Mathematical and Computational Applications

Open AccessFeature PaperArticle

A (p,q)-Averaged Hausdorff Distance for Arbitrary Measurable Sets

by Johan M. Bogoya, Andrés Vargas, Oliver Cuate and Oliver Schütze

Math. Comput. Appl. 2018, 23(3), 51; https://doi.org/10.3390/mca23030051 - 18 September 2018

The Hausdorff distance is a widely used tool to measure the distance between different sets. For the approximation of certain objects via stochastic search algorithms this distance is, however, of limited use as it punishes single outliers. As a remedy in the context

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The Hausdorff distance is a widely used tool to measure the distance between different sets. For the approximation of certain objects via stochastic search algorithms this distance is, however, of limited use as it punishes single outliers. As a remedy in the context of evolutionary multi-objective optimization (EMO), the averaged Hausdorff distance

Δ_{p}

has been proposed that is better suited as an indicator for the performance assessment of EMO algorithms since such methods tend to generate outliers. Later on, the two-parameter indicator

Δ_{p, q}

has been proposed for finite sets as an extension to

Δ_{p}

which also averages distances, but which yields some desired metric properties. In this paper, we extend

Δ_{p, q}

to a continuous function between general bounded subsets of finite measure inside a metric measure space. In particular, this extension applies to bounded subsets of

R^{k}

endowed with the Euclidean metric, which is the natural context for EMO applications. We show that our extension preserves the nice metric properties of the finite case, and finally provide some useful numerical examples that arise in EMO. Full article

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Open AccessCommunication

Left Ventricular Volume in Bovines: The Correlation between Teichholz’s Medical Mathematical Method and the Volume of the Truncated Prolate Spheroid

by José Sérgio Domingues, Ravel Alvarenga Coelho, Francielly dos Santos Bento, Brenda Cristina Almeida and Ayanne Oliveira de Castro

Math. Comput. Appl. 2018, 23(3), 50; https://doi.org/10.3390/mca23030050 - 11 September 2018

Abstract

The main objective of this article is to determine the existing linear correlation between the real left ventricular volume (RV) from the heart of bovines and the volumes obtained by Teichholz’s mathematical model and the volume of the truncated prolate spheroid (TPS) to

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The main objective of this article is to determine the existing linear correlation between the real left ventricular volume (RV) from the heart of bovines and the volumes obtained by Teichholz’s mathematical model and the volume of the truncated prolate spheroid (TPS) to identify which model has a higher proximity to the RV. For that, ten silicon rubber molds of the left ventricle (LV) were manufactured, and their real volumes were obtained through Archimedes’ principle, and their linear dimensions were also obtained. These dimensions were used to feed Teichholz’s and the TPS models. It was verified that, for ventricles of lower volume, the models showed relatively close results, and Teichholz’s model was the most accurate one. The TPS method shows a grave accuracy mistake for higher volume ventricles. Besides, both methods showed strong linear correlations with the RV, and both with high significance. Full article

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Open AccessFeature PaperArticle

Algebraic Operations on Delta-Sigma Bit-Streams

by Axel Klein and Walter Schumacher

Math. Comput. Appl. 2018, 23(3), 49; https://doi.org/10.3390/mca23030049 - 10 September 2018

Abstract

Operations in the Delta-Sigma (

Δ Σ

) domain are a broad field of research. In this article the main, focus is on applications in control systems, nevertheless the results are generally applicable for

Δ Σ

-signal processing (

Δ Σ

SP) in

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Operations in the Delta-Sigma (

Δ Σ

) domain are a broad field of research. In this article the main, focus is on applications in control systems, nevertheless the results are generally applicable for

