Mathematics

16 pages, 499 KiB

Open AccessArticle

Optimizing Three-Dimensional Constrained Ordered Weighted Averaging Aggregation Problem with Bounded Variables

by Hui-Chin Tang and Shen-Tai Yang

Mathematics 2018, 6(9), 172; https://doi.org/10.3390/math6090172 - 19 Sep 2018

Cited by 3 | Viewed by 2829

A single constrained ordered weighted averaging aggregation (COWA) problem is of considerable importance in many disciplines. Two models are considered: the maximization COWA problem with lower bounded variables and the minimization COWA problem with upper bounded variables. For a three-dimensional case of these [...] Read more.

A single constrained ordered weighted averaging aggregation (COWA) problem is of considerable importance in many disciplines. Two models are considered: the maximization COWA problem with lower bounded variables and the minimization COWA problem with upper bounded variables. For a three-dimensional case of these models, we present the explicitly optimal solutions theoretically and empirically. The bounds and weights can affect the optimal solution of the three-dimensional COWA problem with bounded variables. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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21 pages, 290 KiB

Open AccessArticle

Almost Periodic Solutions of First-Order Ordinary Differential Equations

by Seifedine Kadry, Gennady Alferov, Gennady Ivanov and Artem Sharlay

Mathematics 2018, 6(9), 171; https://doi.org/10.3390/math6090171 - 17 Sep 2018

Cited by 13 | Viewed by 3011

Abstract

Approaches to estimate the number of almost periodic solutions of ordinary differential equations are considered. Conditions that allow determination for both upper and lower bounds for these solutions are found. The existence and stability of almost periodic problems are studied. The novelty of [...] Read more.

Approaches to estimate the number of almost periodic solutions of ordinary differential equations are considered. Conditions that allow determination for both upper and lower bounds for these solutions are found. The existence and stability of almost periodic problems are studied. The novelty of this paper lies in the fact that the use of apparatus derivatives allows for the reduction of restrictions on the degree of smoothness of the right parts. In our work, regarding the number of periodic solutions of equations first order, we don’t require a high degree of smoothness and no restriction on the smoothness of the second derivative of the Schwartz equation. We have all of these restrictions lifted. Our new form presented also emphasizes this novelty. Full article

39 pages, 366 KiB

Open AccessArticle

Convergence in Fuzzy Semi-Metric Spaces

by Hsien-Chung Wu

Mathematics 2018, 6(9), 170; https://doi.org/10.3390/math6090170 - 17 Sep 2018

Cited by 4 | Viewed by 2672

Abstract

The convergence using the fuzzy semi-metric and dual fuzzy semi-metric is studied in this paper. The infimum type of dual fuzzy semi-metric and the supremum type of dual fuzzy semi-metric are proposed in this paper. Based on these two types of dual fuzzy [...] Read more.

The convergence using the fuzzy semi-metric and dual fuzzy semi-metric is studied in this paper. The infimum type of dual fuzzy semi-metric and the supremum type of dual fuzzy semi-metric are proposed in this paper. Based on these two types of dual fuzzy semi-metrics, the different types of triangle inequalities can be obtained. We also study the convergence of these two types of dual fuzzy semi-metrics. Full article

(This article belongs to the Special Issue Nonlinear Analysis Using Fuzzy Mathematics)

44 pages, 950 KiB

Open AccessFeature PaperArticle

A Nonlinear Systems Framework for Cyberattack Prevention for Chemical Process Control Systems ^†

by Helen Durand

Mathematics 2018, 6(9), 169; https://doi.org/10.3390/math6090169 - 14 Sep 2018

Cited by 26 | Viewed by 4191

Abstract

Recent cyberattacks against industrial control systems highlight the criticality of preventing future attacks from disrupting plants economically or, more critically, from impacting plant safety. This work develops a nonlinear systems framework for understanding cyberattack-resilience of process and control designs and indicates through an [...] Read more.

