Mathematics

23 pages, 365 KiB

Open AccessEditorial

Acknowledgement to Reviewers of Mathematics in 2019

by Mathematics Editorial Office

Mathematics 2020, 8(1), 145; https://doi.org/10.3390/math8010145 - 20 Jan 2020

Viewed by 3338

The editorial team greatly appreciates the reviewers who have dedicated their considerable time and expertise to the journal’s rigorous editorial process over the past 12 months, regardless of whether the papers are finally published or not [...] Full article

43 pages, 1041 KiB

Open AccessArticle

On Degenerate Truncated Special Polynomials

by Ugur Duran and Mehmet Acikgoz

Mathematics 2020, 8(1), 144; https://doi.org/10.3390/math8010144 - 20 Jan 2020

Cited by 12 | Viewed by 3359

Abstract

The main aim of this paper is to introduce the degenerate truncated forms of multifarious special polynomials and numbers and is to investigate their various properties and relationships by using the series manipulation method and diverse special proof techniques. The degenerate truncated exponential [...] Read more.

The main aim of this paper is to introduce the degenerate truncated forms of multifarious special polynomials and numbers and is to investigate their various properties and relationships by using the series manipulation method and diverse special proof techniques. The degenerate truncated exponential polynomials are first considered and their several properties are given. Then the degenerate truncated Stirling polynomials of the second kind are defined and their elementary properties and relations are proved. Also, the degenerate truncated forms of the bivariate Fubini and Bell polynomials and numbers are introduced and various relations and formulas for these polynomials and numbers, which cover several summation formulas, addition identities, recurrence relationships, derivative property and correlations with the degenerate truncated Stirling polynomials of the second kind, are acquired. Thereafter, the truncated degenerate Bernoulli and Euler polynomials are considered and multifarious correlations and formulas including summation formulas, derivation rules and correlations with the degenerate truncated Stirling numbers of the second are derived. In addition, regarding applications, by introducing the degenerate truncated forms of the classical Bernstein polynomials, we obtain diverse correlations and formulas. Some interesting surface plots of these polynomials in the special cases are provided. Full article

(This article belongs to the Special Issue Special Functions and Applications)

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Surface plot of <inline-formula><mml:math id="mm525" display="block"><mml:semantics><mml:mrow><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>;</mml:mo><mml:mi>λ</mml:mi></mml:mrow></mml:msub><mml:mfenced open="(" close=")"><mml:mi>x</mml:mi></mml:mfenced></mml:mrow></mml:semantics></mml:math></inline-formula>. Full article ">Figure 4
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17 pages, 2180 KiB

Open AccessArticle

An Inequality Approach to Approximate Solutions of Set Optimization Problems in Real Linear Spaces

by Elisabeth Köbis, Markus A. Köbis and Xiaolong Qin

Mathematics 2020, 8(1), 143; https://doi.org/10.3390/math8010143 - 20 Jan 2020

Viewed by 2610

Abstract

This paper explores new notions of approximate minimality in set optimization using a set approach. We propose characterizations of several approximate minimal elements of families of sets in real linear spaces by means of general functionals, which can be unified in an inequality [...] Read more.

This paper explores new notions of approximate minimality in set optimization using a set approach. We propose characterizations of several approximate minimal elements of families of sets in real linear spaces by means of general functionals, which can be unified in an inequality approach. As particular cases, we investigate the use of the prominent Tammer–Weidner nonlinear scalarizing functionals, without assuming any topology, in our context. We also derive numerical methods to obtain approximate minimal elements of families of finitely many sets by means of our obtained results. Full article

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

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20 pages, 2415 KiB

Open AccessArticle

A New Divergence Measure of Pythagorean Fuzzy Sets Based on Belief Function and Its Application in Medical Diagnosis

by Qianli Zhou, Hongming Mo and Yong Deng

Mathematics 2020, 8(1), 142; https://doi.org/10.3390/math8010142 - 20 Jan 2020

Cited by 103 | Viewed by 5359

Abstract

As the extension of the fuzzy sets (FSs) theory, the intuitionistic fuzzy sets (IFSs) play an important role in handling the uncertainty under the uncertain environments. The Pythagoreanfuzzy sets (PFSs) proposed by Yager in 2013 can deal with more uncertain situations than intuitionistic [...] Read more.

As the extension of the fuzzy sets (FSs) theory, the intuitionistic fuzzy sets (IFSs) play an important role in handling the uncertainty under the uncertain environments. The Pythagoreanfuzzy sets (PFSs) proposed by Yager in 2013 can deal with more uncertain situations than intuitionistic fuzzy sets because of its larger range of describing the membership grades. How to measure the distance of Pythagorean fuzzy sets is still an open issue. Jensen–Shannon divergence is a useful distance measure in the probability distribution space. In order to efficiently deal with uncertainty in practical applications, this paper proposes a new divergence measure of Pythagorean fuzzy sets, which is based on the belief function in Dempster–Shafer evidence theory, and is called PFSDM distance. It describes the Pythagorean fuzzy sets in the form of basic probability assignments (BPAs) and calculates the divergence of BPAs to get the divergence of PFSs, which is the step in establishing a link between the PFSs and BPAs. Since the proposed method combines the characters of belief function and divergence, it has a more powerful resolution than other existing methods. Additionally, an improved algorithm using PFSDM distance is proposed in medical diagnosis, which can avoid producing counter-intuitive results especially when a data conflict exists. The proposed method and the magnified algorithm are both demonstrated to be rational and practical in applications. Full article

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

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12 pages, 1186 KiB

Open AccessArticle

A Rectifying Acceptance Sampling Plan Based on the Process Capability Index

by Ching-Ho Yen, Chun-Chia Lee, Kuo-Hung Lo, Yeou-Ren Shiue and Shu-Hua Li

Mathematics 2020, 8(1), 141; https://doi.org/10.3390/math8010141 - 20 Jan 2020

Cited by 12 | Viewed by 5105

Abstract

The acceptance sampling plan and process capability index (PCI) are critical decision tools for quality control. Recently, numerous research papers have examined the acceptance sampling plan in combination with the PCI. However, most of these papers have not considered the aspect of rectifying [...] Read more.

