Mathematics doi: 10.3390/math6070127

Authors: Prakash Khomami Golnar Khomami Fernando 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 probabilityPL is given by, PL = 1−(1-Pe)(1-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.

]]>Mathematics doi: 10.3390/math6050074

Authors: Young Bae Jun Florentin Smarandache Seok-Zun Song Hashem Bordbar

The concept of a ( &isin; , &isin; ) -neutrosophic ideal is introduced, and its characterizations are established. The notions of neutrosophic permeable values are introduced, and related properties are investigated. Conditions for the neutrosophic level sets to be energetic, right stable, and right vanished are discussed. Relations between neutrosophic permeable S- and I-values are considered.

]]>Mathematics doi: 10.3390/math6050073

Authors: Yuri S. Popkov

This paper develops an upper bound design method of the Lipschitz constant for the generalized Fermi&ndash;Dirac information entropy operator with a polyhedral admissible set. We introduce the concept of a normal operator from this class in which the constraint matrix has normalized columns. Next, we establish a connection between the normal and original operator. Finally, we demonstrate that the normal operator is majorized by the linear one and find numerical characteristics of this majorant.

]]>Mathematics doi: 10.3390/math6050072

Authors: Shuang Liu Xinfeng Liu

The systems of reaction-diffusion equations coupled with moving boundaries defined by Stefan condition have been widely used to describe the dynamics of spreading population and with competition of two species. To solve these systems numerically, new numerical challenges arise from the competition of two species due to the interaction of their free boundaries. On the one hand, extremely small time steps are usually needed due to the stiffness of the system. On the other hand, it is always difficult to efficiently and accurately handle the moving boundaries especially with competition of two species. To overcome these numerical difficulties, we introduce a front tracking method coupled with an implicit solver for the 1D model. For the general 2D model, we use a level set approach to handle the moving boundaries to efficiently treat complicated topological changes. Several numerical examples are examined to illustrate the efficiency, accuracy and consistency for different approaches.

]]>Mathematics doi: 10.3390/math6050071

Authors: Nasrin Jahan Nasu Md. Abdullah Al Mahbub Shahid Hussain Haibiao Zheng

In this paper, a two-level finite element method for Oseen viscoelastic fluid flow obeying an Oldroyd-B type constitutive law is presented. With the newly proposed algorithm, solving a large system of the constitutive equations will not be much more complex than the solution of one linearized equation. The viscoelastic fluid flow constitutive equation consists of nonlinear terms, which are linearized by taking a known velocity b ( x ) , and transforms into the Oseen viscoelastic fluid flow model. Since Oseen viscoelastic fluid flow is already linear, we use a two-level method with a new technique. The two-level approach is consistent and efficient to study the coupled system which contains nonlinear terms. In the first step, the solution on the coarse grid is derived, and the result is used to determine the solution on the fine mesh in the second step. The decoupling algorithm takes two steps to solve a linear system on the fine mesh. The stability of the algorithm is derived for the temporal discretization and obtains the desired error bound. Two numerical experiments are executed to show the accuracy of the theoretical analysis. The approximations of the stress tensor, velocity vector, and pressure field are P 1 -discontinuous, P 2 -continuous and P 1 -continuous finite elements respectively.

]]>Mathematics doi: 10.3390/math6050070

Authors: Giuseppina Albano Virginia Giorno

In this paper, we consider a stochastic diffusion process able to model the interest rate evolving with respect to time and propose a first passage time (FPT) approach through a boundary, defined as the &ldquo;alert threshold&rdquo;, in order to evaluate the risk of a proposed loan. Above this alert threshold, the rate is considered at the risk of usury, so new monetary policies have been adopted. Moreover, the mean FPT can be used as an indicator of the &ldquo;goodness&rdquo; of a loan; i.e., when an applicant is to choose between two loan offers, s/he will choose the one with a higher mean exit time from the alert boundary. An application to real data is considered by analyzing the Italian average effect global rate by means of two widely used models in finance, the Ornstein-Uhlenbeck (Vasicek) and Feller (Cox-Ingersoll-Ross) models.