Recent cyberattacks against industrial control systems highlight the criticality of preventing future attacks from disrupting plants economically or, more critically, from impacting plant safety. This work develops a nonlinear systems framework for understanding cyberattack-resilience of process and control designs and indicates through an analysis of three control designs how control laws can be inspected for this property. A chemical process example illustrates that control approaches intended for cyberattack prevention which seem intuitive are not cyberattack-resilient unless they meet the requirements of a nonlinear systems description of this property. Full article

(This article belongs to the Special Issue New Directions on Model Predictive Control)

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10 pages, 275 KiB

Open AccessArticle

Equivalences of Riemann Integral Based on p-Norm

by Ray-Ming Chen

Mathematics 2018, 6(9), 168; https://doi.org/10.3390/math6090168 - 13 Sep 2018

Cited by 1 | Viewed by 3934

Abstract

In the usual Riemann integral setting, the Riemann norm or a mesh is adopted for Riemann sums. In this article, we use the p-norm to define the p-integral and show the equivalences between the Riemann integral and the p-integral. The [...] Read more.

In the usual Riemann integral setting, the Riemann norm or a mesh is adopted for Riemann sums. In this article, we use the p-norm to define the p-integral and show the equivalences between the Riemann integral and the p-integral. The p-norm provides an alternative approach to define the Riemann integral. Based on this norm, we also derive some other equivalences of the Riemann integral and the p-integral. Full article

2 pages, 137 KiB

Open AccessEditorial

Progress in Mathematical Ecology

by Sergei Petrovskii

Mathematics 2018, 6(9), 167; https://doi.org/10.3390/math6090167 - 13 Sep 2018

Cited by 1 | Viewed by 3368

Abstract

n/a Full article

(This article belongs to the Special Issue Progress in Mathematical Ecology)

18 pages, 293 KiB

Open AccessArticle

A Heuristic Method for Certifying Isolated Zeros of Polynomial Systems

by Xiaojie Dou and Jin-San Cheng

Mathematics 2018, 6(9), 166; https://doi.org/10.3390/math6090166 - 11 Sep 2018

Viewed by 2885

Abstract

In this paper, by transforming the given over-determined system into a square system, we prove a necessary and sufficient condition to certify the simple real zeros of the over-determined system by certifying the simple real zeros of the square system. After certifying a [...] Read more.

In this paper, by transforming the given over-determined system into a square system, we prove a necessary and sufficient condition to certify the simple real zeros of the over-determined system by certifying the simple real zeros of the square system. After certifying a simple real zero of the related square system with the interval methods, we assert that the certified zero is a local minimum of sum of squares of the input polynomials. If the value of sum of squares of the input polynomials at the certified zero is equal to zero, it is a zero of the input system. As an application, we also consider the heuristic verification of isolated zeros of polynomial systems and their multiplicity structures. Full article

(This article belongs to the Special Issue Computer Algebra in Scientific Computing)

17 pages, 363 KiB

Open AccessArticle

Payoff Distribution in a Multi-Company Extraction Game with Uncertain Duration

by Ekaterina Gromova, Anastasiya Malakhova and Arsen Palestini

Mathematics 2018, 6(9), 165; https://doi.org/10.3390/math6090165 - 11 Sep 2018

Cited by 8 | Viewed by 3292

Abstract

A nonrenewable resource extraction game model is analyzed in a differential game theory framework with random duration. If the cumulative distribution function (c.d.f.) of the final time is discontinuous, the related subgames are differentiated based on the position of the initial instant with [...] Read more.

A nonrenewable resource extraction game model is analyzed in a differential game theory framework with random duration. If the cumulative distribution function (c.d.f.) of the final time is discontinuous, the related subgames are differentiated based on the position of the initial instant with respect to the jump. We investigate properties of optimal trajectories and of imputation distribution procedures if the game is played cooperatively. Full article

(This article belongs to the Special Issue Mathematical Game Theory)

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24 pages, 1014 KiB

Open AccessArticle

Introducing Weights Restrictions in Data Envelopment Analysis Models for Mutual Funds

by Antonella Basso and Stefania Funari

Mathematics 2018, 6(9), 164; https://doi.org/10.3390/math6090164 - 10 Sep 2018

Cited by 5 | Viewed by 5152

Abstract

Data envelopment analysis has been applied in a number of papers to measure the performance of mutual funds, besides a great many applications on the more diverse fields of performance evaluation. The data envelopment analysis models proposed in the mutual funds literature do [...] Read more.