The acceptance sampling plan and process capability index (PCI) are critical decision tools for quality control. Recently, numerous research papers have examined the acceptance sampling plan in combination with the PCI. However, most of these papers have not considered the aspect of rectifying inspections. In this paper, we propose a quality cost model of repetitive sampling to develop a rectifying acceptance sampling plan based on the one-sided PCI. This proposed model minimizes the total quality cost (TQC) of sentencing one lot, including inspection cost, internal failure cost, and external failure cost. In addition, sensitivity analysis is conducted to investigate the behavior of relevant parameters against TQC. To demonstrate the advantages of the proposed methodology, a comparison is implemented with the existing rectifying sampling plan in terms of TQC and average outgoing quality limit. This comparison reveals that our proposed methodology exhibits superior performance. Full article

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16 pages, 948 KiB

Open AccessArticle

Venture Capital Contracting with Ambiguity Sharing and Effort Complementarity Effect

by Jiajia Chang, Zhijun Hu and Hui Yang

Mathematics 2020, 8(1), 140; https://doi.org/10.3390/math8010140 - 19 Jan 2020

Cited by 3 | Viewed by 2328

Abstract

In this paper, we established a continuous-time agency model in which an ambiguity-averse venture capitalist (VC) employs an ambiguity-neutral entrepreneur (EN) to manage an innovative project. We analyzed the connection between ambiguity sharing and incentives under double moral hazard. Applying a stochastic dynamic [...] Read more.

In this paper, we established a continuous-time agency model in which an ambiguity-averse venture capitalist (VC) employs an ambiguity-neutral entrepreneur (EN) to manage an innovative project. We analyzed the connection between ambiguity sharing and incentives under double moral hazard. Applying a stochastic dynamic programming approach, we solved the VC’s maximization problem and obtained the Hamilton–Jacobi–Bellman (HJB) equation under a special form of the value function. We showed that the optimal pay-performance sensitivity was a fixed point of a nonlinear equation. The model ambiguity on the probability measure induced a tradeoff between ambiguity sharing and the incentive compensation that improved the EN’s pay-performance sensitivity level. Besides, we simulated the model and showed that when two efforts were complementary, the VC’s effort did not monotonically decrease with respect to the pay-performance sensitivity, while the EN’s effort did not monotonically increase in the pay-performance sensitivity level. More importantly, we found that as efforts tended to be more complementary, the optimal pay-performance sensitivity tended to approach those that maximized the efforts exerted by the EN and the VC. Full article

(This article belongs to the Section Financial Mathematics)

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26 pages, 479 KiB

Open AccessArticle

Nash Equilibrium Investment-Reinsurance Strategies for an Insurer and a Reinsurer with Intertemporal Restrictions and Common Interests

by Yanfei Bai, Zhongbao Zhou, Rui Gao and Helu Xiao

Mathematics 2020, 8(1), 139; https://doi.org/10.3390/math8010139 - 19 Jan 2020

Cited by 5 | Viewed by 2434

Abstract

This paper investigates the generalized multi-period mean-variance investment-reinsurance optimization model in a discrete-time framework for a general insurance company that contains a reinsurer and an insurer. The intertemporal restrictions and the common interests of the reinsurer and the insurer are considered. The common [...] Read more.

This paper investigates the generalized multi-period mean-variance investment-reinsurance optimization model in a discrete-time framework for a general insurance company that contains a reinsurer and an insurer. The intertemporal restrictions and the common interests of the reinsurer and the insurer are considered. The common goal of the reinsurer and the insurer is to maximize the expectation of the weighted sum of their wealth processes and minimize the corresponding variance. Based on the game method, we obtain the Nash equilibrium investment-reinsurance strategies for the above-proposed model and find out the equilibrium strategies when unilateral interest is considered. In addition, the Nash equilibrium investment-reinsurance strategies are deduced under two special premium calculated principles (i.e., the expected value premium principle and the variance value premium principle). We theoretically study the effect of the intertemporal restrictions on Nash equilibrium investment-reinsurance strategies and find the effect depends on the value of some parameters, which differs from the previous researches that generally believed that intertemporal restrictions would make investors avoid risks. Finally, we perform corresponding numerical analyses to verify our theoretical results. Full article

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16 pages, 316 KiB

Open AccessArticle

The Basic Algorithm for the Constrained Zero-One Quadratic Programming Problem with k-diagonal Matrix and Its Application in the Power System

by Shenshen Gu and Xinyi Chen

Mathematics 2020, 8(1), 138; https://doi.org/10.3390/math8010138 - 19 Jan 2020

Cited by 2 | Viewed by 2593

Abstract

Zero-one quadratic programming is a classical combinatorial optimization problem that has many real-world applications. However, it is well known that zero-one quadratic programming is non-deterministic polynomial-hard (NP-hard) in general. On one hand, the exact solution algorithms that can guarantee the global optimum are [...] Read more.