]]>Mathematics doi: 10.3390/math6050069

Authors: Zhe Wu Helen Durand Panagiotis D. Christofides

Process operational safety plays an important role in designing control systems for chemical processes. Motivated by this, in this work, we develop a process Safeness Index-based economic model predictive control system for a broad class of stochastic nonlinear systems with input constraints. A stochastic Lyapunov-based controller is first utilized to characterize a region of the state-space surrounding the origin, starting from which the origin is rendered asymptotically stable in probability. Using this stability region characterization and a process Safeness Index function that characterizes the region in state-space in which it is safe to operate the process, an economic model predictive control system is then developed using Lyapunov-based constraints to ensure economic optimality, as well as process operational safety and closed-loop stability in probability. A chemical process example is used to demonstrate the applicability and effectiveness of the proposed approach.

]]>Mathematics doi: 10.3390/math6050068

Authors: Badr Alqahtani Andreea Fulga Erdal Karapınar

In this paper, we investigate the existence of fixed points that are not necessarily unique in the setting of extended b-metric space. We state some examples to illustrate our results.

]]>Mathematics doi: 10.3390/math6050067

Authors: Surapati Pramanik Shyamal Dalapati Shariful Alam Florentin Smarandache Tapan Kumar Roy

The objective of the paper is to introduce a new cross entropy measure in a neutrosophic cubic set (NCS) environment, which we call NC-cross entropy measure. We prove its basic properties. We also propose weighted NC-cross entropy and investigate its basic properties. We develop a novel multi attribute decision-making (MADM) strategy based on a weighted NC-cross entropy measure. To show the feasibility and applicability of the proposed multi attribute decision-making strategy, we solve an illustrative example of the multi attribute decision-making problem.

]]>Mathematics doi: 10.3390/math6050066

Authors: Farzad Fatehi Yuliya N. Kyrychko Konstantin B. Blyuss

A major contribution to the onset and development of autoimmune disease is known to come from infections. An important practical problem is identifying the precise mechanism by which the breakdown of immune tolerance as a result of immune response to infection leads to autoimmunity. In this paper, we develop a mathematical model of immune response to a viral infection, which includes T cells with different activation thresholds, regulatory T cells (Tregs), and a cytokine mediating immune dynamics. Particular emphasis is made on the role of time delays associated with the processes of infection and mounting the immune response. Stability analysis of various steady states of the model allows us to identify parameter regions associated with different types of immune behaviour, such as, normal clearance of infection, chronic infection, and autoimmune dynamics. Numerical simulations are used to illustrate different dynamical regimes, and to identify basins of attraction of different dynamical states. An important result of the analysis is that not only the parameters of the system, but also the initial level of infection and the initial state of the immune system determine the progress and outcome of the dynamics.

]]>Mathematics doi: 10.3390/math6050065

Authors: Su Liu Jinfeng Liu

In this work, we propose a framework for economic model predictive control (EMPC) with zone tracking. A zone tracking stage cost is incorporated into the existing EMPC framework to form a multi-objective optimization problem. We provide sufficient conditions for asymptotic stability of the optimal steady state and characterize the exact penalty for the zone tracking cost which prioritizes zone tracking objective over economic objective. Moreover, an algorithm to modify the target zone based on the economic performance and reachability of the optimal steady state is proposed. The modified target zone effectively decouples the dynamic zone tracking and economic objectives and simplifies parameter tuning.