Δ Σ

-signal processing (

Δ Σ

SP) in other fields. As the bit-stream does not have an instantaneous value, algebraic operations cannot be executed directly. The first approaches were made in the 1980s based on small-scale integration logic chips by Kouvaras and by Lagoyannis. Further algebraic operations and other implementations were introduced by Zrilic, by Ng, by Bradshaw and by Homann. Other publications utilize complex networks and operations to achieve the desired algebraic operations. These presented operations can be divided into different operation classes by the based implementation idea. In this paper, the known algebraic operation classes are further developed and new operation classes are presented. All implementations are compared and evaluated. For linear operations in control applications, the introduced Bipolar Interpretation is best rated. It compensates for the signal offset of bipolar bit-streams and results in the best signal quality by mapping the logic values true and false of bit-stream to plus and minus one before the algebraic operation. The output of the algebraic operation is a multibit value, to achieve a bit-stream as output value a third step is taken. The result is modulated by a digital

Δ Σ

-modulator (

Δ Σ

-M). For nonlinear operations the most universal implementation is also based on three steps. In the first stage, the bit-streams are processed with short sinc

^{3}

filters, resulting in multibit values. This signal is processed by digital signal processing (DSP). The output stage is a

Δ Σ

-M. For some nonlinear algebraic operations there can be better solutions than DSP, like shown for limiting. In short, this paper gives a detailed overview about different

Δ Σ

SP classes for linear and nonlinear operations. Newly presented are the scaling with Bit-Stream Modification, the Bipolar Interpretation class, the nonlinear operation class based on digital signal processing (DSP), the modified multiplication based on Delta Adder and benchmarks of all presented operations. Full article

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Open AccessFeature PaperArticle

First Order Hadamard Variation of the Harmonic Navigation Function on a Sphere World

by Isabelle Santos, Stéphane Puechmorel and Guillaume Dufour

Math. Comput. Appl. 2018, 23(3), 48; https://doi.org/10.3390/mca23030048 - 10 September 2018

Abstract

Planning conflict-free trajectories is a long-standing problem in Air Traffic Management. Navigation functions designed specifically to produce flyable trajectories have been previously considered, but lack the robustness to uncertain weather conditions needed for use in an operational context. These uncertainties can be taken

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Planning conflict-free trajectories is a long-standing problem in Air Traffic Management. Navigation functions designed specifically to produce flyable trajectories have been previously considered, but lack the robustness to uncertain weather conditions needed for use in an operational context. These uncertainties can be taken into account be modifying the boundary of the domain on which the navigation function is computed. In the following work, we present a method for efficiently taking into account boundary variations, using the Hadamard variation. Full article

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Open AccessArticle

TL-Moments for Type-I Censored Data with an Application to the Weibull Distribution

by Hager A. Ibrahim, Mahmoud Riad Mahmoud, Fatma A. Khalil and Ghada A. El-Kelany

Math. Comput. Appl. 2018, 23(3), 47; https://doi.org/10.3390/mca23030047 - 9 September 2018

Abstract

This paper aims to provide an adaptation of the trimmed L (TL)-moments method to censored data. The present study concentrates on Type-I censored data. The idea of using TL-moments with censored data may seem conflicting. However, our perspective is that we can use

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This paper aims to provide an adaptation of the trimmed L (TL)-moments method to censored data. The present study concentrates on Type-I censored data. The idea of using TL-moments with censored data may seem conflicting. However, our perspective is that we can use data censored from one side and trimmed from the other side. This study is applied to estimate the two unknown parameters of the Weibull distribution. The suggested point is compared with direct L-moments and maximum likelihood (ML) methods. A Monte Carlo simulation study is carried out to compare these methods in terms of estimate average, root of mean square error (RMSE), and relative absolute biases (RABs). Full article

Open AccessFeature PaperArticle

An Authentication Code over Galois Rings with Optimal Impersonation and Substitution Probabilities

by Juan Carlos Ku-Cauich, Guillermo Morales-Luna and Horacio Tapia-Recillas

Math. Comput. Appl. 2018, 23(3), 46; https://doi.org/10.3390/mca23030046 - 6 September 2018

Abstract

Two new systematic authentication codes based on the Gray map over a Galois ring are introduced. The first introduced code attains optimal impersonation and substitution probabilities. The second code improves space sizes, but it does not attain optimal probabilities. Additionally, it is conditioned