Zero-one quadratic programming is a classical combinatorial optimization problem that has many real-world applications. However, it is well known that zero-one quadratic programming is non-deterministic polynomial-hard (NP-hard) in general. On one hand, the exact solution algorithms that can guarantee the global optimum are very time consuming. And on the other hand, the heuristic algorithms that generate the solution quickly can only provide local optimum. Due to this reason, identifying polynomially solvable subclasses of zero-one quadratic programming problems and their corresponding algorithms is a promising way to not only compromise these two sides but also offer theoretical insight into the complicated nature of the problem. By combining the basic algorithm and dynamic programming method, we propose an effective algorithm in this paper to solve the general linearly constrained zero-one quadratic programming problem with a k-diagonal matrix. In our algorithm, the value of k is changeable that covers different subclasses of the problem. The theoretical analysis and experimental results reveal that our proposed algorithm is reasonably effective and efficient. In addition, the placement of the phasor measurement units problem in the power system is adopted as an example to illustrate the potential real-world applications of this algorithm. Full article

(This article belongs to the Special Issue Optimization for Decision Making II)

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11 pages, 280 KiB

Open AccessArticle

Geodesic Vector Fields on a Riemannian Manifold

by Sharief Deshmukh, Patrik Peska and Nasser Bin Turki

Mathematics 2020, 8(1), 137; https://doi.org/10.3390/math8010137 - 19 Jan 2020

Cited by 12 | Viewed by 5930

Abstract

A unit geodesic vector field on a Riemannian manifold is a vector field whose integral curves are geodesics, or in other worlds have zero acceleration. A geodesic vector field on a Riemannian manifold is a smooth vector field with acceleration of each of [...] Read more.

A unit geodesic vector field on a Riemannian manifold is a vector field whose integral curves are geodesics, or in other worlds have zero acceleration. A geodesic vector field on a Riemannian manifold is a smooth vector field with acceleration of each of its integral curves is proportional to velocity. In this paper, we show that the presence of a geodesic vector field on a Riemannian manifold influences its geometry. We find characterizations of n-spheres as well as Euclidean spaces using geodesic vector fields. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Structures)

23 pages, 369 KiB

Open AccessArticle

On the Fractional Order Rodrigues Formula for the Shifted Legendre-Type Matrix Polynomials

by Mohra Zayed, Mahmoud Abul-Ez, Mohamed Abdalla and Nasser Saad

Mathematics 2020, 8(1), 136; https://doi.org/10.3390/math8010136 - 18 Jan 2020

Cited by 14 | Viewed by 3142

Abstract

The generalization of Rodrigues’ formula for orthogonal matrix polynomials has attracted the attention of many researchers. This generalization provides new integral and differential representations in addition to new mathematical results that are useful in theoretical and numerical computations. Using a recently studied operational [...] Read more.

The generalization of Rodrigues’ formula for orthogonal matrix polynomials has attracted the attention of many researchers. This generalization provides new integral and differential representations in addition to new mathematical results that are useful in theoretical and numerical computations. Using a recently studied operational matrix for shifted Legendre polynomials with the variable coefficients fractional differential equations, the present work introduces the shifted Legendre-type matrix polynomials of arbitrary (fractional) orders utilizing some Rodrigues matrix formulas. Many interesting mathematical properties of these matrix polynomials are investigated and reported in this paper, including recurrence relations, differential properties, hypergeometric function representation, and integral representation. Furthermore, the orthogonality property of these polynomials is examined in some particular cases. The developed results provide a matrix framework that generalizes and enhances the corresponding scalar version and introduces some new properties with proposed applications. Some of these applications are explored in the present work. Full article

17 pages, 454 KiB

Open AccessArticle

A New Three-Parameter Exponential Distribution with Variable Shapes for the Hazard Rate: Estimation and Applications

by Ahmed Z. Afify and Osama Abdo Mohamed

Mathematics 2020, 8(1), 135; https://doi.org/10.3390/math8010135 - 16 Jan 2020

Cited by 50 | Viewed by 4627

Abstract

In this paper, we study a new flexible three-parameter exponential distribution called the extended odd Weibull exponential distribution, which can have constant, decreasing, increasing, bathtub, upside-down bathtub and reversed-J shaped hazard rates, and right-skewed, left-skewed, symmetrical, and reversed-J shaped densities. Some mathematical properties [...] Read more.

In this paper, we study a new flexible three-parameter exponential distribution called the extended odd Weibull exponential distribution, which can have constant, decreasing, increasing, bathtub, upside-down bathtub and reversed-J shaped hazard rates, and right-skewed, left-skewed, symmetrical, and reversed-J shaped densities. Some mathematical properties of the proposed distribution are derived. The model parameters are estimated via eight frequentist estimation methods called, the maximum likelihood estimators, least squares and weighted least-squares estimators, maximum product of spacing estimators, Cramér-von Mises estimators, percentiles estimators, and Anderson-Darling and right-tail Anderson-Darling estimators. Extensive simulations are conducted to compare the performance of these estimation methods for small and large samples. Four practical data sets from the fields of medicine, engineering, and reliability are analyzed, proving the usefulness and flexibility of the proposed distribution. Full article

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22 pages, 1084 KiB

Open AccessArticle

Parameter and State Estimation of One-Dimensional Infiltration Processes: A Simultaneous Approach

by Song Bo, Soumya R. Sahoo, Xunyuan Yin, Jinfeng Liu and Sirish L. Shah

Mathematics 2020, 8(1), 134; https://doi.org/10.3390/math8010134 - 16 Jan 2020

Cited by 15 | Viewed by 2791

Abstract

The Richards equation plays an important role in the study of agro-hydrological systems. It models the water movement in soil in the vadose zone, which is driven by capillary and gravitational forces. Its states (capillary potential) and parameters (hydraulic conductivity, saturated and residual [...] Read more.