]]>Mathematics doi: 10.3390/math6040064

Authors: Mei Huang Ron Kerman Susanna Spektor

Let f be a band-limited function in L 2 ( R ) . Fix T &gt; 0 , and suppose f ′ exists and is integrable on [ − T , T ] . This paper gives a concrete estimate of the error incurred when approximating f in the root mean square by a partial sum of its Hermite series. Specifically, we show, that for K = 2 n , n ∈ Z + , 1 2 T ∫ − T T [ f ( t ) − ( S K f ) ( t ) ] 2 d t 1 / 2 ≤ 1 + 1 K 1 2 T ∫ | t | &gt; T f ( t ) 2 d t 1 / 2 + 1 2 T ∫ | ω | &gt; N | f ^ ( ω ) | 2 d ω 1 / 2 + 1 K 1 2 T ∫ | t | ≤ T f N ( t ) 2 d t 1 / 2 + 1 π 1 + 1 2 K S a ( K , T ) , in which S K f is the K-th partial sum of the Hermite series of f , f ^ is the Fourier transform of f, N = 2 K + 1 + 2 K + 3 2 and f N = ( f ^ χ ( − N , N ) ) ∨ ( t ) = 1 π ∫ − ∞ ∞ sin ( N ( t − s ) ) t − s f ( s ) d s . An explicit upper bound is obtained for S a ( K , T ) .

]]>Mathematics doi: 10.3390/math6040063

Authors: Mina Torabi Mohammad-Mehdi Hosseini

In this paper, three-step Taylor expansion, which is equivalent to third-order Taylor expansion, is used as a mathematical base of the new descent method. At each iteration of this method, three steps are performed. Each step has a similar structure to the steepest descent method, except that the generalized search direction, step length, and next iterative point are applied. Compared with the steepest descent method, it is shown that the proposed algorithm has higher convergence speed and lower computational cost and storage.

]]>Mathematics doi: 10.3390/math6040062

Authors: John N. Mordeson Sunil Mathew

It is the case that, in certain applications of fuzzy graphs, a t-norm, instead of a minimum, is more suitable. This requires the development of a new theory of fuzzy graphs involving an arbitrary t-norm in the basic definition of a fuzzy graph. There is very little known about this type of fuzzy graph. The purpose of this paper is to further develop this type of fuzzy graph. We concentrate on the relatively new concept of fuzzy incidence graphs.

]]>Mathematics doi: 10.3390/math6040061

Authors: Ye Jin Yuehong Sun Hongjiao Ma

The Artificial Bee Colony (ABC) algorithm is a bionic intelligent optimization method. The cloud model is a kind of uncertainty conversion model between a qualitative concept T &tilde; that is presented by nature language and its quantitative expression, which integrates probability theory and the fuzzy mathematics. A developed ABC algorithm based on cloud model is proposed to enhance accuracy of the basic ABC algorithm and avoid getting trapped into local optima by introducing a new select mechanism, replacing the onlooker bees&rsquo; search formula and changing the scout bees&rsquo; updating formula. Experiments on CEC15 show that the new algorithm has a faster convergence speed and higher accuracy than the basic ABC and some cloud model based ABC variants.

]]>Mathematics doi: 10.3390/math6040060

Authors: Xinan Zhang Ruigang Wang Jie Bao

With the penetration of grid-connected renewable energy generation, microgrids are facing stability and power quality problems caused by renewable intermittency. To alleviate such problems, demand side management (DSM) of responsive loads, such as building air-conditioning system (BACS), has been proposed and studied. In recent years, numerous control approaches have been published for proper management of single BACS. The majority of these approaches focus on either the control of BACS for attenuating power fluctuations in the grid or the operating cost minimization on behalf of the residents. These two control objectives are paramount for BACS control in microgrids and can be conflicting. As such, they should be considered together in control design. As individual buildings may have different owners/residents, it is natural to control different BACSs in an autonomous and self-interested manner to minimize the operational costs for the owners/residents. Unfortunately, such &ldquo;selfish&rdquo; operation can result in abrupt and large power fluctuations at the point of common coupling (PCC) of the microgrid due to lack of coordination. Consequently, the original objective of mitigating power fluctuations generated by renewable intermittency cannot be achieved. To minimize the operating costs of individual BACSs and simultaneously ensure desirable overall power flow at PCC, this paper proposes a novel distributed control framework based on the dissipativity theory. The proposed method achieves the objective of renewable intermittency mitigation through proper coordination of distributed BACS controllers and is scalable and computationally efficient. Simulation studies are carried out to illustrate the efficacy of the proposed control framework.