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Two new systematic authentication codes based on the Gray map over a Galois ring are introduced. The first introduced code attains optimal impersonation and substitution probabilities. The second code improves space sizes, but it does not attain optimal probabilities. Additionally, it is conditioned to the existence of a special class of bent maps on Galois rings. Full article

Open AccessFeature PaperArticle

Optimal Strategies for Psoriasis Treatment

by Ellina Grigorieva and Evgenii Khailov

Math. Comput. Appl. 2018, 23(3), 45; https://doi.org/10.3390/mca23030045 - 4 September 2018

Abstract

Within a given time interval we consider a nonlinear system of differential equations describing psoriasis treatment. Its phase variables define the concentrations of T-lymphocytes, keratinocytes and dendritic cells. Two scalar bounded controls are introduced into this system to reflect medication dosages aimed at

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Within a given time interval we consider a nonlinear system of differential equations describing psoriasis treatment. Its phase variables define the concentrations of T-lymphocytes, keratinocytes and dendritic cells. Two scalar bounded controls are introduced into this system to reflect medication dosages aimed at suppressing interactions between T-lymphocytes and keratinocytes, and between T-lymphocytes and dendritic cells. For such a controlled system, a minimization problem of the concentration of keratinocytes at the terminal time is considered. For its analysis, the Pontryagin maximum principle is applied. As a result of this analysis, the properties of the optimal controls and their possible types are established. It is shown that each of these controls is either a bang-bang type on the entire time interval or (in addition to bang-bang type) contains a singular arc. The obtained analytical results are confirmed by numerical calculations using the software “BOCOP-2.0.5”. Their detailed analysis and the corresponding conclusions are presented. Full article

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Open AccessRetraction

Retraction: Hang, Y.; Liu, Y.; Xu, X.; Chen, Y.; Mo, S. Sensitivity Analysis Based on Markovian Integration by Parts Formula. Math. Comput. Appl. 2017, 22, 40

by Yongsheng Hang, Yue Liu, Xiaoyang Xu, Yan Chen and Shu Mo

Math. Comput. Appl. 2018, 23(3), 44; https://doi.org/10.3390/mca23030044 - 30 August 2018

Abstract

We, the authors, have requested that the title paper [1] is retracted[...] Full article

Open AccessArticle

A Complex Variable Solution for Lining Stress and Deformation in a Non-Circular Deep Tunnel II Practical Application and Verification

by Yansong Li and Shougen Chen

Math. Comput. Appl. 2018, 23(3), 43; https://doi.org/10.3390/mca23030043 - 30 August 2018

Abstract

A new complex variable method is presented for stress and displacement problems in a non-circular deep tunnel with certain given boundary conditions at infinity. In order to overcome the complex problems caused by non-circular geometric configurations and the multiply-connected region, a complex variable

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A new complex variable method is presented for stress and displacement problems in a non-circular deep tunnel with certain given boundary conditions at infinity. In order to overcome the complex problems caused by non-circular geometric configurations and the multiply-connected region, a complex variable method and continuity boundary conditions are used to determine stress and displacement within the tunnel lining and within the surrounding rock. The coefficients in the conformal mapping function and stress functions are determined by the optimal design and complex variable method, respectively. The new method is validated by FLAC3D finite difference software through an example. Both the new method and the numerical simulation obtained similar results for the stress concentration and the minimum radial displacement occurred at a similar place in the tunnel. It is demonstrated that the new complex variable method is reliable and reasonable. The new method also provides another way to solve non-circular tunnel excavation problems in a faster and more accurate way. Full article

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Open AccessArticle

Specific Types of Pythagorean Fuzzy Graphs and Application to Decision-Making

by Muhammad Akram, Amna Habib, Farwa Ilyas and Jawaria Mohsan Dar

Math. Comput. Appl. 2018, 23(3), 42; https://doi.org/10.3390/mca23030042 - 23 August 2018

Abstract

The purpose of this research study is to present some new operations, including rejection, symmetric difference, residue product, and maximal product of Pythagorean fuzzy graphs (PFGs), and to explore some of their properties. This research article introduces certain notions, including intuitionistic fuzzy graphs

