Mathematics doi: 10.3390/math6100174

Authors: Allen D. Parks

It is shown that the set of all networks of fixed order n form a semigroup that is isomorphic to the semigroup BX of binary relations on a set X of cardinality n. Consequently, BX provides for Green’s L,R,H, and D equivalence classifications of all networks of fixed order n. These classifications reveal that a fixed-order network which evolves within a Green’s equivalence class maintains certain structural invariants during its evolution. The “Green’s symmetry problem” is introduced and is defined as the determination of all symmetries (i.e., transformations) that produce an evolution between an initial and final network within an L or an R class such that each symmetry preserves the required structural invariants. Such symmetries are shown to be solutions to special Boolean equations specific to each class. The satisfiability and computational complexity of the “Green’s symmetry problem” are discussed and it is demonstrated that such symmetries encode information about which node neighborhoods in the initial network can be joined to form node neighborhoods in the final network such that the structural invariants required by the evolution are preserved, i.e., the internal dynamics of the evolution. The notion of “propensity” is also introduced. It is a measure of the tendency of node neighborhoods to join to form new neighborhoods during a network evolution and is used to define “energy”, which quantifies the complexity of the internal dynamics of a network evolution.

]]>Mathematics doi: 10.3390/math6100173

Authors: Zhe Wu Fahad Albalawi Junfeng Zhang Zhihao Zhang Helen Durand Panagiotis D. Christofides

Since industrial control systems are usually integrated with numerous physical devices, the security of control systems plays an important role in safe operation of industrial chemical processes. However, due to the use of a large number of control actuators and measurement sensors and the increasing use of wireless communication, control systems are becoming increasingly vulnerable to cyber-attacks, which may spread rapidly and may cause severe industrial incidents. To mitigate the impact of cyber-attacks in chemical processes, this work integrates a neural network (NN)-based detection method and a Lyapunov-based model predictive controller for a class of nonlinear systems. A chemical process example is used to illustrate the application of the proposed NN-based detection and LMPC methods to handle cyber-attacks.

]]>Mathematics doi: 10.3390/math6090172

Authors: Hui-Chin Tang Shen-Tai Yang

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.

]]>Mathematics doi: 10.3390/math6090171

Authors: Seifedine Kadry Gennady Alferov Gennady Ivanov Artem Sharlay

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&rsquo;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.

]]>Mathematics doi: 10.3390/math6090170

Authors: Hsien-Chung Wu

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.

]]>Mathematics doi: 10.3390/math6090169

Authors: Helen Durand

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.

]]>Mathematics doi: 10.3390/math6090168

Authors: Ray-Ming Chen

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.

]]>Mathematics doi: 10.3390/math6090167

Authors: Sergei Petrovskii

n/a

]]>Mathematics doi: 10.3390/math6090166

Authors: Xiaojie Dou Jin-San Cheng

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.

]]>Mathematics doi: 10.3390/math6090165

Authors: Ekaterina Gromova Anastasiya Malakhova Arsen Palestini

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.

]]>Mathematics doi: 10.3390/math6090164

Authors: Antonella Basso Stefania Funari

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.

]]>Mathematics doi: 10.3390/math6090163

Authors: Dana Smetanová

The aim of this paper is to report some recent results regarding second order Lagrangians corresponding to 2nd and 3rd order Euler&ndash;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&ndash;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.

]]>Mathematics doi: 10.3390/math6090162

Authors: Zhenhua Feng Jaimie W. Lien Jie Zheng

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&rsquo; 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.

]]>Mathematics doi: 10.3390/math6090161

Authors: Mario Albert Werner M. Seiler

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.

]]>Mathematics doi: 10.3390/math6090160

Authors: Thananya Kaewnoi Montakarn Petapirak Ronnason Chinram

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.

]]>Mathematics doi: 10.3390/math6090159

Authors: Giacomo Ascione Nikolai Leonenko Enrica Pirozzi

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.

]]>Mathematics doi: 10.3390/math6090158

Authors: Farzaneh Pourahmadi Payman Dehghanian

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.

]]>Mathematics doi: 10.3390/math6090157

Authors: Hari Mohan Srivastava

n/a

]]>Mathematics doi: 10.3390/math6090156

Authors: Anna Rettieva

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&rsquo; strategies and the payoffs are obtained under cooperative and noncooperative behavior.

]]>Mathematics doi: 10.3390/math6090155

Authors: Giovanni Modanese

In relativistic quantum field theory with local interactions, charge is locally conserved. This implies local conservation of probability for the Dirac and Klein&ndash;Gordon wavefunctions, as special cases; and in turn for non-relativistic quantum field theory and for the Schr&ouml;dinger and Ginzburg&ndash;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 Schr&ouml;dinger 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 Schr&ouml;dinger equation and the non-local Schr&ouml;dinger 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.

]]>Mathematics doi: 10.3390/math6090154

Authors: Paride O. Lolika Steady Mushayabasa

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.

]]>Mathematics doi: 10.3390/math6090153

Authors: Xiujun Zhang Muhammad Kamran Siddiqui Muhammad Naeem Abdul Qudair Baig

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 &epsilon; 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 &xi; ( G ) = &sum; v &isin; V ( G ) d v / &epsilon; v , Eccentric Connective Index &xi; ( G ) = &sum; v &isin; V ( G ) d v &epsilon; v and eccentric Zagreb index of Octagonal Grid O n m , where d v represents the degree of the vertex v in G.

]]>Mathematics doi: 10.3390/math6090152

Authors: Anna Vysotskaya

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.

]]>Mathematics doi: 10.3390/math6090151

Authors: Yurii Averboukh

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.

]]>Mathematics doi: 10.3390/math6090150

Authors: Hong Yang Muhammad Kamran Siddiqui Muhammad Ibrahim Sarfraz Ahmad Ali Ahmad

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.

]]>Mathematics doi: 10.3390/math6090149

Authors: Young Bae Jun Eun Hwan Roh Mehmet Ali Öztürk

The concepts of a positive implicative ( &isin; , &isin;)-intuitionistic fuzzy ideal and a positive implicative falling intuitionistic fuzzy ideal are introduced, and several properties are investigated. Characterizations of a positive implicative ( &isin; , &isin;)-intuitionistic fuzzy ideal are obtained, and relations between a positive implicative ( &isin; , &isin;)-intuitionistic fuzzy ideal and an intuitionistic fuzzy ideal are discussed. Conditions for an intuitionistic fuzzy ideal to be a positive implicative ( &isin; , &isin;)-intuitionistic fuzzy ideal are provided, and relations between a positive implicative ( &isin; , &isin;)-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.

]]>Mathematics doi: 10.3390/math6090148

Authors: Brendon Smeresky Alexa Rizzo Timothy Sands

Modern kinematics derives directly from developments in the 1700s, and in their current instantiation, have been adopted as standard realizations&hellip;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 &ldquo;information age&rdquo; 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 &ldquo;313 sequence&rdquo;) has degraded relative accuracy measured by mean and standard deviations of errors, but may be calculated faster than the other ubiquitous sequence (the &ldquo;321 sequence&rdquo;), while a lesser known &ldquo;231 sequence&rdquo; 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.

]]>Mathematics doi: 10.3390/math6090147

Authors: Toshikazu Kuniya

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&ndash;Hurwitz criterion, we numerically show that the endemic equilibrium is asymptotically stable in some epidemiologically relevant parameter settings.