]]>Mathematics doi: 10.3390/math6040059

Authors: Weam Alharbi Sergei Petrovskii

A telegraph equation is believed to be an appropriate model of population dynamics as it accounts for the directional persistence of individual animal movement. Being motivated by the problem of habitat fragmentation, which is known to be a major threat to biodiversity that causes species extinction worldwide, we consider the reaction&ndash;telegraph equation (i.e., telegraph equation combined with the population growth) on a bounded domain with the goal to establish the conditions of species survival. We first show analytically that, in the case of linear growth, the expression for the domain&rsquo;s critical size coincides with the critical size of the corresponding reaction&ndash;diffusion model. We then consider two biologically relevant cases of nonlinear growth, i.e., the logistic growth and the growth with a strong Allee effect. Using extensive numerical simulations, we show that in both cases the critical domain size of the reaction&ndash;telegraph equation is larger than the critical domain size of the reaction&ndash;diffusion equation. Finally, we discuss possible modifications of the model in order to enhance the positivity of its solutions.

]]>Mathematics doi: 10.3390/math6040058

Authors: Christophe Guyeux Jean-François Couchot Arnaud Le Rouzic Jacques M. Bahi Luigi Marangio

Predicting how a genetic change affects a given character is a major challenge in biology, and being able to tackle this problem relies on our ability to develop realistic models of gene networks. However, such models are rarely tractable mathematically. In this paper, we propose a mathematical analysis of the sigmoid variant of the Wagner gene-network model. By considering the simplest case, that is, one unique self-regulating gene, we show that numerical simulations are not the only tool available to study such models: theoretical studies can be done too, by mathematical analysis of discrete dynamical systems. It is first shown that the particular sigmoid function can be theoretically investigated. Secondly, we provide an illustration of how to apply such investigations in the case of the dynamical system representing the one self-regulating gene. In this context, we focused on the composite function f a ( m . x ) where f a is the parametric sigmoid function and m is a scalar not in { 0 , 1 } and we have proven that the number of fixed-point can be deduced theoretically, according to the values of a and m.

]]>Mathematics doi: 10.3390/math6040057

Authors: Hossein Moradi Mohammad Reza Darafsheh Ali Iranmanesh

Let G be a finite group. The prime graph &Gamma; ( G ) of G is defined as follows: The set of vertices of &Gamma; ( G ) is the set of prime divisors of | G | and two distinct vertices p and p &prime; are connected in &Gamma; ( G ) , whenever G contains an element of order p p &prime; . A non-abelian simple group P is called recognizable by prime graph if for any finite group G with &Gamma; ( G ) = &Gamma; ( P ) , G has a composition factor isomorphic to P. It is been proved that finite simple groups 2 D n ( q ) , where n &ne; 4 k , are quasirecognizable by prime graph. Now in this paper we discuss the quasirecognizability by prime graph of the simple groups 2 D 2 k ( q ) , where k &ge; 9 and q is a prime power less than 10 5 .

]]>Mathematics doi: 10.3390/math6040056

Authors: Matt Visser

The Lambert W function, implicitly defined by W ( x ) e W ( x ) = x , is a relatively “new” special function that has recently been the subject of an extended upsurge in interest and applications. In this note, I point out that the Lambert W function can also be used to gain a new perspective on the distribution of the prime numbers.

]]>Mathematics doi: 10.3390/math6040055

Authors: Alexander Nazarov Yakov Nikitin

We study the exact small deviation asymptotics with respect to the Hilbert norm for some mixed Gaussian processes. The simplest example here is the linear combination of the Wiener process and the Brownian bridge. We get the precise final result in this case and in some examples of more complicated processes of similar structure. The proof is based on Karhunen–Loève expansion together with spectral asymptotics of differential operators and complex analysis methods.

