article

Fractional differential equations with impulses arise in modeling real world phenomena where the state changes instantaneously at some moments. Often, these instantaneous changes occur at random moments. In this situation the theory of Differential equations has to be combined with Probability theory to set up the problem correctly and to study the properties of the solutions. We study the case when the time between two consecutive moments of impulses is exponentially distributed. In connection with the application of the Riemann–Liouville fractional derivative in the equation, we define in an appropriate way both the initial condition and the impulsive conditions. We consider the case when the lower limit of the Riemann–Liouville fractional derivative is fixed at the initial time. We define the so called p-moment Mittag–Leffler stability in time of the model. In the case of integer order derivative the introduced type of stability reduces to the p–moment exponential stability. Sufficient conditions for p–moment Mittag–Leffler stability in time are obtained. The argument is based on Lyapunov functions with the help of the defined fractional Dini derivative. The main contributions of the suggested model is connected with the implementation of impulses occurring at random times and the application of the Riemann–Liouville fractional derivative of order between 0 and 1. For this model the p-moment Mittag–Leffler stability in time of the model is defined and studied by Lyapunov functions once one defines in an appropriate way their Dini fractional derivative. Full article

Fractional order calculus has been used to generalize various types of controllers, including internal model controllers (IMC). The focus of this manuscript is towards fractional order IMCs for first order plus dead-time (FOPDT) processes, including delay and lag dominant ones. The design is novel at it is based on a new approximation approach, the non-rational transfer function method. This allows for a more accurate approximation of the process dead-time and ensures an improved closed loop response. The main problem with fractional order controllers is concerned with their implementation as higher order transfer functions. In cases where central processing unit CPU, bandwidth allocation, and energy usage are limited, resources need to be efficiently managed. This can be achieved using an event-based implementation. The novelty of this paper resides in such an event-based algorithm for fractional order IMC (FO-IMC) controllers. Numerical results are provided for lag and delay dominant FOPDT processes. For comparison purposes, an integer order PI controller, tuned according to the same performance specifications as the FO-IMC, is also implemented as an event-based control strategy. The numerical results show that the proposed event-based implementation for the FO-IMC controller is suitable and provides for a smaller computational effort, thus being more suitable in various industrial applications. Full article

The discrete wavelet transform (DWT) is unable to represent the directional features of an image. Similarly, a fixed embedding strength is not able to establish an ideal balance between imperceptibility and robustness of a watermarked image. In this work, we propose an adaptive embedding strength watermarking algorithm based on shearlets’ capture directional features (S-AES). We improve the watermarking algorithm in the domain of DWT using non-subsampled shearlet transform (NSST). The improvement is made in terms of coping with anti-geometric attacks. The embedding strength is optimized by artificial bee colony (ABC) to achieve higher robustness under the premise of satisfying imperceptibility. The principle components (PC) of the watermark are embedded into the host image to overcome the false positive problem. The simulation results show that the proposed algorithm has better imperceptibility and strong robustness against multi-attacks, especially those of high intensity. Full article

This paper presents an insurance contract that the supplier of wind power may subscribe to with an insurance company in order to immunize his/her revenue against the volatility of wind power and prices. Based on empirical evidence, we found that wind power and electricity prices are correlated. Then, we adopted a joint stochastic process to model both time series, which is based on indexed semi-Markov chains for the wind power generation process and on a general Markovian process for the electricity price process. Using a joint stochastic model allows the insurance company to compute the fair premium that the wind power producer has to pay in order to hedge the risk against inadequate revenues. Recursive type equations are obtained for the prospective mathematical reserves of the insurance contract. The model and the validity of the results are illustrated through a real data application. Full article

Segregation is an important step in health care waste management. If done incorrectly, the risk of preventable infections, toxic effects, and injuries to care and non-care staff, waste handlers, patients, visitors, and the community at large, is increased. It also increases the risk of environmental pollution and prevents recyclable waste from being recovered. Despite its importance, it is acknowledged that poor waste segregation occurs in most health care organizations. This study therefore intends to produce, for the first time, a classification of failure modes related to segregation in the Nuclear Medicine Department of a health care organization. This will be done using Failure Mode and Effects Analysis (FMEA), by combining an intuitionistic fuzzy hybrid weighted Euclidean distance operator, and the multicriteria method Potentially All Pairwise RanKings of all possible Alternatives (PAPRIKA). Subjective and objective weights of risk factors were considered simultaneously. The failure modes identified in the top three positions are: improper storage of waste (placing items in the wrong bins), improper labeling of containers, and bad waste management (inappropriate collection periods and bin set-up). Full article