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The purpose of this research study is to present some new operations, including rejection, symmetric difference, residue product, and maximal product of Pythagorean fuzzy graphs (PFGs), and to explore some of their properties. This research article introduces certain notions, including intuitionistic fuzzy graphs of 3-type (IFGs3T), intuitionistic fuzzy graphs of 4-type (IFGs4T), and intuitionistic fuzzy graphs of n-type (IFGsnT), and proves that every IFG(n − 1)T is an IFGnT (for

n \geq

2). Moreover, this study discusses the application of Pythagorean fuzzy graphs in decision making. Full article

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Open AccessArticle

The Average Hull Dimension of Negacyclic Codes over Finite Fields

by Somphong Jitman and Ekkasit Sangwisut

Math. Comput. Appl. 2018, 23(3), 41; https://doi.org/10.3390/mca23030041 - 20 August 2018

Abstract

Hulls of linear codes have been extensively studied due to their wide applications and links with the efficiency of some algorithms in coding theory. In this paper, the average dimension of the Euclidean hull of negacyclic codes of length n over finite fields

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Hulls of linear codes have been extensively studied due to their wide applications and links with the efficiency of some algorithms in coding theory. In this paper, the average dimension of the Euclidean hull of negacyclic codes of length n over finite fields

F_{q}

, denoted by

E (n, - 1, q)

, has been investigated. The formula for

E (n, - 1, q)

has been determined. Some upper and lower bounds of

E (n, - 1, q)

have been given as well. Asymptotically, it has been shown that either

E (n, - 1, q)

is zero or it grows the same rate as n. Full article

Open AccessArticle

An Improved Differential Evolution Algorithm for Crop Planning in the Northeastern Region of Thailand

by Udompong Ketsripongsa, Rapeepan Pitakaso, Kanchana Sethanan and Tassin Srivarapongse

Math. Comput. Appl. 2018, 23(3), 40; https://doi.org/10.3390/mca23030040 - 10 August 2018

Abstract

This research aimed to solve the economic crop planning problem, considering transportation logistics to maximize the profit from cultivated activities. Income is derived from the selling price and production rate of the plants; costs are due to operating and transportation expenses. Two solving

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This research aimed to solve the economic crop planning problem, considering transportation logistics to maximize the profit from cultivated activities. Income is derived from the selling price and production rate of the plants; costs are due to operating and transportation expenses. Two solving methods are presented: (1) developing a mathematical model and solving it using Lingo v.11, and (2) using three improved Differential Evolution (DE) Algorithms—I-DE-SW, I-DE-CY, and I-DE-KV—which are DE with swap, cyclic moves (CY), and K-variables moves (KV) respectively. The algorithms were tested by 16 test instances, including this case study. The computational results showed that Lingo v.11 and all DE algorithms can find the optimal solution eight out of 16 times. Regarding the remaining test instances, Lingo v.11 was unable to find the optimal solution within 400 h. The results for the DE algorithms were compared with the best solution generated within that time. The DE solutions were 1.196–1.488% better than the best solution generated by Lingo v.11 and used 200 times less computational time. Comparing the three DE algorithms, MDE-KV was the DE that was the most flexible, with the biggest neighborhood structure, and outperformed the other DE algorithms. Full article

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Open AccessArticle

Rapid Modeling and Parameter Estimation of Partial Differential Algebraic Equations by a Functional Spreadsheet Paradigm

by Chahid Kamel Ghaddar

Math. Comput. Appl. 2018, 23(3), 39; https://doi.org/10.3390/mca23030039 - 3 August 2018

Abstract

We present a systematic spreadsheet method for modeling and optimizing general partial differential algebraic equations (PDAE). The method exploits a pure spreadsheet PDAE solver function design that encapsulates the Method of Lines and permits seamless integration with an Excel spreadsheet nonlinear programming solver.