The Richards equation plays an important role in the study of agro-hydrological systems. It models the water movement in soil in the vadose zone, which is driven by capillary and gravitational forces. Its states (capillary potential) and parameters (hydraulic conductivity, saturated and residual soil moistures and van Genuchten-Mualem parameters) are essential for the accuracy of mathematical modeling, yet difficult to obtain experimentally. In this work, an estimation approach is developed to estimate the parameters and states of Richards equation simultaneously. In the proposed approach, parameter identifiability and sensitivity analysis are used to determine the most important parameters for estimation purpose. Three common estimation schemes (extended Kalman filter, ensemble Kalman filter and moving horizon estimation) are investigated. The estimation performance is compared and analyzed based on extensive simulations. Full article

(This article belongs to the Special Issue Mathematics and Engineering)

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A schematic diagram of an agro-hydrological system. Full article ">Figure 2
A flowchart of the procedure to determine the significant parameters and number of sensors. Full article ">Figure 3
A schematic diagram of the investigated loam soil column. Full article ">Figure 4
Selected trajectories of the process state and estimated states using MHE, extended Kalman filter (EKF) and ensemble Kalman filter (EnKF). Full article ">Figure 5
Trajectories of estimated parameters using MHE, EKF and EnKF, compared with their actual values. Full article ">Figure 6
Trajectories of RMSE measuring the estimation performance of MHE, EKF, and EnKF. Full article ">Figure 7
Trajectories of RMSE measuring the error between actual model and estimated states and parameters of MHE using 4, 8 and 12 measurements. Full article ">Figure 8
Trajectories of RMSE measuring actual model and estimated states and parameters of MHE with window sizes of 8, 12, 16 and 20. Full article ">

11 pages, 730 KiB

Open AccessArticle

Approximation Algorithm for the Single Machine Scheduling Problem with Release Dates and Submodular Rejection Penalty

by Xiaofei Liu and Weidong Li

Mathematics 2020, 8(1), 133; https://doi.org/10.3390/math8010133 - 16 Jan 2020

Cited by 12 | Viewed by 2166

Abstract

In this paper, we consider the single machine scheduling problem with release dates and nonmonotone submodular rejection penalty. We are given a single machine and multiple jobs with probably different release dates and processing times. For each job, it is either accepted and [...] Read more.

In this paper, we consider the single machine scheduling problem with release dates and nonmonotone submodular rejection penalty. We are given a single machine and multiple jobs with probably different release dates and processing times. For each job, it is either accepted and processed on the machine or rejected. The objective is to minimize the sum of the makespan of the accepted jobs and the rejection penalty of the rejected jobs which is determined by a nonmonotone submodular function. We design a combinatorial algorithm based on the primal-dual framework to deal with the problem, and study its property under two cases. For the general case where the release dates can be different, the proposed algorithm have an approximation ratio of 2. When all the jobs release at the same time, the proposed algorithm becomes a polynomial-time exact algorithm. Full article

21 pages, 956 KiB

Open AccessArticle

Dynamic Modelling of Interactions between Microglia and Endogenous Neural Stem Cells in the Brain during a Stroke

by Awatif Jahman Alqarni, Azmin Sham Rambely and Ishak Hashim

Mathematics 2020, 8(1), 132; https://doi.org/10.3390/math8010132 - 16 Jan 2020

Cited by 6 | Viewed by 2885

Abstract

In this paper, we study the interactions between microglia and neural stem cells and the impact of these interactions on the brain cells during a stroke. Microglia cells, neural stem cells, the damage on brain cells from the stroke and the impacts these [...] Read more.

In this paper, we study the interactions between microglia and neural stem cells and the impact of these interactions on the brain cells during a stroke. Microglia cells, neural stem cells, the damage on brain cells from the stroke and the impacts these interactions have on living brain cells are considered in the design of mathematical models. The models consist of ordinary differential equations describing the effects of microglia on brain cells and the interactions between microglia and neural stem cells in the case of a stroke. Variables considered include: resident microglia, classically activated microglia, alternatively activated microglia, neural stem cells, tissue damage on cells in the brain, and the impacts these interactions have on living brain cells. The first model describes what happens in the brain at the stroke onset during the first three days without the generation of any neural stem cells. The second model studies the dynamic effect of microglia and neural stem cells on the brain cells following the generation of neural stem cells and potential recovery after this stage. We look at the stability and the instability of the models which are both studied analytically. The results show that the immune cells can help the brain by cleaning dead cells and stimulating the generation of neural stem cells; however, excessive activation may cause damage and affect the injured region. Microglia have beneficial and harmful functions after ischemic stroke. The microglia stimulate neural stem cells to generate new cells that substitute dead cells during the recovery stage but sometimes the endogenous neural stem cells are highly sensitive to inflammatory in the brain. Full article

(This article belongs to the Special Issue Mathematical Methods in Applied Sciences)

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14 pages, 833 KiB

Open AccessFeature PaperReview

Analysis of the Cryptographic Tools for Blockchain and Bitcoin

by Víctor Gayoso Martínez, Luis Hernández-Álvarez and Luis Hernández Encinas

Mathematics 2020, 8(1), 131; https://doi.org/10.3390/math8010131 - 15 Jan 2020

Cited by 29 | Viewed by 10903

Abstract

Blockchain is one of the most interesting emerging technologies nowadays, with applications ranging from cryptocurrencies to smart contracts. This paper presents a review of the cryptographic tools necessary to understand the fundamentals of this technology and the foundations of its security. Among other [...] Read more.