]]>Mathematics doi: 10.3390/math6090146

Authors: Erich Peter Klement Radko Mesiar

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 &ldquo;intuitionistic&rdquo; and &ldquo;Pythagorean&rdquo; fuzzy sets.

]]>Mathematics doi: 10.3390/math6090145

Authors: Francesco Mainardi

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

]]>Mathematics doi: 10.3390/math6090144

Authors: Ravi Agarwal Snezhana Hristova Donal O’Regan Peter Kopanov

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.

]]>Mathematics doi: 10.3390/math6090143

Authors: Antonio Gómez-Corral Martín López-García

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&ndash;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.

]]>Mathematics doi: 10.3390/math6090142

Authors: Xiujun Zhang Muhammad Ibrahim Syed Ahtsham ul Haq Bokhary Muhammad Kamran Siddiqui

In graph theory, a graph is given names&mdash;generally a whole number&mdash;to edges, vertices, or both in a chart. Formally, given a graph G = ( V , E ) , a vertex naming is a capacity from V to an arrangement of marks. A diagram with such a capacity characterized defined is known as a vertex-marked graph. Similarly, an edge naming is a mapping of an element of E to an arrangement of marks. In this case, the diagram is called an edge-marked graph. We consider an edge irregular reflexive k-labeling for the disjoint association of wheel-related diagrams and deduce the correct estimation of the reflexive edge strength for the disjoint association of m copies of some wheel-related graphs, specifically gear graphs and prism graphs.

]]>Mathematics doi: 10.3390/math6080141

Authors: Manzoor Ahmed Zahid Abdul Qudair Baig Muhammad Naeem Muhammad Razwan Azhar

In this article, we study the chemical graph of a cyclic octahedron structure of dimension n and compute the eccentric connectivity polynomial, the eccentric connectivity index, the total eccentricity, the average eccentricity, the first Zagreb index, the second Zagreb index, the third Zagreb index, the atom bond connectivity index and the geometric arithmetic index of the cyclic octahedron structure. Furthermore, we give the analytically closed formulas of these indices which are helpful for studying the underlying topologies.

]]>Mathematics doi: 10.3390/math6080140

Authors: Edoardo Ballico

Let X &sub; P r be an integral and non-degenerate variety. We study when a finite set S &sub; X evinces the X-rank of the general point of the linear span of S. We give a criterion when X is the order d Veronese embedding X n , d of P n and | S | &le; ( n + &lfloor; d / 2 &rfloor; n ) . For the tensor rank, we describe the cases with | S | &le; 3 . For X n , d , we raise some questions of the maximum rank for d ≫ 0 (for a fixed n) and for n ≫ 0 (for a fixed d).

]]>Mathematics doi: 10.3390/math6080139

Authors: Juan Pablo Gomez Derya Akleman Ergun Akleman Ioannis Pavlidis

In this paper, we demonstrate that interventions and stressors do not necessarily cause the same distractions in all people; therefore, it is impossible to evaluate the impacts of interventions and stressors on traffic accidents. We analyzed publicly available multimodal data that was collected through one of the largest controlled experiments on distracted driving. A crossover design was used to examine the effects of actual and perceived interventions and stressors in driving behaviors and parallel designs on reactivity to a startling event. To analyze this data and make recommendations, we developed and compared a wide variety of mixed effects statistical models and machine learning methods to evaluate the effects of interventions and stressors on driving behaviors.

]]>Mathematics doi: 10.3390/math6080138

Authors: Young Bae Jun Seok-Zun Song Kyoung Ja Lee

Intuitionistic falling shadow is introduced, and applied to B C K / B C I -algebras. Falling intuitionistic subalgebra and falling intuitionistic ideal of B C K / B C I -algebras are introduced, and related properties are investigated. Relations between falling intuitionistic subalgebra and falling intuitionistic ideal are discussed. A characterization of falling intuitionistic ideal is established.

]]>Mathematics doi: 10.3390/math6080137

Authors: Shahid Imran Muhammad Kamran Siddiqui Muhammad Imran Muhammad Faisal Nadeem

A topological index is a number related to the atomic index that allows quantitative structure&ndash;action/property/toxicity connections. All the more vital topological indices correspond to certain physico-concoction properties like breaking point, solidness, strain vitality, and so forth, of synthetic mixes. The idea of the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials was set up in the substance diagram hypothesis in light of vertex degrees. These indices are valuable in the investigation of calming exercises of certain compound systems. In this paper, we computed the first and second Zagreb index, the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials of the line graph of wheel and ladder graphs by utilizing the idea of subdivision.

]]>Mathematics doi: 10.3390/math6080136

Authors: Muhammad Akram Sumera Naz

Pythagorean fuzzy sets (PFSs), an extension of intuitionistic fuzzy sets (IFSs), inherit the duality property of IFSs and have a more powerful ability than IFSs to model the obscurity in practical decision-making problems. In this research study, we compute the energy and Laplacian energy of Pythagorean fuzzy graphs (PFGs) and Pythagorean fuzzy digraphs (PFDGs). Moreover, we derive the lower and upper bounds for the energy and Laplacian energy of PFGs. Finally, we present numerical examples, including the design of a satellite communication system and the evaluation of the schemes of reservoir operation to illustrate the applications of our proposed concepts in decision making.

]]>Mathematics doi: 10.3390/math6080135

Authors: Alexander Van-Brunt Matt Visser

The Baker&ndash;Campbell&ndash;Hausdorff (BCH) expansion is a general purpose tool of use in many branches of mathematics and theoretical physics. Only in some special cases can the expansion be evaluated in closed form. In an earlier article we demonstrated that whenever [X,Y]=uX+vY+cI, BCH expansion reduces to the tractable closed-form expression Z(X,Y)=ln(eXeY)=X+Y+f(u,v)[X,Y], where f(u,v)=f(v,u) is explicitly given by the the function f(u,v)=(u&minus;v)eu+v&minus;(ueu&minus;vev)uv(eu&minus;ev)=(u&minus;v)&minus;(ue&minus;v&minus;ve&minus;u)uv(e&minus;v&minus;e&minus;u). This result is much more general than those usually presented for either the Heisenberg commutator, [P,Q]=&minus;iℏI, or the creation-destruction commutator, [a,a&dagger;]=I. In the current article, we provide an explicit and pedagogical exposition and further generalize and extend this result, primarily by relaxing the input assumptions. Under suitable conditions, to be discussed more fully in the text, and taking LAB=[A,B] as usual, we obtain the explicit result ln(eXeY)=X+Y+Ie&minus;LX&minus;e+LYI&minus;e&minus;LXLX+I&minus;e+LYLY[X,Y]. We then indicate some potential applications.

]]>Mathematics doi: 10.3390/math6080134

Authors: Chollawat Pookpienlert Preeyanuch Honyam Jintana Sanwong

Let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed nonempty subset Y of X. For an equivalence relation &rho; on X, let &rho;^ be the restriction of &rho; on Y, R a cross-section of Y/&rho;^ and define T(X,Y,&rho;,R) to be the set of all total transformations &alpha; from X into Y such that &alpha; preserves both &rho; (if (a,b)&isin;&rho;, then (a&alpha;,b&alpha;)&isin;&rho;) and R (if r&isin;R, then r&alpha;&isin;R). T(X,Y,&rho;,R) is then a subsemigroup of T(X,Y). In this paper, we give descriptions of Green&rsquo;s relations on T(X,Y,&rho;,R), and these results extend the results on T(X,Y) and T(X,&rho;,R) when taking &rho; to be the identity relation and Y=X, respectively.