]]>Mathematics doi: 10.3390/math6040054

Authors: Prakash Veeraraghavan

In this paper, we introduce the notion of g-convex weight sequence (gcws) for connected graphs based on the concept of g-convexity and g-weight. g-weight is a natural generalization of the notion of branch weight for trees. We investigate the various questions of realization of an integer sequence as a g-convex weight sequence for trees and some special classes of graphs such as complete graphs, windmill and degenerate windmill graphs and wheels.

]]>Mathematics doi: 10.3390/math6040053

Authors: Florentin Smarandache Mehmet Şahin Abdullah Kargın

In this study, the neutrosophic triplet G-module is introduced and the properties of neutrosophic triplet G-module are studied. Furthermore, reducible, irreducible, and completely reducible neutrosophic triplet G-modules are defined, and relationships of these structures with each other are examined. Also, it is shown that the neutrosophic triplet G-module is different from the G-module.

]]>Mathematics doi: 10.3390/math6040052

Authors: Leopold Verstraelen

The main purpose of the present paper is to well define Minkowskian angles and pseudo-angles between the two null directions and between a null direction and any non-null direction, respectively. Moreover, in a kind of way that will be tried to be made clear at the end of the paper, these new sorts of angles and pseudo-angles can similarly to the previously known angles be seen as (combinations of) Minkowskian lengths of arcs on a Minkowskian unit circle together with Minkowskian pseudo-lengths of parts of the straight null lines.

]]>Mathematics doi: 10.3390/math6040051

Authors: Masoud Kheradmandi Prashant Mhaskar

This manuscript addresses the problem of data driven model based economic model predictive control (MPC) design. To this end, first, a data-driven Lyapunov-based MPC is designed, and shown to be capable of stabilizing a system at an unstable equilibrium point. The data driven Lyapunov-based MPC utilizes a linear time invariant (LTI) model cognizant of the fact that the training data, owing to the unstable nature of the equilibrium point, has to be obtained from closed-loop operation or experiments. Simulation results are first presented demonstrating closed-loop stability under the proposed data-driven Lyapunov-based MPC. The underlying data-driven model is then utilized as the basis to design an economic MPC. The economic improvements yielded by the proposed method are illustrated through simulations on a nonlinear chemical process system example.

]]>Mathematics doi: 10.3390/math6040050

Authors: Mohamadtaghi Rahimi Pranesh Kumar Gholamhossein Yari

Credibility measures in vague environments are to quantify the approximate chance of occurrence of fuzzy events. This paper presents a novel definition about credibility for intuitionistic fuzzy variables. We axiomatize this credibility measure and to clarify, give some examples. Based on the notion of these concepts, we provide expected values, entropy, and general formulae for the central moments and discuss them through examples.

]]>Mathematics doi: 10.3390/math6040049

Authors: Rita Giuliano Claudio Macci

In this paper, we prove large deviation results for some sequences of weighted sums of random variables. These sequences have applications to the probabilistic generalized Cramér model for products of primes in arithmetic progressions; they could lead to new conjectures concerning the (non-random) set of products of primes in arithmetic progressions, a relevant topic in number theory.

]]>Mathematics doi: 10.3390/math6040048

Authors: Muhammed Syam Mohammed Abu Omar

In this paper, we study a class of fractional nonlinear second order Volterra integro-differential type of singularly perturbed problems with fractional order. We divide the problem into two subproblems. The first subproblems is the reduced problem when ϵ = 0 . The second subproblems is fractional Volterra integro-differential problem. We use the finite difference method to solve the first problem and the reproducing kernel method to solve the second problem. In addition, we use the pade’ approximation. The results show that the proposed analytical method can achieve excellent results in predicting the solutions of such problems. Theoretical results are presented. Numerical results are presented to show the efficiency of the proposed method.