Blockchain is one of the most interesting emerging technologies nowadays, with applications ranging from cryptocurrencies to smart contracts. This paper presents a review of the cryptographic tools necessary to understand the fundamentals of this technology and the foundations of its security. Among other elements, hash functions, digital signatures, elliptic curves, and Merkle trees are reviewed in the scope of their usage as building blocks of this technology. Full article

(This article belongs to the Special Issue Mathematical Models in Security, Defense, Cyber Security and Cyber Defense)

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18 pages, 1053 KiB

Open AccessArticle

Impacts of Thermal Radiation and Heat Consumption/Generation on Unsteady MHD Convection Flow of an Oldroyd-B Fluid with Ramped Velocity and Temperature in a Generalized Darcy Medium

by Talha Anwar, Ilyas Khan, Poom Kumam and Wiboonsak Watthayu

Mathematics 2020, 8(1), 130; https://doi.org/10.3390/math8010130 - 15 Jan 2020

Cited by 19 | Viewed by 2903

Abstract

This article analyzes the time-dependent magnetohydrodynamic flow of Oldroyed-B fluid in the presence of heat consumption/generation and thermal radiation. The flow is restricted to a vertical infinite plate saturated in porous material along with ramp wall velocity and ramp wall temperature conditions. This [...] Read more.

This article analyzes the time-dependent magnetohydrodynamic flow of Oldroyed-B fluid in the presence of heat consumption/generation and thermal radiation. The flow is restricted to a vertical infinite plate saturated in porous material along with ramp wall velocity and ramp wall temperature conditions. This flow also incorporates the generalized Darcy’s law. In this paper, accurate equation of velocity field is presented first and then solutions of mass and energy equation are derived in Laplace domain. Real-time domain solutions are obtained by tackling the complexity of Laplace domain expressions through numerical Laplace inversion. Skin friction coefficient and Nusselt number are also calculated. A comparison for ramp wall temperature condition and isothermal temperature condition is also drawn to investigate the difference. A graphical study is conducted to analyze the influence of parameters on fluid flow and heat transfer. It is found that radiation parameter and heat generation elevate the energy profile, while flow is accelerated by increasing the retardation time and porosity parameter and an opposite behavior is noted for increasing relaxation time and magnetic parameter. Furthermore, heat transfer rate is higher for increasing Prandtl number and velocity on plate decreases with increase in relaxation time

λ_{1}

. Full article

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30 pages, 631 KiB

Open AccessArticle

Design and Analysis of a Non-Iterative Estimator for Target Location in Multistatic Sonar Systems with Sensor Position Uncertainties

by Xin Wang, Zhi Yu, Le Yang and Ji Li

Mathematics 2020, 8(1), 129; https://doi.org/10.3390/math8010129 - 15 Jan 2020

Viewed by 2127

Abstract

Target location is the basic application of a multistatic sonar system. Determining the position/velocity vector of a target from the related sonar observations is a nonlinear estimation problem. The presence of possible sensor position uncertainties turns this problem into a more challenging hybrid [...] Read more.

Target location is the basic application of a multistatic sonar system. Determining the position/velocity vector of a target from the related sonar observations is a nonlinear estimation problem. The presence of possible sensor position uncertainties turns this problem into a more challenging hybrid parameter estimation problem. Conventional gradient-based iterative estimators suffer from the problems of initialization difficulties and local convergence. Even if there is no problem with initialization and convergence, a large computational cost is required in most cases. In view of these drawbacks, we develop a computationally efficient non-iterative position/velocity estimator. The main numerical computation involved is the weighted least squares optimization, which makes the estimator computationally efficient. Parameter transformation, model linearization and two-stage processing are exploited to prevent the estimator from iterative computation. Through performance analysis and experimental verification, we find that the proposed estimator reaches the hybrid Cramér–Rao bound and has linear computational complexity. Full article

(This article belongs to the Section Engineering Mathematics)

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15 pages, 309 KiB

Open AccessArticle

Multiplicity of Radially Symmetric Small Energy Solutions for Quasilinear Elliptic Equations Involving Nonhomogeneous Operators

by Jun Ik Lee and Yun-Ho Kim

Mathematics 2020, 8(1), 128; https://doi.org/10.3390/math8010128 - 15 Jan 2020

Cited by 4 | Viewed by 1737

Abstract

We investigate the multiplicity of radially symmetric solutions for the quasilinear elliptic equation of Kirchhoff type. This paper is devoted to the study of the

L^{\infty}

-bound of solutions to the problem above by applying De Giorgi’s iteration method and the localization [...] Read more.

We investigate the multiplicity of radially symmetric solutions for the quasilinear elliptic equation of Kirchhoff type. This paper is devoted to the study of the

L^{\infty}

-bound of solutions to the problem above by applying De Giorgi’s iteration method and the localization method. Employing this, we provide the existence of multiple small energy radially symmetric solutions whose

L^{\infty}

-norms converge to zero. We utilize the modified functional method and the dual fountain theorem as the main tools. Full article

(This article belongs to the Section Difference and Differential Equations)

20 pages, 361 KiB

Open AccessArticle

Banach Lattice Structures and Concavifications in Banach Spaces

by Lucia Agud, Jose Manuel Calabuig, Maria Aranzazu Juan and Enrique A. Sánchez Pérez

Mathematics 2020, 8(1), 127; https://doi.org/10.3390/math8010127 - 14 Jan 2020

Viewed by 3048

Abstract

Let

(Ω, Σ, μ)

be a finite measure space and consider a Banach function space

Y (μ)

. We say that a Banach space E is representable by

Y (μ)

if there is a continuous [...] Read more.