Data envelopment analysis has been applied in a number of papers to measure the performance of mutual funds, besides a great many applications on the more diverse fields of performance evaluation. The data envelopment analysis models proposed in the mutual funds literature do not generally set restrictions on the weights assigned to the input and output variables. In this paper, we study the effects of the introduction of different weight restrictions on the results of the performance evaluation of mutual funds. In addition, we provide a unified matrix representation for three widely used approaches on weight restrictions: virtual weight restrictions with constraints on all decision-making units (DMUs) (on all funds); virtual weight restrictions with constraints only on the target unit; assurance regions. Using the unified matrix representation of the weights constraints, we formulate the data envelopment analysis (DEA ) efficiency model and express the efficient frontier in a unified way for the different weight restrictions considered. We investigate the effects of the different weight restrictions on the performance evaluation by means of an empirical application on a set of European mutual funds. Moreover, we study the behaviour of the fund performance scores as the restrictions on the weights become increasingly strict. Full article

(This article belongs to the Special Issue Financial Mathematics)

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15 pages, 247 KiB

Open AccessArticle

Higher Order Hamiltonian Systems with Generalized Legendre Transformation

by Dana Smetanová

Mathematics 2018, 6(9), 163; https://doi.org/10.3390/math6090163 - 10 Sep 2018

Cited by 3 | Viewed by 3000

Abstract

The aim of this paper is to report some recent results regarding second order Lagrangians corresponding to 2nd and 3rd order Euler–Lagrange forms. The associated 3rd order Hamiltonian systems are found. The generalized Legendre transformation and geometrical correspondence between solutions of the Hamilton [...] Read more.

The aim of this paper is to report some recent results regarding second order Lagrangians corresponding to 2nd and 3rd order Euler–Lagrange forms. The associated 3rd order Hamiltonian systems are found. The generalized Legendre transformation and geometrical correspondence between solutions of the Hamilton equations and the Euler–Lagrange equations are studied. The theory is illustrated on examples of Hamiltonian systems satisfying the following conditions: (a) the Hamiltonian system is strongly regular and the Legendre transformation exists; (b) the Hamiltonian system is strongly regular and the Legendre transformation does not exist; (c) the Legendre transformation exists and the Hamiltonian system is not regular but satisfies a weaker condition. Full article

(This article belongs to the Special Issue Differential Geometry of Special Mappings)

14 pages, 636 KiB

Open AccessArticle

Keeping up with the Neighbors: Social Interaction in a Production Economy

by Zhenhua Feng, Jaimie W. Lien and Jie Zheng

Mathematics 2018, 6(9), 162; https://doi.org/10.3390/math6090162 - 9 Sep 2018

Cited by 5 | Viewed by 4621

Abstract

It is well-documented that individuals care about how others around them are doing. This paper studies a production economy in which consumers provide labor supply to a representative firm to earn income for consumption, and their utility depends on their own leisure time, [...] Read more.

It is well-documented that individuals care about how others around them are doing. This paper studies a production economy in which consumers provide labor supply to a representative firm to earn income for consumption, and their utility depends on their own leisure time, their own consumption level, as well as their neighbors’ consumption levels. We characterize the unique equilibrium for such an economy, allowing for three different types of effects of the neighborhood size: linear effect, zero effect, and nonlinear effect. Four network structures (empty network, ring network, star network, and core-periphery network) with different production technologies are analyzed. Our work contributes to a better understanding of the general equilibrium effect of social preferences and network structures. Full article

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17 pages, 346 KiB

Open AccessArticle

Resolving Decompositions for Polynomial Modules

by Mario Albert and Werner M. Seiler

Mathematics 2018, 6(9), 161; https://doi.org/10.3390/math6090161 - 7 Sep 2018

Cited by 1 | Viewed by 2783

Abstract

We introduce the novel concept of a resolving decomposition of a polynomial module as a combinatorial structure that allows for the effective construction of free resolutions. It provides a unifying framework for recent results of the authors for different types of bases. Full article

(This article belongs to the Special Issue Computer Algebra in Scientific Computing)

7 pages, 234 KiB

Open AccessArticle

On Magnifying Elements in E-Preserving Partial Transformation Semigroups

by Thananya Kaewnoi, Montakarn Petapirak and Ronnason Chinram

Mathematics 2018, 6(9), 160; https://doi.org/10.3390/math6090160 - 6 Sep 2018

Cited by 6 | Viewed by 2975

Abstract

Let S be a semigroup. An element a of S is called a right [left] magnifying element if there exists a proper subset M of S satisfying

S = M a

[S = a M] .

Let E be an equivalence [...] Read more.

Let S be a semigroup. An element a of S is called a right [left] magnifying element if there exists a proper subset M of S satisfying

S = M a

[S = a M] .