]]>Mathematics doi: 10.3390/math6040047

Authors: Dheeraj Kumar Joshi Ismat Beg Sanjay Kumar

Uncertainties due to randomness and fuzziness comprehensively exist in control and decision support systems. In the present study, we introduce notion of occurring probability of possible values into hesitant fuzzy linguistic element (HFLE) and define hesitant probabilistic fuzzy linguistic set (HPFLS) for ill structured and complex decision making problem. HPFLS provides a single framework where both stochastic and non-stochastic uncertainties can be efficiently handled along with hesitation. We have also proposed expected mean, variance, score and accuracy function and basic operations for HPFLS. Weighted and ordered weighted aggregation operators for HPFLS are also defined in the present study for its applications in multi-criteria group decision making (MCGDM) problems. We propose a MCGDM method with HPFL information which is illustrated by an example. A real case study is also taken in the present study to rank State Bank of India, InfoTech Enterprises, I.T.C., H.D.F.C. Bank, Tata Steel, Tata Motors and Bajaj Finance using real data. Proposed HPFLS-based MCGDM method is also compared with two HFL-based decision making methods.

]]>Mathematics doi: 10.3390/math6040046

Authors: Mumtaz Ali Florentin Smarandache Mohsin Khan

Rings and fields are significant algebraic structures in algebra and both of them are based on the group structure. In this paper, we attempt to extend the notion of a neutrosophic triplet group to a neutrosophic triplet ring and a neutrosophic triplet field. We introduce a neutrosophic triplet ring and study some of its basic properties. Further, we define the zero divisor, neutrosophic triplet subring, neutrosophic triplet ideal, nilpotent integral neutrosophic triplet domain, and neutrosophic triplet ring homomorphism. Finally, we introduce a neutrosophic triplet field.

]]>Mathematics doi: 10.3390/math6040045

Authors: Frédéric Magoulès Guillaume Gbikpi-Benissan Qinmeng Zou

Spatial domain decomposition methods have been largely investigated in the last decades, while time domain decomposition seems to be contrary to intuition and so is not as popular as the former. However, many attractive methods have been proposed, especially the parareal algorithm, which showed both theoretical and experimental efficiency in the context of parallel computing. In this paper, we present an original model of asynchronous variant based on the parareal scheme, applied to the European option pricing problem. Some numerical experiments are given to illustrate the convergence performance and computational efficiency of such a method.

]]>Mathematics doi: 10.3390/math6030044

Authors: Adela Mihai Ion Mihai

We consider statistical submanifolds of Hessian manifolds of constant Hessian curvature. For such submanifolds we establish a Euler inequality and a Chen-Ricci inequality with respect to a sectional curvature of the ambient Hessian manifold.

]]>Mathematics doi: 10.3390/math6030043

Authors: Tien Nguyen Hien Tran Andy Guillen Tung Bui Sumner Matsunaga

This paper provides a high-level discussion and propositions of frameworks and models for acquisition strategy of complex systems. In particular, it presents an innovative system engineering approach to model the Department of Defense (DoD) acquisition process and offers several optimization modules including simulation models using game theory and war-gaming concepts. Our frameworks employ Advanced Game-based Mathematical Framework (AGMF) and Unified Game-based Acquisition Framework (UGAF), and related advanced simulation and mathematical models that include a set of War-Gaming Engines (WGEs) implemented in MATLAB statistical optimization models. WGEs are defined as a set of algorithms, characterizing the Program and Technical Baseline (PTB), technology enablers, architectural solutions, contract type, contract parameters and associated incentives, and industry bidding position. As a proof of concept, Aerospace, in collaboration with the North Carolina State University (NCSU) and University of Hawaii (UH), successfully applied and extended the proposed frameworks and decision models to determine the optimum contract parameters and incentives for a Cost Plus Incentive Fee (CPIF) contract. As a result, we can suggest a set of acquisition strategies that ensure the optimization of the PTB.