Let

(Ω, Σ, μ)

be a finite measure space and consider a Banach function space

Y (μ)

. We say that a Banach space E is representable by

Y (μ)

if there is a continuous bijection

I : Y (μ) \to E

. In this case, it is possible to define an order and, consequently, a lattice structure for E in such a way that we can identify it as a Banach function space, at least regarding some local properties. General and concrete applications are shown, including the study of the notion of the pth power of a Banach space, the characterization of spaces of operators that are isomorphic to Banach lattices of multiplication operators, and the representation of certain spaces of homogeneous polynomials on Banach spaces as operators acting in function spaces. Full article

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

22 pages, 316 KiB

Open AccessArticle

Existence of Solutions for Fractional Multi-Point Boundary Value Problems on an Infinite Interval at Resonance

by Wei Zhang and Wenbin Liu

Mathematics 2020, 8(1), 126; https://doi.org/10.3390/math8010126 - 14 Jan 2020

Cited by 7 | Viewed by 2243

Abstract

This paper aims to investigate a class of fractional multi-point boundary value problems at resonance on an infinite interval. New existence results are obtained for the given problem using Mawhin’s coincidence degree theory. Moreover, two examples are given to illustrate the main results. [...] Read more.

This paper aims to investigate a class of fractional multi-point boundary value problems at resonance on an infinite interval. New existence results are obtained for the given problem using Mawhin’s coincidence degree theory. Moreover, two examples are given to illustrate the main results. Full article

11 pages, 286 KiB

Open AccessArticle

The Simultaneous Strong Resolving Graph and the Simultaneous Strong Metric Dimension of Graph Families

by Ismael González Yero

Mathematics 2020, 8(1), 125; https://doi.org/10.3390/math8010125 - 14 Jan 2020

Cited by 1 | Viewed by 1957

Abstract

We consider in this work a new approach to study the simultaneous strong metric dimension of graphs families, while introducing the simultaneous version of the strong resolving graph. In concordance, we consider here connected graphs G whose vertex sets are represented as [...] Read more.

We consider in this work a new approach to study the simultaneous strong metric dimension of graphs families, while introducing the simultaneous version of the strong resolving graph. In concordance, we consider here connected graphs G whose vertex sets are represented as

V (G)

, and the following terminology. Two vertices

u, v \in V (G)

are strongly resolved by a vertex

w \in V (G)

, if there is a shortest

w - v

path containing u or a shortest

w - u

containing v. A set A of vertices of the graph G is said to be a strong metric generator for G if every two vertices of G are strongly resolved by some vertex of A. The smallest possible cardinality of any strong metric generator (SSMG) for the graph G is taken as the strong metric dimension of the graph G. Given a family

F

of graphs defined over a common vertex set V, a set

S \subset V

is an SSMG for

F

, if such set S is a strong metric generator for every graph

G \in F

. The simultaneous strong metric dimension of

F

is the minimum cardinality of any strong metric generator for

F

, and is denoted by

{Sd}_{s} (F)

. The notion of simultaneous strong resolving graph of a graph family

F

is introduced in this work, and its usefulness in the study of

{Sd}_{s} (F)

is described. That is, it is proved that computing

{Sd}_{s} (F)

is equivalent to computing the vertex cover number of the simultaneous strong resolving graph of

F

. Several consequences (computational and combinatorial) of such relationship are then deduced. Among them, we remark for instance that we have proved the NP-hardness of computing the simultaneous strong metric dimension of families of paths, which is an improvement (with respect to the increasing difficulty of the problem) on the results known from the literature. Full article

(This article belongs to the Special Issue Distances and Domination in Graphs)

8 pages, 382 KiB

Open AccessFeature PaperArticle

On the Diophantine Equation z(n) = (2 − 1/k)n Involving the Order of Appearance in the Fibonacci Sequence

by Eva Trojovská

Mathematics 2020, 8(1), 124; https://doi.org/10.3390/math8010124 - 14 Jan 2020

Cited by 8 | Viewed by 3476

Abstract

Let

{(F_{n})}_{n \geq 0}

be the sequence of the Fibonacci numbers. The order (or rank) of appearance

z (n)

of a positive integer n is defined as the smallest positive integer m such that n divides [...] Read more.

Let

{(F_{n})}_{n \geq 0}

be the sequence of the Fibonacci numbers. The order (or rank) of appearance

z (n)

of a positive integer n is defined as the smallest positive integer m such that n divides

F_{m}

. In 1975, Sallé proved that

z (n) \leq 2 n

, for all positive integers n. In this paper, we shall solve the Diophantine equation

z (n) = (2 - 1 / k) n

for positive integers n and k. Full article

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14 pages, 646 KiB

Open AccessArticle

A Class of Itô Diffusions with Known Terminal Value and Specified Optimal Barrier

by Bernardo D’Auria and Alessandro Ferriero

Mathematics 2020, 8(1), 123; https://doi.org/10.3390/math8010123 - 14 Jan 2020

Cited by 1 | Viewed by 1972

Abstract

In this paper, we study the optimal stopping-time problems related to a class of Itô diffusions, modeling for example an investment gain, for which the terminal value is a priori known. This could be the case of an insider trading or of the [...] Read more.

In this paper, we study the optimal stopping-time problems related to a class of Itô diffusions, modeling for example an investment gain, for which the terminal value is a priori known. This could be the case of an insider trading or of the pinning at expiration of stock options. We give the explicit solution to these optimization problems and in particular we provide a class of processes whose optimal barrier has the same form as the one of the Brownian bridge. These processes may be a possible alternative to the Brownian bridge in practice as they could better model real applications. Moreover, we discuss the existence of a process with a prescribed curve as optimal barrier, for any given (decreasing) curve. This gives a modeling approach for the optimal liquidation time, i.e., the optimal time at which the investor should liquidate a position to maximize the gain. Full article

(This article belongs to the Special Issue Diffusion Processes Associated with Growth Curves: Probabilistic and Inferential Analysis)

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23 pages, 1437 KiB

Open AccessArticle

The Trapezoidal Fuzzy Two-Dimensional Linguistic Power Generalized Hamy Mean Operator and Its Application in Multi-Attribute Decision-Making

by Yisheng Liu and Ye Li

Mathematics 2020, 8(1), 122; https://doi.org/10.3390/math8010122 - 13 Jan 2020

Cited by 6 | Viewed by 2033

Abstract

As a common information aggregation tool, the Hamy mean (HM) operator can consider the relationships among multiple input elements, but cannot adjust the effect of elements. In this paper, we integrate the idea of generalized a weighted average (GWA) operator into the HM [...] Read more.