]]>Mathematics doi: 10.3390/math6080133

Authors: Irina Georgescu

In this paper, several portfolio choice models are studied: a purely possibilistic model in which the return of the risky is a fuzzy number, and four models in which the background risk appears in addition to the investment risk. In these four models, risk is a bidimensional vector whose components are random variables or fuzzy numbers. Approximate formulas of the optimal allocation are obtained for all models, expressed in terms of some probabilistic or possibilistic moments, depending on the indicators of the investor preferences (risk aversion, prudence).

]]>Mathematics doi: 10.3390/math6080132

Authors: Harwinder Singh Sidhu Prashanth Siddhamshetty Joseph S. Kwon

Hydraulic fracturing has played a crucial role in enhancing the extraction of oil and gas from deep underground sources. The two main objectives of hydraulic fracturing are to produce fractures with a desired fracture geometry and to achieve the target proppant concentration inside the fracture. Recently, some efforts have been made to accomplish these objectives by the model predictive control (MPC) theory based on the assumption that the rock mechanical properties such as the Young&rsquo;s modulus are known and spatially homogenous. However, this approach may not be optimal if there is an uncertainty in the rock mechanical properties. Furthermore, the computational requirements associated with the MPC approach to calculate the control moves at each sampling time can be significantly high when the underlying process dynamics is described by a nonlinear large-scale system. To address these issues, the current work proposes an approximate dynamic programming (ADP) based approach for the closed-loop control of hydraulic fracturing to achieve the target proppant concentration at the end of pumping. ADP is a model-based control technique which combines a high-fidelity simulation and function approximator to alleviate the &ldquo;curse-of-dimensionality&rdquo; associated with the traditional dynamic programming (DP) approach. A series of simulations results is provided to demonstrate the performance of the ADP-based controller in achieving the target proppant concentration at the end of pumping at a fraction of the computational cost required by MPC while handling the uncertainty in the Young&rsquo;s modulus of the rock formation.

]]>Mathematics doi: 10.3390/math6080131

Authors: Grigory Belyavsky Natalya Danilova Guennady Ougolnitsky

This paper considers resource allocation among producers (agents) in the case where the Principal knows nothing about their cost functions while the agents have Markovian awareness about his/her strategies. We use a dynamic setup of the stochastic inverse Stackelberg game as the model. We suggest an algorithm for solving this game based on Q-learning. The associated Bellman equations contain functions of one variable for the Principal and also for the agents. The new results are illustrated by numerical examples.

]]>Mathematics doi: 10.3390/math6080130

Authors: Dae Won Yoon Dong-Soo Kim Young Ho Kim Jae Won Lee

In this paper, we study submanifolds in a Euclidean space with a generalized 1-type Gauss map. The Gauss map, G, of a submanifold in the n-dimensional Euclidean space, En, is said to be of generalized 1-type if, for the Laplace operator, &Delta;, on the submanifold, it satisfies &Delta;G=fG+gC, where C is a constant vector and f and g are some functions. The notion of a generalized 1-type Gauss map is a generalization of both a 1-type Gauss map and a pointwise 1-type Gauss map. With the new definition, first of all, we classify conical surfaces with a generalized 1-type Gauss map in E3. Second, we show that the Gauss map of any cylindrical surface in E3 is of the generalized 1-type. Third, we prove that there are no tangent developable surfaces with generalized 1-type Gauss maps in E3, except planes. Finally, we show that cylindrical hypersurfaces in En+2 always have generalized 1-type Gauss maps.

]]>Mathematics doi: 10.3390/math6080129

Authors: Panumart Sawangtong Kamonchat Trachoo Wannika Sawangtong Benchawan Wiwattanapataphee

It is well known that the Black-Scholes model is used to establish the behavior of the option pricing in the financial market. In this paper, we propose the modified version of Black-Scholes model with two assets based on the Liouville-Caputo fractional derivative. The analytical solution of the proposed model is investigated by the Laplace transform homotopy perturbation method.

]]>Mathematics doi: 10.3390/math6080128

Authors: Maria Gamboa Maria Jesus Lopez-Herrero

This paper deals with an infective process of type SIS, taking place in a closed population of moderate size that is inspected periodically. Our aim is to study the number of inspections that find the epidemic process still in progress. As the underlying mathematical model involves a discrete time Markov chain (DTMC) with a single absorbing state, the number of inspections in an outbreak is a first-passage time into this absorbing state. Cumulative probabilities are numerically determined from a recursive algorithm and expected values came from explicit expressions.

]]>Mathematics doi: 10.3390/math6070127

Authors: Prakash Veeraraghavan Golnar Khomami Fernando Perez Fontan

The broadcast performance of the 802.11 wireless protocol depends on several factors. One of the important factor is the number of nodes simultaneously contending for the shared channel. The Medium Access Control (MAC) technique of 802.11 is called the Distributed Coordination Function (DCF). DCF is a Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) scheme with binary slotted exponential backoff. A collision is the result of two or more stations transmitting simultaneously. Given the simplicity of the DCF scheme, it was adapted for Dedicated Short Range Communication (DSRC) based vehicular communication. A broadcast mechanism is used to disseminate emergency and safety related messages in a vehicular network. Emergency and safety related messages have a strict end-to-end latency of 100 ms and a Packet Delivery Ratio (PDR) of 90% and above. The PDR can be evaluated through the packet loss probability. The packet loss probability PL is given by, PL = 1 &minus; (1&minus;Pe)(1&minus;PC), where Pe is the probability of channel error and PC is the probability of collision. Pe depends on several environmental and operating factors and thus cannot be improved. The only way to reduce PL is by reducing PC. Currently, expensive radio hardware are used to measure PL. Several adaptive algorithms are available to reduce PC. In this paper, we establish a closed relation between PC and the Stirling number of the second kind. Simulation results are presented and compared with the analytical model for accuracy. Our simulation results show an accuracy of 99.9% compared with the analytical model. Even on a smaller sample size, our simulation results show an accuracy of 95% and above. Based on our analytical model, vehicles can precisely estimate these real-time requirements with the least expensive hardware available. Also, once the distribution of PC and PL are known, one can precisely determine the distribution of Pe.

]]>Mathematics doi: 10.3390/math6070126

Authors: Muhammad Imran Muhammad Siddiqui Amna Abunamous Dana Adi Saida Rafique Abdul Baig

Graph theory has much great advances in the field of mathematical chemistry. Chemical graph theory has become very popular among researchers because of its wide applications in mathematical chemistry. The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful for predicting their bioactivity. A great variety of such indices are studied and used in theoretical chemistry, pharmaceutical researchers, in drugs and in different other fields. In this article, we study the chemical graph of an oxide network and compute the total eccentricity, average eccentricity, eccentricity based Zagreb indices, atom-bond connectivity (ABC) index and geometric arithmetic index of an oxide network. Furthermore, we give analytically closed formulas of these indices which are helpful in studying the underlying topologies.