The application of echo state networks to time series prediction has provided notable results, favored by their reduced computational cost, since the connection weights require no learning. However, there is a need for general methods that guide the choice of parameters (particularly the reservoir size and ridge regression coefficient), improve the prediction accuracy, and provide an assessment of the uncertainty of the estimates. In this paper we propose such a mechanism for uncertainty quantification based on Monte Carlo dropout, where the output of a subset of reservoir units is zeroed before the computation of the output. Dropout is only performed at the test stage, since the immediate goal is only the computation of a measure of the goodness of the prediction. Results show that the proposal is a promising method for uncertainty quantification, providing a value that is either strongly correlated with the prediction error or reflects the prediction of qualitative features of the time series. This mechanism could eventually be included into the learning algorithm in order to obtain performance enhancements and alleviate the burden of parameter choice. Full article

In 2020, Kang et al. introduced the concept of a multipolar intuitionistic fuzzy set of finite degree, which is a generalization of a k-polar fuzzy set, and applied it to a BCK/BCI-algebra. The specific purpose of this study was to apply the concept of a multipolar intuitionistic fuzzy set of finite degree to a hyper BCK-algebra. The notions of the k-polar intuitionistic fuzzy hyper BCK-ideal, the k-polar intuitionistic fuzzy weak hyper BCK-ideal, the k-polar intuitionistic fuzzy s-weak hyper BCK-ideal, the k-polar intuitionistic fuzzy strong hyper BCK-ideal and the k-polar intuitionistic fuzzy reflexive hyper BCK-ideal are introduced herein, and their relations and properties are investigated. These concepts are discussed in connection with the k-polar lower level set and the k-polar upper level set. Full article

The Choquet capacity and integral is an eminent scheme to represent the interaction knowledge among multiple decision criteria and deal with the independent multiple sources preference information. In this paper, we enhance this scheme’s decision pattern learning ability by combining it with another powerful machine learning tool, the random forest of decision trees. We first use the capacity fitting method to train the Choquet capacity and integral-based decision trees and then compose them into the capacity random forest (CRF) to better learn and explain the given decision pattern. The CRF algorithms of solving the correlative multiple criteria based ranking and sorting decision problems are both constructed and discussed. Two illustrative examples are given to show the feasibilities of the proposed algorithms. It is shown that on the one hand, CRF method can provide more detailed explanation information and a more reliable collective prediction result than the main existing capacity fitting methods; on the other hand, CRF extends the applicability of the traditional random forest method into solving the multiple criteria ranking and sorting problems with a relatively small pool of decision learning data. Full article

This paper presents a lumped perturbation observer-based robust control method using an extended multiple sliding surface for a system with matched and unmatched uncertainties. The fundamental methodology is to apply the multiple surfaces to approximate the unknown lumped perturbations simultaneously influencing on a nonlinear single input–single output (SISO) system. Subsequently, a robust controller, based on the proposed multi-surface and the approximated values, is designed to highly improve the control performance of the system. A general stability of the lumped perturbation observer and closed-loop control system is obtained through the Lyapunov theory. Results of a numerical simulation of an illustrative example demonstrate the soundness of the proposed algorithm. Full article

This study proposes a novel evaluation model for lawyer selection incorporating the lawyer’s backbone leadership attitude employing the hybrid multi-criteria decision-making (MCDM) approach. In the proposed approach, the lawyer’s backbone leadership attitude is employed as an evaluation factor in the evaluation model for lawyer selection from law firms’ perspective. In this paper, a hybrid approach based on the Delphi technique and analytic hierarchy process (AHP) is proposed to manage qualitative and quantitative criteria for selecting the best alternative lawyer for law firms in China. Finally, a law firm in China is carried out to verify the feasibility of the proposed approach. Based on the result, the backbone leadership does provide valuable information in the evaluation model for lawyer selection. The results also revealed that the proposed approach would help law firms and human resource managers to understand and develop strategies to hire a lawyer. Full article