As a common information aggregation tool, the Hamy mean (HM) operator can consider the relationships among multiple input elements, but cannot adjust the effect of elements. In this paper, we integrate the idea of generalized a weighted average (GWA) operator into the HM operator, and reduce the influence of related elements by adjusting the value of the parameter. In addition, considering that extreme input data may lead to a deviation in the results, we further combine the power average (PA) operator with HM, and propose the power generalized Hamy mean (PGHM) operator. Then, we extend the PGHM operator to the trapezoidal fuzzy two-dimensional linguistic environment, and propose two new information aggregation tools, the trapezoidal fuzzy two-dimensional linguistic power generalized Hamy mean (TF2DLPGHM) operator and the weighted TF2DLPGHM (WTF2DLPGHM) operator. Some properties and special cases of these operators are discussed. Furthermore, based on the proposed WTF2DLPGHM operator, a new multi-attribute decision-making method is proposed for lean management evaluation of industrial residential projects. Finally, an example is given to show the specific steps, effectiveness, and superiority of the method. Full article

(This article belongs to the Special Issue Recent Advances in Game and Decision Theory: Structures, Models, Applications and Software Implementation 2019)

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17 pages, 1621 KiB

Open AccessArticle

Mathematical Model and Evaluation Function for Conflict-Free Warranted Makespan Minimization of Mixed Blocking Constraint Job-Shop Problems

by Christophe Sauvey, Wajdi Trabelsi and Nathalie Sauer

Mathematics 2020, 8(1), 121; https://doi.org/10.3390/math8010121 - 13 Jan 2020

Cited by 6 | Viewed by 3278

Abstract

In this paper, we consider a job-shop scheduling problem with mixed blocking constraints. Contrary to most previous studies, where no blocking or only one type of blocking constraint was used among successive operations, we assume that, generally, we may address several different blocking [...] Read more.

In this paper, we consider a job-shop scheduling problem with mixed blocking constraints. Contrary to most previous studies, where no blocking or only one type of blocking constraint was used among successive operations, we assume that, generally, we may address several different blocking constraints in the same scheduling problem depending on the intermediate storage among machines, the characteristics of the machines, the technical constraints, and even the jobs. Our objective was to schedule a set of jobs to minimize the makespan. Thus, we propose, for the first time, a mathematical model of the job-shop problem taking into account the general case of mixed blocking constraints, and the results were obtained using Mosel Xpress software. Then, after explaining why and how groups of jobs have to be processed, a blocking constraint conflict-free warranted evaluation function is proposed and tested with the particle swarm optimization and genetic algorithm methods. The results prove that we obtained a near-optimal solution to this problem in a very short time. Full article

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27 pages, 11681 KiB

Open AccessArticle

Analyzing the Causality and Dependence between Gold Shocks and Asian Emerging Stock Markets: A Smooth Transition Copula Approach

by Woraphon Yamaka and Paravee Maneejuk

Mathematics 2020, 8(1), 120; https://doi.org/10.3390/math8010120 - 13 Jan 2020

Cited by 15 | Viewed by 3709

Abstract

This study aims to investigate the causality and dependence structure of gold shocks and Asian emerging stock markets. The positive and negative shocks of gold prices are quantified, and Granger causality-based Vector autoregressive and Copula approaches are employed to measure the causality and [...] Read more.

This study aims to investigate the causality and dependence structure of gold shocks and Asian emerging stock markets. The positive and negative shocks of gold prices are quantified, and Granger causality-based Vector autoregressive and Copula approaches are employed to measure the causality and contagion effect, respectively, between the positive and negative gold shocks and Asian emerging stock markets’ volatilities. In addition, the nonlinear link between gold and stock markets is of concern and this motivates us to propose a Smooth Transition Dynamic Copula that allows for the structural change in time-varying dependence between gold shocks and Asian stock markets’ volatilities. Several Copula families are also considered, and the best-fit Copula model is used to explain the correlation or contagion effects. The findings of the study show that there is some significant causality between gold shocks and Asian stock markets’ volatilities in some parts of the sample period. We also observe a stronger correlation during the global financial crisis when compared to the pre- and post-crisis periods. In addition, the tail dependence is found between Indian stock and negative gold shock and between Korean stock and negative gold shock, which indicated the existence of the risk contagion effects between gold and these two stock markets. Full article

(This article belongs to the Special Issue Advanced Methods in Mathematical Finance)

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20 pages, 10035 KiB

Open AccessArticle

Improving Initial Guess for the Iterative Solution of Linear Equation Systems in Incompressible Flow

by Shuai Ye, Yufei Lin, Liyang Xu and Jiaming Wu

Mathematics 2020, 8(1), 119; https://doi.org/10.3390/math8010119 - 13 Jan 2020

Cited by 3 | Viewed by 4297

Abstract

The pressure equation, generated while solving the incompressible Navier–Stokes equations with the segregated iterative algorithm such as PISO, produces a series of linear equation systems as the time step advances. In this paper, we target at accelerating the iterative solution of these linear [...] Read more.