]]>Mathematics doi: 10.3390/math6070125

Authors: Hafsa Masood Malik Muhammad Akram Florentin Smarandache

In this paper, we apply the notion of soft rough neutrosophic sets to graph theory. We develop certain new concepts, including soft rough neutrosophic graphs, soft rough neutrosophic influence graphs, soft rough neutrosophic influence cycles and soft rough neutrosophic influence trees. We illustrate these concepts with examples, and investigate some of their properties. We solve the decision-making problem by using our proposed algorithm.

]]>Mathematics doi: 10.3390/math6070124

Authors: Elena Barton Basad Al-Sarray Stéphane Chrétien Kavya Jagan

In this note, we present a component-wise algorithm combining several recent ideas from signal processing for simultaneous piecewise constants trend, seasonality, outliers, and noise decomposition of dynamical time series. Our approach is entirely based on convex optimisation, and our decomposition is guaranteed to be a global optimiser. We demonstrate the efficiency of the approach via simulations results and real data analysis.

]]>Mathematics doi: 10.3390/math6070123

Authors: Krassimir Atanassov

The definition of the most extended modal operator of first type over interval-valued intuitionistic fuzzy sets is given, and some of its basic properties are studied.

]]>Mathematics doi: 10.3390/math6070122

Authors: Shin Min Kang Ghulam Abbas Ghulam Farid Waqas Nazeer

In this paper, we obtain a version of the Fej&eacute;r&ndash;Hadamard inequality for harmonically convex functions via generalized fractional integral operator. In addition, we establish an integral identity and some Fej&eacute;r&ndash;Hadamard type integral inequalities for harmonically convex functions via a generalized fractional integral operator. Being generalizations, our results reproduce some known results.

]]>Mathematics doi: 10.3390/math6070121

Authors: Qaisar Khan Peide Liu Tahir Mahmood

Neutrosophic sets (NSs) are used to illustrate uncertain, inconsistent, and indeterminate information existing in real-world problems. Double-valued neutrosophic sets (DVNSs) are an alternate form of NSs, in which the indeterminacy has two distinct parts: indeterminacy leaning toward truth membership, and indeterminacy leaning toward falsity membership. The aim of this article is to propose novel Dice measures and generalized Dice measures for DVNSs, and to specify Dice measures and asymmetric measures (projection measures) as special cases of generalized Dice measures via specific parameter values. Finally, the proposed generalized Dice measures and generalized weighted Dice measures were applied to pattern recognition and medical diagnosis to show their effectiveness.

]]>Mathematics doi: 10.3390/math6070120

Authors: Farnoosh Hajati Ali Iranmanesh Abolfazl Tehranian

Let G be a finite group and &omega;(G) be the set of element orders of G. Let k&isin;&omega;(G) and mk be the number of elements of order k in G. Let nse(G)={mk|k&isin;&omega;(G)}. In this paper, we prove that if G is a finite group such that nse(G) = nse(H), where H=PSU(3,3) or PSL(3,3), then G&cong;H.

]]>Mathematics doi: 10.3390/math6070119

Authors: Aisling J. Daly Jan M. Baetens Bernard De Baets

Diversity is a concept central to ecology, and its measurement is essential for any study of ecosystem health. But summarizing this complex and multidimensional concept in a single measure is problematic. Dozens of mathematical indices have been proposed for this purpose, but these can provide contradictory results leading to misleading or incorrect conclusions about a community&rsquo;s diversity. In this review, we summarize the key conceptual issues underlying the measurement of ecological diversity, survey the indices most commonly used in ecology, and discuss their relative suitability. We advocate for indices that: (i) satisfy key mathematical axioms; (ii) can be expressed as so-called effective numbers; (iii) can be extended to account for disparity between types; (iv) can be parameterized to obtain diversity profiles; and (v) for which an estimator (preferably unbiased) can be found so that the index is useful for practical applications.

]]>Mathematics doi: 10.3390/math6070118

Authors: Ahmed Elaiw Taofeek Alade Saud Alsulami

In this paper, we study the stability analysis of two within-host virus dynamics models with antibody immune response. We assume that the virus infects n classes of target cells. The second model considers two types of infected cells: (i) latently infected cells; and (ii) actively infected cells that produce the virus particles. For each model, we derive a biological threshold number R0. Using the method of Lyapunov function, we establish the global stability of the steady states of the models. The theoretical results are confirmed by numerical simulations.

]]>Mathematics doi: 10.3390/math6070117

Authors: Wei-Shih Du Erdal Karapınar Zhenhua He

In this paper, we first establish a new fixed point theorem that generalizes and unifies a number of well-known fixed point results, including the Banach contraction principle, Kannan&rsquo;s fixed point theorem, Chatterjea fixed point theorem, Du-Rassias fixed point theorem and many others. The presented results not only unify and generalize the existing results, but also yield several new fixed point theorems, which are different from the well-known results in the literature.

]]>Mathematics doi: 10.3390/math6070116

Authors: Tharmalingam Gunasekar Saravanan Karpagam Boyan Zlatanov

A cyclic map with a contractive type of condition called p-cyclic orbital M-Kcontraction is introduced in a partial metric space. Sufficient conditions for the existence and uniqueness of fixed points and the best proximity points for these maps in complete partial metric spaces are obtained. Furthermore, a necessary and sufficient condition for the completeness of partial metric spaces is given. The results are illustrated with an example.

]]>Mathematics doi: 10.3390/math6070115

Authors: Chalermpon Bunpog

This paper presents an alternative methodology for finding the solution of the boundary value problem (BVP) for the linear partial differential operator. We are particularly interested in the linear operator &oplus;k, where &oplus;k=♡k♢k, ♡k is the biharmonic operator iterated k-times and ♢k is the diamond operator iterated k-times. The solution is built on the Green&rsquo;s identity of the operators ♡k and &oplus;k, in which their derivations are also provided. To illustrate our findings, the example with prescribed boundary conditions is exhibited.

]]>Mathematics doi: 10.3390/math6070114

Authors: George Livadiotis

Maxwell equations have two types of asymmetries between the electric and magnetic fields. The first asymmetry is the inhomogeneity induced by the absence of magnetic charge sources. The second asymmetry is due to parity. We show how both asymmetries are naturally resolved under an alternative formulation of Maxwell equations for fields or potentials that uses a compact complex vector operator representation. The developed complex symmetric operator formalism can be easily applied to performing the continuity equation, the field wave equations, the Maxwell equations for potentials, the gauge transformations, and the 4-momentum representation; in general, the developed formalism constitutes a simple way of unfolding the Maxwell theory. Finally, we provide insights for extending the presented analysis within the context of (i) bicomplex numbers and tessarine algebra; and (ii) Lp-spaces in nonlinear Maxwell equations.

]]>Mathematics doi: 10.3390/math6070113

Authors: Fadi Barbara Valentina La Morgia Valerio Parodi Giuseppe Toscano Ezio Venturino

A model for the interactions of the invasive grey squirrel species as asymptomatic carriers of the poxvirus with the native red squirrel is presented and analyzed. Equilibria of the dynamical system are assessed, and their sensitivity in terms of the ecosystem parameters is investigated through numerical simulations. The findings are in line with both field and theoretical research. The results indicate that mainly the reproduction rate of the alien population should be drastically reduced to repel the invasion, and to achieve disease eradication, actions must be performed to reduce the intraspecific transmission rate; also, the native species mortality plays a role: if grey squirrels are controlled, increasing it may help in the red squirrel preservation, while the invaders vanish; on the contrary, decreasing it in favorable situations, the coexistence of the two species may occur. Preservation or restoration of the native red squirrel requires removal of the grey squirrels or keeping them at low values. Wildlife managers should exert a constant effort to achieve a harsh reduction of the grey squirrel growth rate and to protect the remnant red squirrel population.