]]>Mathematics doi: 10.3390/math6030042

Authors: Musavarah Sarwar Muhammad Akram

Real data and measures are usually uncertain and cannot be satisfactorily described by accurate real numbers. The imprecision and vagueness should be modeled and represented in data using the concept of fuzzy numbers. Fuzzy splines are proposed as an integrated approach to uncertainty in mathematical interpolation models. In the context of surface modeling, fuzzy tensor product Bézier surfaces are suitable for representing and simplifying both crisp and imprecise surface data with fuzzy numbers. The framework of this research paper is concerned with various properties of fuzzy tensor product surface patches by means of fuzzy numbers including fuzzy parametric curves, affine invariance, fuzzy tangents, convex hull and fuzzy iso-parametric curves. The fuzzification and defuzzification processes are applied to obtain the crisp Beziér curves and surfaces from fuzzy data points. The degree elevation and de Casteljau’s algorithms for fuzzy Bézier curves and fuzzy tensor product Bézier surfaces are studied in detail with numerical examples.

]]>Mathematics doi: 10.3390/math6030041

Authors: Malay Banerjee Nayana Mukherjee Vitaly Volpert

Spatiotemporal pattern formation in integro-differential equation models of interacting populations is an active area of research, which has emerged through the introduction of nonlocal intra- and inter-specific interactions. Stationary patterns are reported for nonlocal interactions in prey and predator populations for models with prey-dependent functional response, specialist predator and linear intrinsic death rate for predator species. The primary goal of our present work is to consider nonlocal consumption of resources in a spatiotemporal prey-predator model with bistable reaction kinetics for prey growth in the absence of predators. We derive the conditions of the Turing and of the spatial Hopf bifurcation around the coexisting homogeneous steady-state and verify the analytical results through extensive numerical simulations. Bifurcations of spatial patterns are also explored numerically.

]]>Mathematics doi: 10.3390/math6030040

Authors: Ali Rezaiguia Smail Kelaiaia

In this paper, we study a third-order differential inclusion with three-point boundary conditions. We prove the existence of a solution under convexity conditions on the multi-valued right-hand side; the proof is based on a nonlinear alternative of Leray-Schauder type. We also study the compactness of the set of solutions and establish some Filippov’s- type results for this problem.

]]>Mathematics doi: 10.3390/math6030039

Authors: Ricardo Bioni Liberalquino Maurizio Monge Stefano Galatolo Luigi Marangio

We consider a random dynamical system arising as a model of the behavior of a macrovariable related to a more complicated model of associative memory. This system can be seen as a small (stochastic and deterministic) perturbation of a determinstic system having two weak attractors which are destroyed after the perturbation. We show, with a computer aided proof, that the system has a kind of chaotic itineracy. Typical orbits are globally chaotic, while they spend a relatively long time visiting the attractor’s ruins.

]]>Mathematics doi: 10.3390/math6030038

Authors: Marouane Airouss Mohamed Tahiri Amale Lahlou Abdelhak Hassouni

In this paper, we have analyzed and tested the Expected Tail Loss (ETL) approach for the Value at Risk (VaR) on the Moroccan stock market portfolio. We have compared the results with the general approaches for the standard VaR, which has been the most suitable method for Moroccan stock investors up to now. These methods calculate the maximum loss that a portfolio is likely to experience over a given time span. Our work advances those modeling methods with supplementation by inputs from the ETL approach for application to the Moroccan stock market portfolio—the Moroccan All Shares Index (MASI). We calculate these indicators using several methods, according to an easy and fast implementation with a high-level probability and with accommodation for extreme risks; this is in order to numerically simulate and study their behavior to better understand investment opportunities and, thus, form a clear view of the Moroccan financial landscape.