Let E be an equivalence relation on a nonempty set X. In this paper, we consider the semigroup

P (X, E)

consisting of all E-preserving partial transformations, which is a subsemigroup of the partial transformation semigroup

P (X)

. The main propose of this paper is to show the necessary and sufficient conditions for elements in

P (X, E)

to be right or left magnifying. Full article

26 pages, 321 KiB

Open AccessArticle

Fractional Queues with Catastrophes and Their Transient Behaviour

by Giacomo Ascione, Nikolai Leonenko and Enrica Pirozzi

Mathematics 2018, 6(9), 159; https://doi.org/10.3390/math6090159 - 6 Sep 2018

Cited by 19 | Viewed by 3956

Abstract

Starting from the definition of fractional M/M/1 queue given in the reference by Cahoy et al. in 2015 and M/M/1 queue with catastrophes given in the reference by Di Crescenzo et al. in 2003, we define and study a fractional M/M/1 queue with [...] Read more.

Starting from the definition of fractional M/M/1 queue given in the reference by Cahoy et al. in 2015 and M/M/1 queue with catastrophes given in the reference by Di Crescenzo et al. in 2003, we define and study a fractional M/M/1 queue with catastrophes. In particular, we focus our attention on the transient behaviour, in which the time-change plays a key role. We first specify the conditions for the global uniqueness of solutions of the corresponding linear fractional differential problem. Then, we provide an alternative expression for the transient distribution of the fractional M/M/1 model, the state probabilities for the fractional queue with catastrophes, the distributions of the busy period for fractional queues without and with catastrophes and, finally, the distribution of the time of the first occurrence of a catastrophe. Full article

(This article belongs to the Special Issue Stochastic Processes with Applications)

14 pages, 2624 KiB

Open AccessArticle

A Game-Theoretic Loss Allocation Approach in Power Distribution Systems with High Penetration of Distributed Generations

by Farzaneh Pourahmadi and Payman Dehghanian

Mathematics 2018, 6(9), 158; https://doi.org/10.3390/math6090158 - 6 Sep 2018

Cited by 8 | Viewed by 3833

Abstract

Allocation of the power losses to distributed generators and consumers has been a challenging concern for decades in restructured power systems. This paper proposes a promising approach for loss allocation in power distribution systems based on a cooperative concept of game-theory, named Shapley [...] Read more.

Allocation of the power losses to distributed generators and consumers has been a challenging concern for decades in restructured power systems. This paper proposes a promising approach for loss allocation in power distribution systems based on a cooperative concept of game-theory, named Shapley Value allocation. The proposed solution is a generic approach, applicable to both radial and meshed distribution systems as well as those with high penetration of renewables and DG units. With several different methods for distribution system loss allocation, the suggested method has been shown to be a straight-forward and efficient criterion for performance comparisons. The suggested loss allocation approach is numerically investigated, the results of which are presented for two distribution systems and its performance is compared with those obtained by other methodologies. Full article

(This article belongs to the Special Issue Mathematical Game Theory)

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2 pages, 141 KiB

Open AccessEditorial

Operators of Fractional Calculus and Their Applications

by Hari Mohan Srivastava

Mathematics 2018, 6(9), 157; https://doi.org/10.3390/math6090157 - 5 Sep 2018

Cited by 2 | Viewed by 3034

Abstract

n/a Full article

(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)

9 pages, 266 KiB

Open AccessArticle

Dynamic Multicriteria Games with Finite Horizon

by Anna Rettieva

Mathematics 2018, 6(9), 156; https://doi.org/10.3390/math6090156 - 5 Sep 2018

Cited by 16 | Viewed by 2931

Abstract

The approaches to construct optimal behavior in dynamic multicriteria games with finite horizon are presented. To obtain a multicriteria Nash equilibrium, the bargaining construction (Nash product) is adopted. To construct a multicriteria cooperative equilibrium, a Nash bargaining scheme is applied. Dynamic multicriteria bioresource [...] Read more.