]]>Mathematics doi: 10.3390/math6070112

Authors: Noor Rehman Choonkil Park Syed Inayat Ali Shah Abbas Ali

The generalized roughness in LA-semigroups is introduced, and several properties of lower and upper approximations are discussed. We provide examples to show that the lower approximation of a subset of an LA-semigroup may not be an LA-subsemigroup/ideal of LA-semigroup under a set valued homomorphism.

]]>Mathematics doi: 10.3390/math6070111

Authors: Miguel Simón Álvaro Buendía J. Muga

We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation AHA† that leaves the Hamiltonian H unchanged represents a symmetry of the Hamiltonian, which implies the commutativity [H,A]=0 and, if A is linear and time-independent, a conservation law, namely the invariance of expectation values of A. For non-Hermitian Hamiltonians, H† comes into play as a distinct operator that complements H in generalized unitarity relations. The above description of symmetries has to be extended to include also A-pseudohermiticity relations of the form AH=H†A. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of A operators that form Klein’s 4-group: parity, time-reversal, parity&amp;time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be richer and subtler for non-Hermitian than for Hermitian Hamiltonians.

]]>Mathematics doi: 10.3390/math6070110

Authors: Rudolf Seising

In the 1950s, the mathematically oriented electrical engineer, Lotfi A. Zadeh, investigated system theory, and in the mid-1960s, he established the theory of Fuzzy sets and systems based on the mathematical theorem of linear separability and the pattern classification problem. Contemporaneously, the psychologist, Frank Rosenblatt, developed the theory of the perceptron as a pattern recognition machine based on the starting research in so-called artificial intelligence, and especially in research on artificial neural networks, until the book of Marvin L. Minsky and Seymour Papert disrupted this research program. In the 1980s, the Parallel Distributed Processing research group requickened the artificial neural network technology. In this paper, we present the interwoven historical developments of the two mathematical theories which opened up into fuzzy pattern classification and fuzzy clustering.

]]>Mathematics doi: 10.3390/math6070109

Authors: Muhammed Syam Azza Alsuwaidi Asia Alneyadi Safa Al Refai Sondos Al Khaldi

In this article, a modified implicit hybrid method for solving the fractional Bagley-Torvik boundary (BTB) value problem is investigated. This approach is of a higher order. We study the convergence, zero stability, consistency, and region of absolute stability of the modified implicit hybrid method. Three of our numerical examples are presented.

]]>Mathematics doi: 10.3390/math6070108

Authors: Hsien-Chung Wu

The fuzzy numbers are fuzzy sets owning some elegant mathematical structures. The space consisting of all fuzzy numbers cannot form a vector space because it lacks the concept of the additive inverse element. In other words, the space of fuzzy numbers cannot be a normed space even though the normed structure can be defined on this space. This also says that the fixed point theorems established in the normed space cannot apply directly to the space of fuzzy numbers. The purpose of this paper is to propose the concept of near fixed point in the space of fuzzy numbers and to study its existence. In order to consider the contraction of fuzzy-number-valued function, the concepts of near metric space and near normed space of fuzzy numbers are proposed based on the almost identical concept. The concepts of Cauchy sequences in near metric space and near normed space of fuzzy numbers are also proposed. Under these settings, the existence of near fixed points of fuzzy-number-valued contraction function in complete near metric space and near Banach space of fuzzy numbers are established.

]]>Mathematics doi: 10.3390/math6070107

Authors: Morteza Baniasad Azad Behrooz Khosravi

In this paper we prove that if M is a simple K3-group, then M&times;M is uniquely determined by its order and some information on irreducible character degrees and as a consequence of our results we show that M&times;M is uniquely determined by the structure of its complex group algebra.

]]>Mathematics doi: 10.3390/math6070106

Authors: Hsien-Chung Wu

The T1-spaces induced by the fuzzy semi-metric spaces endowed with the special kind of triangle inequality are investigated in this paper. The limits in fuzzy semi-metric spaces are also studied to demonstrate the consistency of limit concepts in the induced topologies.

]]>Mathematics doi: 10.3390/math6060105

Authors: Y. Mahendra Singh Mohammad Saeed Khan Shin Min Kang

In this paper, we introduce F-convex contraction via admissible mapping in the sense of Wardowski [Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl., 94 (2012), 6 pages] which extends convex contraction mapping of type-2 of Istrǎţescu [Some fixed point theorems for convex contraction mappings and convex non-expansive mappings (I), Libertas Mathematica, 1(1981), 151&ndash;163] and establish a fixed point theorem in the setting of metric space. Our result extends and generalizes some other similar results in the literature. As an application of our main result, we establish an existence theorem for the non-linear Fredholm integral equation and give a numerical example to validate the application of our obtained result.

]]>Mathematics doi: 10.3390/math6060104

Authors: Da Xue Nael El-Farra

This paper addresses the problem of fault-tolerant stabilization of nonlinear processes subject to input constraints, control actuator faults and limited sensor–controller communication. A fault-tolerant Lyapunov-based model predictive control (MPC) formulation that enforces the fault-tolerant stabilization objective with reduced sensor–controller communication needs is developed. In the proposed formulation, the control action is obtained through the online solution of a finite-horizon optimal control problem based on an uncertain model of the plant. The optimization problem is solved in a receding horizon fashion subject to appropriate Lyapunov-based stability constraints which are designed to ensure that the desired stability and performance properties of the closed-loop system are met in the presence of faults. The state-space region where fault-tolerant stabilization is guaranteed is explicitly characterized in terms of the fault magnitude, the size of the plant-model mismatch and the choice of controller design parameters. To achieve the control objective with minimal sensor–controller communication, a forecast-triggered communication strategy is developed to determine when sensor–controller communication can be suspended and when it should be restored. In this strategy, transmission of the sensor measurement at a given sampling time over the sensor–controller communication channel to update the model state in the predictive controller is triggered only when the Lyapunov function or its time-derivative are forecasted to breach certain thresholds over the next sampling interval. The communication-triggering thresholds are derived from a Lyapunov stability analysis and are explicitly parameterized in terms of the fault size and a suitable fault accommodation parameter. Based on this characterization, fault accommodation strategies that guarantee closed-loop stability while simultaneously optimizing control and communication system resources are devised. Finally, a simulation case study involving a chemical process example is presented to illustrate the implementation and evaluate the efficacy of the developed fault-tolerant MPC formulation.

]]>Mathematics doi: 10.3390/math6060103

Authors: André H. Erhardt

In this paper, we study the dynamics of a certain Hodgkin-Huxley model describing the action potential (AP) of a cardiac muscle cell for a better understanding of the occurrence of a special type of cardiac arrhythmia, the so-called early afterdepolarisations (EADs). EADs are pathological voltage oscillations during the repolarisation or plateau phase of cardiac APs. They are considered as potential precursors to cardiac arrhythmia and are often associated with deficiencies in potassium currents or enhancements in the calcium or sodium currents, e.g., induced by ion channel diseases, drugs or stress. Our study is focused on the enhancement in the calcium current to identify regions, where EADs related to enhanced calcium current appear. To this aim, we study the dynamics of the model using bifurcation theory and numerical bifurcation analysis. Furthermore, we investigate the interaction of the potassium and calcium current. It turns out that a suitable increasing of the potassium current adjusted the EADs related to an enhanced calcium current. Thus, one can use our result to balance the EADs in the sense that an enhancement in the potassium currents may compensate the effect of enhanced calcium currents.