The pressure equation, generated while solving the incompressible Navier–Stokes equations with the segregated iterative algorithm such as PISO, produces a series of linear equation systems as the time step advances. In this paper, we target at accelerating the iterative solution of these linear systems by improving their initial guesses. We propose a weighted group extrapolation method to obtain a superior initial guess instead of a general one, the solution of the previous linear equation system. In this method, the previous solutions that are used to extrapolate the predicted solutions are carefully organized to address the oscillatory solution on each grid. The proposed method uses a weighted average of the predicted solutions as the new initial guess to avoid over extrapolating. Three numerical test results show that the proposed method can accelerate the iterative solution of most linear equation systems and reduce the simulation time up to 61.3%. Full article

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16 pages, 3338 KiB

Open AccessArticle

On the Number of Shortest Weighted Paths in a Triangular Grid

by Benedek Nagy and Bashar Khassawneh

Mathematics 2020, 8(1), 118; https://doi.org/10.3390/math8010118 - 13 Jan 2020

Cited by 3 | Viewed by 3123

Abstract

Counting the number of shortest paths in various graphs is an important and interesting combinatorial problem, especially in weighted graphs with various applications. We consider a specific infinite graph here, namely the honeycomb grid. Changing to its dual, the triangular grid, paths between [...] Read more.

Counting the number of shortest paths in various graphs is an important and interesting combinatorial problem, especially in weighted graphs with various applications. We consider a specific infinite graph here, namely the honeycomb grid. Changing to its dual, the triangular grid, paths between triangle pixels (we abbreviate this term to trixels) are counted. The number of shortest weighted paths between any two trixels of the triangular grid is discussed. For each trixel, there are three different types of neighbor trixels, 1-, 2- and 3-neighbours, depending the Euclidean distance of their midpoints. When considering weighted distances, the positive values α, β and γ are assigned to the ‘steps’ to various neighbors. We gave formulae for the number of shortest weighted paths between any two trixels in various cases by the respective weight values. The results are nicely connected to various numbers well-known in combinatorics, e.g., to binomial coefficients and Fibonacci numbers. Full article

(This article belongs to the Special Issue Discrete and Computational Geometry)

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Graphical abstract

Graphical abstract
Full article ">Figure 1
Coordinate system for the triangular grid with the Origin and the axes. Full article ">Figure 2
Three types of neighbors of trixel 0. 1-neighbors are red, 2-neighbors are yellow and 3-neighbors are green. Full article ">Figure 3
The number of shortest weighed paths from trixel (0,0,0) to other trixels of the grid with the condition 2α < β and 3α < γ. Full article ">Figure 4
The number of shortest weighed paths between trixel (0,0,0) and some other trixels in the case 2α > β and 3α < γ. Full article ">Figure 5
The number of shortest weighed paths from trixel (0,0,0) to some other trixels with the condition 2α < β < γ < 3α. Full article ">Figure 6
The number of the shortest weighed paths from trixel the origin to some other trixels with the condition 2α = β and 3α < γ. Full article ">Figure 7
The number of shortest weighed paths from trixel (0,0,0) to all other displayed trixels in the case of 2α < β and 3α = γ. Full article ">

20 pages, 2323 KiB

Open AccessArticle

Nonlocal Reaction–Diffusion Model of Viral Evolution: Emergence of Virus Strains

by Nikolai Bessonov, Gennady Bocharov, Andreas Meyerhans, Vladimir Popov and Vitaly Volpert

Mathematics 2020, 8(1), 117; https://doi.org/10.3390/math8010117 - 12 Jan 2020

Cited by 10 | Viewed by 5053

Abstract

This work is devoted to the investigation of virus quasi-species evolution and diversification due to mutations, competition for host cells, and cross-reactive immune responses. The model consists of a nonlocal reaction–diffusion equation for the virus density depending on the genotype considered to be [...] Read more.

This work is devoted to the investigation of virus quasi-species evolution and diversification due to mutations, competition for host cells, and cross-reactive immune responses. The model consists of a nonlocal reaction–diffusion equation for the virus density depending on the genotype considered to be a continuous variable and on time. This equation contains two integral terms corresponding to the nonlocal effects of virus interaction with host cells and with immune cells. In the model, a virus strain is represented by a localized solution concentrated around some given genotype. Emergence of new strains corresponds to a periodic wave propagating in the space of genotypes. The conditions of appearance of such waves and their dynamics are described. Full article

(This article belongs to the Special Issue Mathematical Modelling in Biomedicine)

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35 pages, 473 KiB

Open AccessArticle

Cohomology of Presheaves of Monoids

by Pilar Carrasco and Antonio M. Cegarra

Mathematics 2020, 8(1), 116; https://doi.org/10.3390/math8010116 - 12 Jan 2020

Cited by 1 | Viewed by 2345

Abstract

The purpose of this work is to extend Leech cohomology for monoids (and so Eilenberg-Mac Lane cohomology of groups) to presheaves of monoids on an arbitrary small category. The main result states and proves a cohomological classification of monoidal prestacks on a category [...] Read more.

The purpose of this work is to extend Leech cohomology for monoids (and so Eilenberg-Mac Lane cohomology of groups) to presheaves of monoids on an arbitrary small category. The main result states and proves a cohomological classification of monoidal prestacks on a category with values in groupoids with abelian isotropy groups. The paper also includes a cohomological classification for extensions of presheaves of monoids, which is useful to the study of

H

-extensions of presheaves of regular monoids. The results apply directly in several settings such as presheaves of monoids on a topological space, simplicial monoids, presheaves of simplicial monoids on a topological space, monoids or simplicial monoids on which a fixed monoid or group acts, and so forth. Full article

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Mathematics, Volume 8, Issue 1 (January 2020) – 145 articles

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