The approaches to construct optimal behavior in dynamic multicriteria games with finite horizon are presented. To obtain a multicriteria Nash equilibrium, the bargaining construction (Nash product) is adopted. To construct a multicriteria cooperative equilibrium, a Nash bargaining scheme is applied. Dynamic multicriteria bioresource management problem with finite harvesting times is considered. The players’ strategies and the payoffs are obtained under cooperative and noncooperative behavior. Full article

(This article belongs to the Special Issue Mathematical Game Theory)

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14 pages, 447 KiB

Open AccessArticle

Time in Quantum Mechanics and the Local Non-Conservation of the Probability Current

by Giovanni Modanese

Mathematics 2018, 6(9), 155; https://doi.org/10.3390/math6090155 - 4 Sep 2018

Cited by 14 | Viewed by 4241

Abstract

In relativistic quantum field theory with local interactions, charge is locally conserved. This implies local conservation of probability for the Dirac and Klein–Gordon wavefunctions, as special cases; and in turn for non-relativistic quantum field theory and for the Schrödinger and Ginzburg–Landau equations, regarded [...] Read more.

In relativistic quantum field theory with local interactions, charge is locally conserved. This implies local conservation of probability for the Dirac and Klein–Gordon wavefunctions, as special cases; and in turn for non-relativistic quantum field theory and for the Schrödinger and Ginzburg–Landau equations, regarded as low energy limits. Quantum mechanics, however, is wider than quantum field theory, as an effective model of reality. For instance, fractional quantum mechanics and Schrödinger equations with non-local terms have been successfully employed in several applications. The non-locality of these formalisms is strictly related to the problem of time in quantum mechanics. We explicitly compute, for continuum wave packets, the terms of the fractional Schrödinger equation and the non-local Schrödinger equation by Lenzi et al. that break local current conservation. Additionally, we discuss the physical significance of these terms. The results are especially relevant for the electromagnetic coupling of these wavefunctions. A connection with the non-local Gorkov equation for superconductors and their proximity effect is also outlined. Full article

(This article belongs to the Special Issue Time and Time Dependence in Quantum Mechanics)

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18 pages, 9231 KiB

Open AccessArticle

On the Role of Short-Term Animal Movements on the Persistence of Brucellosis

by Paride O. Lolika and Steady Mushayabasa

Mathematics 2018, 6(9), 154; https://doi.org/10.3390/math6090154 - 4 Sep 2018

Cited by 6 | Viewed by 3510

Abstract

Short-term animal movements play an integral role in the transmission and control of zoonotic infections such as brucellosis, in communal farming zones where animal movements are highly uncontrolled. Such movements need to be incorporated in models that aim at informing animal managers effective [...] Read more.

Short-term animal movements play an integral role in the transmission and control of zoonotic infections such as brucellosis, in communal farming zones where animal movements are highly uncontrolled. Such movements need to be incorporated in models that aim at informing animal managers effective ways to control the spread of zoonotic diseases. We developed, analyzed and simulated a two-patch mathematical model for brucellosis transmission that incorporates short-term animal mobility. We computed the basic reproduction number and demonstrated that it is a sharp threshold for disease dynamics. In particular, we demonstrated that, when the basic reproduction number is less than unity, then the disease dies out. However, if the basic reproduction number is greater than unity, the disease persists. Meanwhile, we applied optimal control theory to the proposed model with the aim of exploring the cost-effectiveness of different culling strategies. The results demonstrate that animal mobility plays an important role in shaping optimal control strategy. Full article

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14 pages, 341 KiB

Open AccessArticle

Computing Eccentricity Based Topological Indices of Octagonal Grid O n m

by Xiujun Zhang, Muhammad Kamran Siddiqui, Muhammad Naeem and Abdul Qudair Baig

Mathematics 2018, 6(9), 153; https://doi.org/10.3390/math6090153 - 31 Aug 2018

Cited by 18 | Viewed by 3041

Abstract

Graph theory is successfully applied in developing a relationship between chemical structure and biological activity. The relationship of two graph invariants, the eccentric connectivity index and the eccentric Zagreb index are investigated with regard to anti-inflammatory activity, for a dataset consisting of 76 [...] Read more.

Graph theory is successfully applied in developing a relationship between chemical structure and biological activity. The relationship of two graph invariants, the eccentric connectivity index and the eccentric Zagreb index are investigated with regard to anti-inflammatory activity, for a dataset consisting of 76 pyrazole carboxylic acid hydrazide analogs. The eccentricity

ε_{v}

of vertex v in a graph G is the distance between v and the vertex furthermost from v in a graph G. The distance between two vertices is the length of a shortest path between those vertices in a graph G. In this paper, we consider the Octagonal Grid