]]>Mathematics doi: 10.3390/math6060102

Authors: Mariam K. A. Al-Moqbali Nasser S. Al-Salti Ibrahim M. Elmojtaba

Prey&ndash;predator models with variable carrying capacity are proposed. These models are more realistic in modeling population dynamics in an environment that undergoes changes. In particular, prey&ndash;predator models with Holling type I and type II functional responses, incorporating the idea of a variable carrying capacity, are considered. The carrying capacity is modeled by a logistic equation that increases sigmoidally between an initial value &kappa;0&gt;&kappa;1 (a lower bound for the carrying capacity) and a final value &kappa;1+&kappa;2 (an upper bound for the carrying capacity). In order to examine the effect of the variable carrying capacity on the prey&ndash;predator dynamics, the two models were analyzed qualitatively using stability analysis and numerical solutions for the prey, and the predator population densities were obtained. Results on global stability and Hopf bifurcation of certain equilibrium points have been also presented. Additionally, the effect of other model parameters on the prey&ndash;predator dynamics has been examined. In particular, results on the effect of the handling parameter and the predator&rsquo;s death rate, which has been taken to be the bifurcation parameter, are presented.

]]>Mathematics doi: 10.3390/math6060101

Authors: G. Ayyappan S. Karpagam

In this paper, we discuss a non-Markovian batch arrival general bulk service single-server queueing system with server breakdown and repair, a stand-by server, multiple vacation and re-service. The main server&rsquo;s regular service time, re-service time, vacation time and stand-by server&rsquo;s service time are followed by general distributions and breakdown and repair times of the main server with exponential distributions. There is a stand-by server which is employed during the period in which the regular server remains under repair. The probability generating function of the queue size at an arbitrary time and some performance measures of the system are derived. Extensive numerical results are also illustrated.

]]>Mathematics doi: 10.3390/math6060100

Authors: Yahui Tian Xiaoli Luan Fei Liu Stevan Dubljevic

Column flotation is an efficient method commonly used in the mineral industry to separate useful minerals from ores of low grade and complex mineral composition. Its main purpose is to achieve maximum recovery while ensuring desired product grade. This work addresses a model predictive control design for a mineral column flotation process modeled by a set of nonlinear coupled heterodirectional hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs), which accounts for the interconnection of well-stirred regions represented by continuous stirred tank reactors (CSTRs) and transport systems given by heterodirectional hyperbolic PDEs, with these two regions combined through the PDEs&rsquo; boundaries. The model predictive control considers both optimality of the process operations and naturally present input and state/output constraints. For the discrete controller design, spatially varying steady-state profiles are obtained by linearizing the coupled ODE&ndash;PDE model, and then the discrete system is obtained by using the Cayley&ndash;Tustin time discretization transformation without any spatial discretization and/or without model reduction. The model predictive controller is designed by solving an optimization problem with input and state/output constraints as well as input disturbance to minimize the objective function, which leads to an online-solvable finite constrained quadratic regulator problem. Finally, the controller performance to keep the output at the steady state within the constraint range is demonstrated by simulation studies, and it is concluded that the optimal control scheme presented in this work makes this flotation process more efficient.

]]>Mathematics doi: 10.3390/math6060099

Authors: Luca Pratelli Pietro Rigo

It is not unusual that Xn⟶distVZ where Xn, V, Z are real random variables, V is independent of Z and Z&sim;N(0,1). An intriguing feature is that PVZ&isin;A=EN(0,V2)(A) for each Borel set A&sub;R, namely, the probability distribution of the limit VZ is a mixture of centered Gaussian laws with (random) variance V2. In this paper, conditions for dTV(Xn,VZ)&rarr;0 are given, where dTV(Xn,VZ) is the total variation distance between the probability distributions of Xn and VZ. To estimate the rate of convergence, a few upper bounds for dTV(Xn,VZ) are given as well. Special attention is paid to the following two cases: (i) Xn is a linear combination of the squares of Gaussian random variables; and (ii) Xn is related to the weighted quadratic variations of two independent Brownian motions.

]]>Mathematics doi: 10.3390/math6060098

Authors: Sylvain Contassot-Vivier Jean-François Couchot Pierre-Cyrille Héam

In previous works, some of the authors have proposed a canonical form of Gray Codes (GCs) in N-cubes (hypercubes of dimension N). This form allowed them to draw an algorithm that theoretically provides exactly all the GCs for a given dimension N. In another work, we first have shown that any of these GC can be used to build the transition function of a Pseudorandom Number Generator (PRNG). Also, we have found a theoretical quadratic upper bound of the mixing time, i.e., the number of iterations that are required to provide a PRNG whose output is uniform. This article, extends these two previous works both practically and theoretically. On the one hand, another algorithm for generating GCs is proposed that provides an efficient generation of subsets of the entire set of GCs related to a given dimension N. This offers a large choice of GC to be used in the construction of Choatic Iterations based PRNGs (CI-PRNGs), leading to a large class of possible PRNGs. On the other hand, the mixing time has been theoretically shown to be in Nlog(N), which was anticipated in the previous article, but not proven.

]]>Mathematics doi: 10.3390/math6060097

Authors: Chenkuan Li Changpin Li Kyle Clarkson

This paper is to study certain types of fractional differential and integral equations, such as &theta; ( x &minus; x 0 ) g ( x ) = 1 &Gamma; ( &alpha; ) &int; 0 x ( x &minus; &zeta; ) &alpha; &minus; 1 f ( &zeta; ) d &zeta; , y ( x ) + &int; 0 x y ( &tau; ) x &minus; &tau; d &tau; = x + &minus; 2 + &delta; ( x ) , and x + k &int; 0 x y ( &tau; ) ( x &minus; &tau; ) &alpha; &minus; 1 d &tau; = &delta; ( m ) ( x ) in the distributional sense by Babenko&rsquo;s approach and fractional calculus. Applying convolutions and products of distributions in the Schwartz sense, we obtain generalized solutions for integral and differential equations of fractional order by using the Mittag-Leffler function, which cannot be achieved in the classical sense including numerical analysis methods, or by the Laplace transform.

]]>Mathematics doi: 10.3390/math6060096

Authors: Dimplekumar Chalishajar Avadhesh Kumar

In this paper, the existence and uniqueness of the solutions to a fractional order nonlinear coupled system with integral boundary conditions is investigated. Furthermore, Ulam&rsquo;s type stability of the proposed coupled system is studied. Banach&rsquo;s fixed point theorem is used to obtain the existence and uniqueness of the solutions. Finally, an example is provided to illustrate the analytical findings.

]]>Mathematics doi: 10.3390/math6060095

Authors: Sumera Naz Samina Ashraf Muhammad Akram

A Pythagorean fuzzy set (PFS) is a powerful tool for depicting fuzziness and uncertainty. This model is more flexible and practical as compared to an intuitionistic fuzzy model. This paper proposes a new graph, called Pythagorean fuzzy graph (PFG). We investigate some properties of our proposed graphs. We determine the degree and total degree of a vertex of PFGs. Furthermore, we present the concept of Pythagorean fuzzy preference relations (PFPRs). In particular, we solve decision-making problems, including evaluation of hospitals, partner selection in supply chain management, and electronic learning main factors evaluation by using PFGs.