In this work, an efficient algorithm is proposed for solving the system of Volterra integral equations based on wavelet Galerkin method. This problem is reduced to a set of algebraic equations using the operational matrix of integration and wavelet transform matrix. For linear type, the computational effort decreases by thresholding. The convergence analysis of the proposed scheme has been investigated and it is shown that its convergence is of order $O\left(2^{-Jr}\right)$, where J is the refinement level and r is the multiplicity of multi-wavelets. Several numerical tests are provided to illustrate the ability and efficiency of the method. Full article

In the lane-based signal optimization model, permitted turn directions in the form of lane markings that guide road users to turn at an intersection are optimized with traffic signal settings. The spatial queue requirements of approach lanes should be considered to avoid the overdesigning of the cycle, effective red, and effective green durations. The point-queue system employed in the conventional modeling approach is unrealistic in many practical situations. Overflow conditions cannot be modeled accurately, while vehicle queues are accumulated that block back upstream intersections. In a previous study, a method was developed to manually refine the traffic signal settings by using the results of lane-based optimization. However, the method was inefficient. In the present study, new design constraint sets are proposed to control the effective red and effective green durations, such that traffic enters the road lanes without overflow. The reduced cycle times discharge the accumulated vehicles more frequently. Moreover, queue spillback and residual queues can be avoided. One of the most complicated four-arm intersections in Hong Kong is considered as a case study for demonstration. The existing traffic signal settings are ineffective for controlling the observed traffic demand, and overflow occurs in short lanes. The optimized traffic signal settings applied to the proposed optimization algorithm effectively avoided traffic overflow. The resultant queuing dynamics are simulated using TRANSYT 15 Cell Transmission Model (CTM) to verify the proposed model. The model application is extended to handle the difficult residual queue scenario. It is found that the proposed model can optimize the traffic signal settings in cases where there are short initial residual queues. Full article

Energy is one of the most important drivers of economic growth, but as the population is increasing, in normal circumstances, in all countries of the world, there is a demand for energy produced from conventional resources. Increasing prices of conventional energy and the negative impact on the environment are two of the main reasons for switching to renewable energy sources (RESs). The aim of the paper is to quantify the impact of the RESs, by type, on the sustainable economic growth at the European Union (EU) level. The research was performed for all 28 EU member states, for a time frame from 2004 to 2017, through a panel autoregressive distributed lag (ARDL) approach and causality analysis. Furthermore, Hausman test was performed on the regression model. By estimating the panel data regression model with random effects, we reveal through our results that RESs, namely wind, solar, biomass, geothermal, and hydropower energy, have a positive influence on economic growth at EU level. Moreover, biomass has the highest impact on economic growth among all RES. In fact, a 1% increase in biomass primary production would impact the economic growth by 0.15%. Based on econometric analysis, our findings suggest that public policies at the EU level should be focused on investment in RESs. Full article

Software reliability and quality are crucial in several fields. Related studies have focused on software reliability growth models (SRGMs). Herein, we propose a new SRGM that assumes interdependent software failures. We conduct experiments on real-world datasets to compare the goodness-of-fit of the proposed model with the results of previous nonhomogeneous Poisson process SRGMs using several evaluation criteria. In addition, we determine software reliability using Wald’s sequential probability ratio test (SPRT), which is more efficient than the classical hypothesis test (the latter requires substantially more data and time because the test is performed only after data collection is completed). The experimental results demonstrate the superiority of the proposed model and the effectiveness of the SPRT. Full article

Inspired by the equivalence between isoperimetric inequality and Sobolev inequality, we provide a new connection between geometry and analysis. We define the minimal perimeter of a log-concave function and establish a characteristic theorem of this extremal problem for log-concave functions analogous to convex bodies. Full article

In the present work, the nonlinear oscillations of a pendulum wrapping on two cylinders is studied by means of a new analytical technique, namely the Optimal Auxiliary Functions Method (OAFM). The equation of motion is derived from the Lagrange’s equation. Analytical solutions and natural frequency of the system are calculated. Our results obtained through this new procedure are compared with numerical ones and a very good agreement was found, which proves the accuracy of the method. The presented numerical examples show that the proposed approach is simple, easy to implement and very accurate. Full article