O_{n}^{m}

. We compute Connective Eccentric index

C^{ξ} (G) = \sum_{v \in V (G)} d_{v} / ε_{v}

, Eccentric Connective Index

ξ (G) = \sum_{v \in V (G)} d_{v} ε_{v}

and eccentric Zagreb index of Octagonal Grid

O_{n}^{m}

, where

d_{v}

represents the degree of the vertex v in G. Full article

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9 pages, 462 KiB

Open AccessArticle

Accounting Games: Using Matrix Algebra in Creating the Accounting Models

by Anna Vysotskaya

Mathematics 2018, 6(9), 152; https://doi.org/10.3390/math6090152 - 31 Aug 2018

Viewed by 5422

Abstract

The aim of this paper is to show the mathematical basis for a precise treatment of double-entry bookkeeping, which was first developed in the nineteenth century by Sir William Rowan Hamilton. This is done by using basic notions of matrix algebra founded on [...] Read more.

The aim of this paper is to show the mathematical basis for a precise treatment of double-entry bookkeeping, which was first developed in the nineteenth century by Sir William Rowan Hamilton. This is done by using basic notions of matrix algebra founded on the idea of ordered pairs. We also reveal how complex numbers and rationals (fractions) developed in mainstream accounting science and became a leading platform for the ongoing processes within Industry 4.0. The paper concludes with examples of how accounting operations can be represented by matrix equations with the result of generating a final report. The author presents a mathematical model of accounting which is independent of specific existential forms, but which is capable of undertaking the form of any of them and thus which has the potential of being understood and accepted by specialists globally. Full article

(This article belongs to the Special Issue Mathematical Game Theory)

9 pages, 239 KiB

Open AccessArticle

Inverse Stackelberg Solutions for Games with Many Followers

by Yurii Averboukh

Mathematics 2018, 6(9), 151; https://doi.org/10.3390/math6090151 - 30 Aug 2018

Cited by 1 | Viewed by 2702

Abstract

The paper is devoted to inverse Stackelberg games with many players. We consider both static and differential games. The main assumption of the paper is the compactness of the strategy sets. We obtain the characterization of inverse Stackelberg solutions and under additional concavity [...] Read more.

The paper is devoted to inverse Stackelberg games with many players. We consider both static and differential games. The main assumption of the paper is the compactness of the strategy sets. We obtain the characterization of inverse Stackelberg solutions and under additional concavity conditions, establish the existence theorem. Full article

(This article belongs to the Special Issue Mathematical Game Theory)

14 pages, 281 KiB

Open AccessArticle

Computing The Irregularity Strength of Planar Graphs

by Hong Yang, Muhammad Kamran Siddiqui, Muhammad Ibrahim, Sarfraz Ahmad and Ali Ahmad

Mathematics 2018, 6(9), 150; https://doi.org/10.3390/math6090150 - 30 Aug 2018

Cited by 10 | Viewed by 4034

Abstract

The field of graph theory plays a vital role in various fields. One of the important areas in graph theory is graph labeling used in many applications such as coding theory, X-ray crystallography, radar, astronomy, circuit design, communication network addressing, and data base [...] Read more.

The field of graph theory plays a vital role in various fields. One of the important areas in graph theory is graph labeling used in many applications such as coding theory, X-ray crystallography, radar, astronomy, circuit design, communication network addressing, and data base management. In this paper, we discuss the totally irregular total k labeling of three planar graphs. If such labeling exists for minimum value of a positive integer k, then this labeling is called totally irregular total k labeling and k is known as the total irregularity strength of a graph G. More preciously, we determine the exact value of the total irregularity strength of three planar graphs. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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16 pages, 293 KiB

Open AccessArticle

Positive Implicative Ideals of BCK-Algebras Based on Intuitionistic Falling Shadows

by Young Bae Jun, Eun Hwan Roh and Mehmet Ali Öztürk

Mathematics 2018, 6(9), 149; https://doi.org/10.3390/math6090149 - 29 Aug 2018

Viewed by 2641

Abstract

The concepts of a positive implicative (

\in,

∈)-intuitionistic fuzzy ideal and a positive implicative falling intuitionistic fuzzy ideal are introduced, and several properties are investigated. Characterizations of a positive implicative (

\in,

∈)-intuitionistic fuzzy ideal are obtained, and relations between [...] Read more.