]]>Mathematics doi: 10.3390/math6060094

Authors: Zahra Pirbodaghi Morteza Mirmohammad Rezaii Seyed Mehdi Kazemi Torbaghan

In this paper, &lambda; -harmonic maps from a Finsler manifold to a Riemannian manifold are studied. Then, some properties of this kind of harmonic maps are presented and some examples are given. Finally, the stability of the &lambda; -harmonic maps from a Finsler manifold to the standard unit sphere S n ( n &gt; 2 ) is investigated.

]]>Mathematics doi: 10.3390/math6060093

Authors: Eshagh Hashemi Reza Saadati

We consider a distance function on generalized metric spaces and we get a generalization of Ekeland Variational Principle (EVP). Next, we prove that EVP is equivalent to Caristi&ndash;Kirk fixed point theorem and minimization Takahashi&rsquo;s theorem.

]]>Mathematics doi: 10.3390/math6060092

Authors: Harish Garg Jaspreet Kaur

The objective of this manuscript is to present a novel information measure for measuring the degree of fuzziness in intuitionistic fuzzy sets (IFSs). To achieve it, we define an ( R , S ) -norm-based information measure called the entropy to measure the degree of fuzziness of the set. Then, we prove that the proposed entropy measure is a valid measure and satisfies certain properties. An illustrative example related to a linguistic variable is given to demonstrate it. Then, we utilized it to propose two decision-making approaches to solve the multi-attribute decision-making (MADM) problem in the IFS environment by considering the attribute weights as either partially known or completely unknown. Finally, a practical example is provided to illustrate the decision-making process. The results corresponding to different pairs of ( R , S ) give different choices to the decision-maker to assess their results.

]]>Mathematics doi: 10.3390/math6060091

Authors: Mario Abundo

Let X ( t ) be a continuously time-changed Brownian motion starting from a random position &eta; , S ( t ) a given continuous, increasing boundary, with S ( 0 ) &ge; 0 , P ( &eta; &ge; S ( 0 ) ) = 1 , and F an assigned distribution function. We study the inverse first-passage time problem for X ( t ) , which consists in finding the distribution of &eta; such that the first-passage time of X ( t ) below S ( t ) has distribution F , generalizing the results, valid in the case when S ( t ) is a straight line. Some explicit examples are reported.

]]>Mathematics doi: 10.3390/math6060090

Authors: Hsien-Chung Wu

The hyperspace consists of all the subsets of a vector space. It is well-known that the hyperspace is not a vector space because it lacks the concept of inverse element. This also says that we cannot consider its normed structure, and some kinds of fixed point theorems cannot be established in this space. In this paper, we shall propose the concept of null set that will be used to endow a norm to the hyperspace. This normed hyperspace is clearly not a conventional normed space. Based on this norm, the concept of Cauchy sequence can be similarly defined. In addition, a Banach hyperspace can be defined according to the concept of Cauchy sequence. The main aim of this paper is to study and establish the so-called near fixed point theorems in Banach hyperspace.

]]>Mathematics doi: 10.3390/math6060089

Authors: Fang Fang Richard Clawson Klee Irwin

In geometrically frustrated clusters of polyhedra, gaps between faces can be closed without distorting the polyhedra by the long established method of discrete curvature, which consists of curving the space into a fourth dimension, resulting in a dihedral angle at the joint between polyhedra in 4D. An alternative method&mdash;the twist method&mdash;has been recently suggested for a particular case, whereby the gaps are closed by twisting the cluster in 3D, resulting in an angular offset of the faces at the joint between adjacent polyhedral. In this paper, we show the general applicability of the twist method, for local clusters, and present the surprising result that both the required angle of the twist transformation and the consequent angle at the joint are the same, respectively, as the angle of bending to 4D in the discrete curvature and its resulting dihedral angle. The twist is therefore not only isomorphic, but isogonic (in terms of the rotation angles) to discrete curvature. Our results apply to local clusters, but in the discussion we offer some justification for the conjecture that the isomorphism between twist and discrete curvature can be extended globally. Furthermore, we present examples for tetrahedral clusters with three-, four-, and fivefold symmetry.

]]>Mathematics doi: 10.3390/math6060088

Authors: Lorentz Jäntschi Sorana D. Bolboacă

The correct application of a statistical test is directly connected with information related to the distribution of data. Anderson&ndash;Darling is one alternative used to test if the distribution of experimental data follows a theoretical distribution. The conclusion of the Anderson&ndash;Darling test is usually drawn by comparing the obtained statistic with the available critical value, which did not give any weight to the same size. This study aimed to provide a formula for calculation of p-value associated with the Anderson&ndash;Darling statistic considering the size of the sample. A Monte Carlo simulation study was conducted for sample sizes starting from 2 to 61, and based on the obtained results, a formula able to give reliable probabilities associated to the Anderson&ndash;Darling statistic is reported.

]]>Mathematics doi: 10.3390/math6060087

Authors: Young Jun Seok-Zun Song Seon Kim

To deal with the uncertainties, fuzzy set theory can be considered as one of the mathematical tools by Zadeh. As a mathematical tool to deal with negative information, Jun et al. introduced a new function, which is called a negative-valued function, and constructed N -structures in 2009. Since then, N -structures are applied to algebraic structures and soft sets, etc. Using the N -structures, the notions of (extended) N -hyper sets, N -substructures of type 1, 2, 3 and 4 are introduced, and several related properties are investigated in this research paper.

]]>Mathematics doi: 10.3390/math6050086

Authors: Shan Gao Yi Zheng Shaoyuan Li

This paper considers a class of large-scale systems which is composed of many interacting subsystems, and each of them is controlled by an individual controller. For this type of system, to improve the optimization performance of the entire closed-loop system in a distributed framework without the entire system’s information or too-complicated network information, connectivity is always an important topic. To achieve this purpose, a distributed model predictive control (DMPC) design method is proposed in this paper, where each local model predictive control (MPC) considers the optimization performance of its strong coupling subsystems and communicates with them. A method to determine the strength of the coupling relationship based on the closed-loop system’s performance and subsystem network connectivity is proposed for the selection of each subsystem’s neighbors. Finally, through integrating the steady-state calculation, the designed DMPC is able to guarantee the recursive feasibility and asymptotic stability of the closed-loop system in the cases of both tracking set point and stabilizing system to zeroes. Simulation results show the efficiency of the proposed DMPC.

]]>Mathematics doi: 10.3390/math6050085

Authors: Patricia Román-Román Juan José Serrano-Pérez Francisco Torres-Ruiz

Different versions of the lognormal diffusion process with exogenous factors have been used in recent years to model and study the behavior of phenomena following a given growth curve. In each case considered, the estimation of the model has been addressed, generally by maximum likelihood (ML), as has been the study of several characteristics associated with the type of curve considered. For this process, a unified version of the ML estimation problem is presented, including how to obtain estimation errors and asymptotic confidence intervals for parametric functions when no explicit expression is available for the estimators of the parameters of the model. The Gompertz-type diffusion process is used here to illustrate the application of the methodology.