This study proposes a general bivariate stochastic differential equation model of population growth which includes random forces governing the dynamics of the bivariate distribution of size variables. The dynamics of the bivariate probability density function of the size variables in a population are described by the mixed-effect parameters Vasicek, Gompertz, Bertalanffy, and the gamma-type bivariate stochastic differential equations (SDEs). The newly derived bivariate probability density function and its marginal univariate, as well as the conditional univariate function, can be applied for the modeling of population attributes such as the mean value, quantiles, and much more. The models presented here are the basis for further developments toward the tree diameter–height and height–diameter relationships for general purpose in forest management. The present study experimentally confirms the effectiveness of using bivariate SDEs to reconstruct diameter–height and height–diameter relationships by using measurements obtained from mountain pine tree (Pinus mugo Turra) species dataset in Lithuania. Full article

The recent success of biological engineering is due to a tremendous amount of research effort and the increasing number of market opportunities. Indeed, this has been partially possible due to the contribution of advanced mathematical tools and the application of engineering principles in genetic-circuit development. In this work, we use a rationally designed genetic circuit to show how models can support research and motivate students to apply mathematics in their future careers. A genetic four-state machine is analyzed using three frameworks: deterministic and stochastic modeling through differential and master equations, and a spatial approach via a cellular automaton. Each theoretical framework sheds light on the problem in a complementary way. It helps in understanding basic concepts of modeling and engineering, such as noise, robustness, and reaction–diffusion systems. The designed automaton could be part of a more complex system of modules conforming future bio-computers and it is a paradigmatic example of how models can assist teachers in multidisciplinary education. Full article

Phasor measurement units (PMUs) are important elements in power systems to monitor and know the real network condition. In order to regulate the performance of PMUs, the IEEE Std. C37.118.1 stablishes two classes—P and M, where the phasor estimation is carried out using a quadrature oscillator and a low-pass (LP) filter for modulation and demodulation, respectively. The LP filter plays the most important role since it determines the accuracy, response time and rejection capability of both harmonics and aliased signals. In this regard and by considering that the M-class filters are used for more accurate measurements, the IEEE Std. presents different M-class filters for different reporting rates (when a result is given). However, they can degrade their performance under frequency deviations if the LP frequency response is not properly considered. In this work, a unified model for magnitude compensation under frequency deviations for all the M-class filters is proposed, providing the necessary values of compensation to improve their performance. The model considers the magnitude response of the M-class filters for different reporting rates, a normalized frequency range based on frequency dilation and a fitted two-variable function. The effectiveness of the proposal is verified using both static and dynamic conditions for frequency deviations. Besides that, a real-time simulator to generate test signals is also used to validate the proposed methodology. Full article

The S-λ model is one of the most useful tools for shape designs and geometric representations in computer-aided geometric design (CAGD), which is due to its good geometric properties such as symmetry, shape adjustable property. With the aim to solve the problem that complex S-λ curves and surfaces cannot be constructed by a single curve and surface, the explicit continuity conditions for G¹ connection of S-λ curves and surfaces are investigated in this paper. On the basis of linear independence and terminal properties of S-λ basis functions, the conditions of G¹ geometric continuity between two adjacent S-λ curves and surfaces are proposed, respectively. Modeling examples imply that the continuity conditions proposed in this paper are easy and effective, which indicate that the S-λ curves and surfaces can be used as a powerful supplement of complex curves and surfaces design in computer aided design/computer aided manufacturing (CAD/CAM) system. Full article

Variational integrators for dynamic systems with holonomic constraints are proposed based on Hamilton’s principle. The variational principle is discretized by approximating the generalized coordinates and Lagrange multipliers by Lagrange polynomials, by approximating the integrals by quadrature rules. Meanwhile, constraint points are defined in order to discrete the holonomic constraints. The functional of the variational principle is divided into two parts, i.e., the action of the unconstrained term and the constrained term and the actions of the unconstrained term and the constrained term are integrated separately using different numerical quadrature rules. The influence of interpolation points, quadrature rule and constraint points on the accuracy of the algorithms is analyzed exhaustively. Properties of the proposed algorithms are investigated using examples. Numerical results show that the proposed algorithms have arbitrary high order, satisfy the holonomic constraints with high precision and provide good performance for long-time integration. Full article