The concepts of a positive implicative (

\in,

∈)-intuitionistic fuzzy ideal and a positive implicative falling intuitionistic fuzzy ideal are introduced, and several properties are investigated. Characterizations of a positive implicative (

\in,

∈)-intuitionistic fuzzy ideal are obtained, and relations between a positive implicative (

\in,

∈)-intuitionistic fuzzy ideal and an intuitionistic fuzzy ideal are discussed. Conditions for an intuitionistic fuzzy ideal to be a positive implicative (

\in,

∈)-intuitionistic fuzzy ideal are provided, and relations between a positive implicative (

\in,

∈)-intuitionistic fuzzy ideal, a falling intuitionistic fuzzy ideal and a positive implicative falling intuitionistic fuzzy ideal are considered. Conditions for a falling intuitionistic fuzzy ideal to be positive implicative are given. Full article

12 pages, 1437 KiB

Open AccessArticle

Kinematics in the Information Age

by Brendon Smeresky, Alexa Rizzo and Timothy Sands

Mathematics 2018, 6(9), 148; https://doi.org/10.3390/math6090148 - 27 Aug 2018

Cited by 12 | Viewed by 5660

Abstract

Modern kinematics derives directly from developments in the 1700s, and in their current instantiation, have been adopted as standard realizations…or templates that seem unquestionable. For example, so-called aerospace sequences of rotations are ubiquitously accepted as the norm for aerospace applications, owing from a [...] Read more.

Modern kinematics derives directly from developments in the 1700s, and in their current instantiation, have been adopted as standard realizations…or templates that seem unquestionable. For example, so-called aerospace sequences of rotations are ubiquitously accepted as the norm for aerospace applications, owing from a recent heritage in the space age of the late twentieth century. With the waning of the space-age as a driver for technology development, the information age has risen with the advent of digital computers, and this begs for re-evaluation of assumptions made in the former era. The new context of the digital computer defines the use of the term “information age” in the manuscript title and further highlights the novelty and originality of the research. The effects of selecting different Direction Cosine Matrices (DCM)-to-Euler Angle rotations on accuracy, step size, and computational time in modern digital computers will be simulated and analyzed. The experimental setup will include all twelve DCM rotations and also includes critical analysis of necessary computational step size. The results show that the rotations are classified into symmetric and non-symmetric rotations and that no one DCM rotation outperforms the others in all metrics used, yielding the potential for trade space analysis to select the best DCM for a specific instance. Novel illustrations include the fact that one of the ubiquitous sequences (the “313 sequence”) has degraded relative accuracy measured by mean and standard deviations of errors, but may be calculated faster than the other ubiquitous sequence (the “321 sequence”), while a lesser known “231 sequence” has comparable accuracy and calculation-time. Evaluation of the 231 sequence also illustrates the originality of the research. These novelties are applied to spacecraft attitude control in this manuscript, but equally apply to robotics, aircraft, and surface and subsurface vehicles. Full article

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

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10 pages, 476 KiB

Open AccessArticle

Stability Analysis of an Age-Structured SIR Epidemic Model with a Reduction Method to ODEs

by Toshikazu Kuniya

Mathematics 2018, 6(9), 147; https://doi.org/10.3390/math6090147 - 23 Aug 2018

Cited by 9 | Viewed by 6102 | Correction

Abstract

In this paper, we are concerned with the asymptotic stability of the nontrivial endemic equilibrium of an age-structured susceptible-infective-recovered (SIR) epidemic model. For a special form of the disease transmission function, we perform the reduction of the model into a four-dimensional system of [...] Read more.

In this paper, we are concerned with the asymptotic stability of the nontrivial endemic equilibrium of an age-structured susceptible-infective-recovered (SIR) epidemic model. For a special form of the disease transmission function, we perform the reduction of the model into a four-dimensional system of ordinary differential equations (ODEs). We show that the unique endemic equilibrium of the reduced system exists if the basic reproduction number for the original system is greater than unity. Furthermore, we perform the stability analysis of the endemic equilibrium and obtain a fourth-order characteristic equation. By using the Routh–Hurwitz criterion, we numerically show that the endemic equilibrium is asymptotically stable in some epidemiologically relevant parameter settings. Full article

(This article belongs to the Special Issue Applied Analysis of Ordinary Differential Equations 2018)

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24 pages, 389 KiB

Open AccessFeature PaperArticle

L-Fuzzy Sets and Isomorphic Lattices: Are All the “New” Results Really New? †

by Erich Peter Klement and Radko Mesiar

Mathematics 2018, 6(9), 146; https://doi.org/10.3390/math6090146 - 23 Aug 2018

Cited by 16 | Viewed by 5878

Abstract

We review several generalizations of the concept of fuzzy sets with two- or three-dimensional lattices of truth values and study their relationship. It turns out that, in the two-dimensional case, several of the lattices of truth values considered here are pairwise isomorphic, and [...] Read more.