]]>Mathematics doi: 10.3390/math6050084

Authors: George Kaimakamis Konstantina Panagiotidou Juan de Dios Pérez

In this paper, three-dimensional real hypersurfaces in non-flat complex space forms, whose shape operator satisfies a geometric condition, are studied. Moreover, the tensor field P = ϕ A - A ϕ is given and three-dimensional real hypersurfaces in non-flat complex space forms whose tensor field P satisfies geometric conditions are classified.

]]>Mathematics doi: 10.3390/math6050083

Authors: Elhoucien Elqorachi Michael Th. Rassias

In the present paper we study the generalized Hyers&ndash;Ulam stability of the generalized trigonometric functional equations f ( x y ) + &mu; ( y ) f ( x &sigma; ( y ) ) = 2 f ( x ) g ( y ) + 2 h ( y ) , x , y &isin; S ; f ( x y ) + &mu; ( y ) f ( x &sigma; ( y ) ) = 2 f ( y ) g ( x ) + 2 h ( x ) , x , y &isin; S , where S is a semigroup, &sigma; : S ⟶ S is a involutive morphism, and &mu; : S ⟶ C is a multiplicative function such that &mu; ( x &sigma; ( x ) ) = 1 for all x &isin; S . As an application, we establish the generalized Hyers&ndash;Ulam stability theorem on amenable monoids and when &sigma; is an involutive automorphism of S.

]]>Mathematics doi: 10.3390/math6050082

Authors: Paul Slade

Kingman’s coalescent process is a mathematical model of genealogy in which only pairwise common ancestry may occur. Inter-arrival times between successive coalescence events have a negative exponential distribution whose rate equals the combinatorial term ( n 2 ) where n denotes the number of lineages present in the genealogy. These two standard constraints of Kingman’s coalescent, obtained in the limit of a large population size, approximate the exact ancestral process of Wright-Fisher or Moran models under appropriate parameterization. Calculation of coalescence event probabilities with higher accuracy quantifies the dependence of sample and population sizes that adhere to Kingman’s coalescent process. The convention that probabilities of leading order N − 2 are negligible provided n ≪ N is examined at key stages of the mathematical derivation. Empirically, expected genealogical parity of the single-pair restricted Wright-Fisher haploid model exceeds 99% where n ≤ 1 2 N 3 ; similarly, per expected interval where n ≤ 1 2 N / 6 . The fractional cubic root criterion is practicable, since although it corresponds to perfect parity and to an extent confounds identifiability it also accords with manageable conditional probabilities of multi-coalescence.

]]>Mathematics doi: 10.3390/math6050081

Authors: Antonio Di Crescenzo Virginia Giorno Balasubramanian Krishna Kumar Amelia Nobile

We consider a time-non-homogeneous double-ended queue subject to catastrophes and repairs. The catastrophes occur according to a non-homogeneous Poisson process and lead the system into a state of failure. Instantaneously, the system is put under repair, such that repair time is governed by a time-varying intensity function. We analyze the transient and the asymptotic behavior of the queueing system. Moreover, we derive a heavy-traffic approximation that allows approximating the state of the systems by a time-non-homogeneous Wiener process subject to jumps to a spurious state (due to catastrophes) and random returns to the zero state (due to repairs). Special attention is devoted to the case of periodic catastrophe and repair intensity functions. The first-passage-time problem through constant levels is also treated both for the queueing model and the approximating diffusion process. Finally, the goodness of the diffusive approximating procedure is discussed.

]]>Mathematics doi: 10.3390/math6050080

Authors: Anna Sinitcina Yacov Satin Alexander Zeifman Galina Shilova Alexander Sipin Ksenia Kiseleva Tatyana Panfilova Anastasia Kryukova Irina Gudkova Elena Fokicheva

The model of a two-dimensional birth-death process with possible catastrophes is studied. The upper bounds on the rate of convergence in some weighted norms and the corresponding perturbation bounds are obtained. In addition, we consider the detailed description of two examples with 1-periodic intensities and various types of death (service) rates. The bounds on the rate of convergence and the behavior of the corresponding mathematical expectations are obtained for each example.

]]>Mathematics doi: 10.3390/math6050079

Authors: E. J. Janowski M. R. S. Kulenović

We investigate the nonautonomous difference equation with real initial conditions and coefficients g i , i = 0 , 1 which are in general functions of n and/or the state variables x n , x n &minus; 1 , &hellip; , and satisfy g 0 + g 1 = 1 . We also obtain some global results about the behavior of solutions of the nonautonomous non-homogeneous difference equation where g i , i = 0 , 1 , 2 are functions of n and/or the state variables x n , x n &minus; 1 , &hellip; , with g 0 + g 1 = 1 . Our results are based on the explicit formulas for solutions. We illustrate our results by numerous examples.

]]>Mathematics doi: 10.3390/math6050078

Authors: M. De la Sen

This paper presents and discusses the stability of a discrete multirate sampling system whose sets of sampling rates (or sampling periods) are the integer multiple of those operating on all the preceding substates. Each of such substates is associated with a particular sampling rate. The sufficiency-type stability conditions are derived based on simple conditions on the norm, spectral radius and numerical radius of the matrix of the dynamics of a system parameterized at the largest sampling period.

]]>Mathematics doi: 10.3390/math6050077

Authors: Danish A. Ahmed Sergei V. Petrovskii Paulo F. C. Tilles

Many empirical and theoretical studies indicate that Brownian motion and diffusion models as its mean field counterpart provide appropriate modeling techniques for individual insect movement. However, this traditional approach has been challenged, and conflicting evidence suggests that an alternative movement pattern such as L&eacute;vy walks can provide a better description. L&eacute;vy walks differ from Brownian motion since they allow for a higher frequency of large steps, resulting in a faster movement. Identification of the &lsquo;correct&rsquo; movement model that would consistently provide the best fit for movement data is challenging and has become a highly controversial issue. In this paper, we show that this controversy may be superficial rather than real if the issue is considered in the context of trapping or, more generally, survival probabilities. In particular, we show that almost identical trap counts are reproduced for inherently different movement models (such as the Brownian motion and the L&eacute;vy walk) under certain conditions of equivalence. This apparently suggests that the whole &lsquo;Levy or diffusion&rsquo; debate is rather senseless unless it is placed into a specific ecological context, e.g., pest monitoring programs.

]]>Mathematics doi: 10.3390/math6050076

Authors: Yanisa Chaiya Chollawat Pookpienlert Nuttawoot Nupo Sayan Panma

Let K n be a complete graph on n vertices. Denote by S K n the set of all subgraphs of K n . For each G , H &isin; S K n , the ring sum of G and H is a graph whose vertex set is V ( G ) &cup; V ( H ) and whose edges are that of either G or H, but not of both. Then S K n is a semigroup under the ring sum. In this paper, we study Green&rsquo;s relations on S K n and characterize ideals, minimal ideals, maximal ideals, and principal ideals of S K n . Moreover, maximal subsemigroups and a class of maximal congruences are investigated. Furthermore, we prescribe the natural order on S K n and consider minimal elements, maximal elements and covering elements of S K n under this order.

]]>Mathematics doi: 10.3390/math6050075

Authors: Junya Takahashi

We construct an incomplete Riemannian manifold with positive Ricci curvature that has non-trivial L 2 -harmonic forms and on which the L 2 -Stokes theorem does not hold. Therefore, a Bochner-type vanishing theorem does not hold for incomplete Riemannian manifolds.

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