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We present a systematic spreadsheet method for modeling and optimizing general partial differential algebraic equations (PDAE). The method exploits a pure spreadsheet PDAE solver function design that encapsulates the Method of Lines and permits seamless integration with an Excel spreadsheet nonlinear programming solver. Two alternative least-square dynamical minimization schemes are devised and demonstrated on a complex parameterized PDAE system with discontinues properties and coupled time derivatives. Applying the method involves no more than defining a few formulas that closely parallel the original mathematical equations, without any programming skills. It offers a simpler alternative to more complex environments which require nontrivial programming skill and effort. Full article

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Open AccessArticle

Cost-Effective Analysis of Control Strategies to Reduce the Prevalence of Cutaneous Leishmaniasis, Based on a Mathematical Model

by Dibyendu Biswas, Suman Dolai, Jahangir Chowdhury, Priti K. Roy and Ellina V. Grigorieva

Math. Comput. Appl. 2018, 23(3), 38; https://doi.org/10.3390/mca23030038 - 25 July 2018

Abstract

Leishmaniasis is a neglected tropical vector-borne epidemic disease, and its transmission is a complex process. Zoonotic transmission to humans or animals occurs through the bites of female Phlebotominae sand flies. Here, reservoir is considered as a major source of endemic pathogen pool for

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Leishmaniasis is a neglected tropical vector-borne epidemic disease, and its transmission is a complex process. Zoonotic transmission to humans or animals occurs through the bites of female Phlebotominae sand flies. Here, reservoir is considered as a major source of endemic pathogen pool for disease outbreak, and the role of more than one reservoir animal becomes indispensable. To study the role of the reservoir animals on disease dynamics, a mathematical model was constructed consisting of susceptible and infected populations of humans and two types of reservoir (animal) and vector populations, respectively. Our aim is to prevent the disease by applying a control theoretic approach, when more than one type of reservoir animal exists in the region. We use drugs like sodium stibogluconate and meglumine antimoniate to control the disease for humans and spray insecticide to control the sand fly population. Similarly, drugs are applied for infected reservoir animals of Types A and B. We calculated the cost-effectiveness of all possible combinations of the intervention and control policies. One of our findings is that the most cost-effective case for Leishmania control is the spray of insecticides for infected sand fly vector. Alternate strategic cases were compared to address the critical shortcomings of single strategic cases, and a range of control strategies were estimated for effective control and economical benefit of the overall control strategy. Our findings provide the most innovative techniques available for application to the successful eradication of cutaneous leishmaniasis in the future. Full article

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Open AccessArticle

Applying Computer Algebra Systems in Approximating the Trigonometric Functions

by Le Phuong Quan and Thái Anh Nhan

Math. Comput. Appl. 2018, 23(3), 37; https://doi.org/10.3390/mca23030037 - 14 July 2018

Abstract

We propose numerical algorithms which can be integrated with modern computer algebra systems in a way that is easily implemented to approximate the sine and cosine functions with an arbitrary accuracy. Our approach is based on Taylor’s expansion about a point having a

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We propose numerical algorithms which can be integrated with modern computer algebra systems in a way that is easily implemented to approximate the sine and cosine functions with an arbitrary accuracy. Our approach is based on Taylor’s expansion about a point having a form of

k p

,

k \in Z

and

p = π / 2

, and being chosen such that it is closest to the argument. A full error analysis, which takes advantage of current computer algebra systems in approximating

π

with a very high accuracy, of our proposed methods is provided. A numerical integration application is performed to demonstrate the use of algorithms. Numerical and graphical results are implemented by MAPLE. Full article

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Open AccessArticle

Existence of Solutions for Fractional Integro-Differential Equations with Non-Local Boundary Conditions

by Hamed Bazgir and Bahman Ghazanfari

Math. Comput. Appl. 2018, 23(3), 36; https://doi.org/10.3390/mca23030036 - 14 July 2018

Abstract

In this paper, we study the existence of solutions for a new class of boundary value problems of non-linear fractional integro-differential equations. The existence result is obtained with the aid of Schauder type fixed point theorem while the uniqueness of solution is established