We review several generalizations of the concept of fuzzy sets with two- or three-dimensional lattices of truth values and study their relationship. It turns out that, in the two-dimensional case, several of the lattices of truth values considered here are pairwise isomorphic, and so are the corresponding families of fuzzy sets. Therefore, each result for one of these types of fuzzy sets can be directly rewritten for each (isomorphic) type of fuzzy set. Finally we also discuss some questionable notations, in particular, those of “intuitionistic” and “Pythagorean” fuzzy sets. Full article

(This article belongs to the Special Issue Fuzzy Mathematics)

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4 pages, 164 KiB

Open AccessEditorial

Fractional Calculus: Theory and Applications

by Francesco Mainardi

Mathematics 2018, 6(9), 145; https://doi.org/10.3390/math6090145 - 21 Aug 2018

Cited by 36 | Viewed by 5462

Abstract

Fractional calculus is allowing integrals and derivatives of any positive order (the term fractional is kept only for historical reasons).[...] Full article

(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)

12 pages, 5223 KiB

Open AccessArticle

Stability Analysis of Cohen–Grossberg Neural Networks with Random Impulses

by Ravi Agarwal, Snezhana Hristova, Donal O’Regan and Peter Kopanov

Mathematics 2018, 6(9), 144; https://doi.org/10.3390/math6090144 - 21 Aug 2018

Cited by 5 | Viewed by 4336

Abstract

The Cohen and Grossberg neural networks model is studied in the case when the neurons are subject to a certain impulsive state displacement at random exponentially-distributed moments. These types of impulses significantly change the behavior of the solutions from a deterministic one to [...] Read more.

The Cohen and Grossberg neural networks model is studied in the case when the neurons are subject to a certain impulsive state displacement at random exponentially-distributed moments. These types of impulses significantly change the behavior of the solutions from a deterministic one to a stochastic process. We examine the stability of the equilibrium of the model. Some sufficient conditions for the mean-square exponential stability and mean exponential stability of the equilibrium of general neural networks are obtained in the case of the time-varying potential (or voltage) of the cells, with time-dependent amplification functions and behaved functions, as well as time-varying strengths of connectivity between cells and variable external bias or input from outside the network to the units. These sufficient conditions are explicitly expressed in terms of the parameters of the system, and hence, they are easily verifiable. The theory relies on a modification of the direct Lyapunov method. We illustrate our theory on a particular nonlinear neural network. Full article

(This article belongs to the Special Issue Stochastic Processes with Applications)

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24 pages, 5791 KiB

Open AccessArticle

A Within-Host Stochastic Model for Nematode Infection

by Antonio Gómez-Corral and Martín López-García

Mathematics 2018, 6(9), 143; https://doi.org/10.3390/math6090143 - 21 Aug 2018

Viewed by 3491

Abstract

We propose a stochastic model for the development of gastrointestinal nematode infection in growing lambs under the assumption that nonhomogeneous Poisson processes govern the acquisition of parasites, the parasite-induced host mortality, the natural (no parasite-induced) host mortality and the death of parasites within [...] Read more.

We propose a stochastic model for the development of gastrointestinal nematode infection in growing lambs under the assumption that nonhomogeneous Poisson processes govern the acquisition of parasites, the parasite-induced host mortality, the natural (no parasite-induced) host mortality and the death of parasites within the host. By means of considering a number of age-dependent birth and death processes with killing, we analyse the impact of grazing strategies that are defined in terms of an intervention instant

t_{0}

, which might imply a move of the host to safe pasture and/or anthelmintic treatment. The efficacy and cost of each grazing strategy are defined in terms of the transient probabilities of the underlying stochastic processes, which are computed by means of Strang–Marchuk splitting techniques. Our model, calibrated with empirical data from Uriarte et al and Nasreen et al., regarding the seasonal presence of nematodes on pasture in temperate zones and anthelmintic efficacy, supports the use of dose-and-move strategies in temperate zones during summer and provides stochastic criteria for selecting the exact optimum time instant

t_{0}

when these strategies should be applied. Full article

(This article belongs to the Special Issue Stochastic Processes with Applications)

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Mathematics, Volume 6, Issue 9 (September 2018) – 31 articles

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