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In this paper, we study the existence of solutions for a new class of boundary value problems of non-linear fractional integro-differential equations. The existence result is obtained with the aid of Schauder type fixed point theorem while the uniqueness of solution is established by means of contraction mapping principle. Then, we present some examples to illustrate our results. Full article

Open AccessArticle

Exact Solutions of Non-Linear Evolution Models in Physics and Biosciences Using the Hyperbolic Tangent Method

by Joseph Malinzi and Paul Amarh Quaye

Math. Comput. Appl. 2018, 23(3), 35; https://doi.org/10.3390/mca23030035 - 6 July 2018

Abstract

There has been considerable interest in seeking exact solutions of non-linear evolution equations that describe important physical and biological processes. Nonetheless, it is a difficult undertaking to determine closed form solutions of mathematical models that describe natural phenomena. This is because of their

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There has been considerable interest in seeking exact solutions of non-linear evolution equations that describe important physical and biological processes. Nonetheless, it is a difficult undertaking to determine closed form solutions of mathematical models that describe natural phenomena. This is because of their high non-linearity and the huge number of parameters of which they consist. In this article we determine, using the hyperbolic tangent (tanh) method, travelling wave solutions to non-linear evolution models of interest in biology and physics. These solutions have recognizable properties expected of other solutions and thus can be used to deduce properties of the general solutions. Full article

Open AccessArticle

Modified Differential Evolution Algorithm Solving the Special Case of Location Routing Problem

by Raknoi Akararungruangkul and Sasitorn Kaewman

Math. Comput. Appl. 2018, 23(3), 34; https://doi.org/10.3390/mca23030034 - 3 July 2018

Abstract

This research article aims to solve the special case of the location routing problem (SLRP) when the objective function is the fuel consumption. The fuel consumption depends on the distance of travel and the condition of the road. The condition of the road

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This research article aims to solve the special case of the location routing problem (SLRP) when the objective function is the fuel consumption. The fuel consumption depends on the distance of travel and the condition of the road. The condition of the road causes the vehicle to use a different speed, which affects fuel usage. This turns the original LRP into a more difficult problem. Moreover, the volume of the goods that are produced in each node could be more or less than the capacity of the vehicle, and as the case study requires the transportation of latex, which is a sensitive good and needs to be carried within a reasonable time so that it does not form solid before being used in the latex process, the maximum time that the latex can be in the truck is limited. All of these attributes are added into the LRP and make it a special case of LRP: a so-called SLRP (a special case of location routing problem). The differential evolution algorithms (DE) are proposed to solve the SLRP. We modified two points in the original DE, which are that (1) the mutation formula is introduced and (2) the new rule of a local search is presented. We call this the modified differential evolution algorithm (MDE). From the computational result, we can see that MDE generates a 13.82% better solution than that of the original version of DE in solving the test instances. Full article

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Open AccessArticle

Qualitative Analysis of a Dengue Fever Model

by Babatunde Gbadamosi, Mayowa Micheal Ojo, Segun Isaac Oke and Maba Boniface Matadi

Math. Comput. Appl. 2018, 23(3), 33; https://doi.org/10.3390/mca23030033 - 21 June 2018

Abstract

In this paper, a deterministic mathematical model of the Dengue virus with a nonlinear incidence function in a population is presented and rigorously analysed. The model incorporates control measures at the aquatic and adult stages of the vector (mosquito). The stability of the

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In this paper, a deterministic mathematical model of the Dengue virus with a nonlinear incidence function in a population is presented and rigorously analysed. The model incorporates control measures at the aquatic and adult stages of the vector (mosquito). The stability of the system is analysed for the disease-free equilibrium and the existence of endemic equilibria under certain conditions. The local stability of the Dengue-free equilibrium is investigated via the threshold parameter (reproduction number) that was obtained using the next-generation matrix techniques. The Routh–Hurwitz criterion, along with Descartes’ rule of signs change, established the local asymptotically stability of the model whenever

R_{0} < 1

and was unstable otherwise. The comparison theorem was used to establish the global asymptomatically stability of the model. Full article

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