]]>Mathematics doi: 10.3390/math6030037

Authors: Huseyin Isik Nurcan Gungor Choonkil Park Sun Jang

In this paper, we investigate the existence and uniqueness of a fixed point of almost contractions via simulation functions in metric spaces. Moreover, some examples and an application to integral equations are given to support availability of the obtained results.

]]>Mathematics doi: 10.3390/math6030036

Authors: Nikhil Pal Sudip Samanta Maia Martcheva Joydev Chattopadhyay

Migration is a key ecological process that enables connections between spatially separated populations. Previous studies have indicated that migration can stabilize chaotic ecosystems. However, the role of migration for two similar types of ecosystems, one chaotic and the other stable, has not yet been studied properly. In the present paper, we investigate the stability of ecological systems that are spatially separated but connected through migration. We consider two similar types of ecosystems that are coupled through migration, where one system shows chaotic dynamics, and other shows stable dynamics. We also note that the direction of the migration is bi-directional and is regulated by the population densities. We propose and analyze the coupled system. We also apply our proposed scheme to three different models. Our results suggest that bi-directional migration makes the coupled system more regular. We have performed numerical simulations to illustrate the dynamics of the coupled systems.

]]>Mathematics doi: 10.3390/math6030035

Authors: Stéphane Chrétien Juan-Pablo Ortega

The Hilbert metric is a widely used tool for analysing the convergence of Markov processes and the ergodic properties of deterministic dynamical systems. A useful representation formula for the Hilbert metric was given by Liverani. The goal of the present paper is to extend this formula to the non-compact and multidimensional setting with a different cone, taylored for sub-Gaussian tails.

]]>Mathematics doi: 10.3390/math6030034

Authors: Davide De Gaetano

Realized volatility, building on the theory of a simple continuous time process, has recently received attention as a nonparametric ex-post estimate of the return variation. This paper addresses the problem of parameter instability due to the presence of structural breaks in realized volatility in the context of three HAR-type models. The analysis is conducted on four major U.S. equity indices. More specifically, a recursive testing methodology is performed to evaluate the null hypothesis of constant parameters, and then, the performance of several forecast combinations based on different weighting schemes is compared in an out-of-sample variance forecasting exercise. The main findings are the following: (i) the hypothesis of constant model parameters is rejected for all markets under consideration; (ii) in all cases, the recursive forecasting approach, which is appropriate in the absence of structural changes, is outperformed by forecast combination schemes; and (iii) weighting schemes that assign more weight in most recent observations are superior in the majority of cases.

]]>Mathematics doi: 10.3390/math6030033

Authors: Mohammed Salaheldeen Abdelgader Chunxiang Wang Sarra Abdalrhman Mohammed

There are several chemical indices that have been introduced in theoretical chemistry to measure the properties of molecular topology, such as distance-based topological indices, degree-based topological indices and counting-related topological indices. Among the degree-based topological indices, the atom-bond connectivity ( A B C ) index and geometric–arithmetic ( G A ) index are the most important, because of their chemical significance. Certain physicochemical properties, such as the boiling point, stability and strain energy, of chemical compounds are correlated by these topological indices. In this paper, we study the molecular topological properties of some special graphs. The indices ( A B C ) , ( A B C 4 ) , ( G A ) and ( G A 5 ) of these special graphs are computed.

]]>Mathematics doi: 10.3390/math6030032

Authors: Chenkuan Li Kyle Clarkson

The goal of this paper is to investigate the following Abel’s integral equation of the second kind: y ( t ) + λ Γ ( α ) ∫ 0 t ( t − τ ) α − 1 y ( τ ) d τ = f ( t ) , ( t &gt; 0 ) and its variants by fractional calculus. Applying Babenko’s approach and fractional integrals, we provide a general method for solving Abel’s integral equation and others with a demonstration of different types of examples by showing convergence of series. In particular, we extend this equation to a distributional space for any arbitrary α ∈ R by fractional operations of generalized functions for the first time and obtain several new and interesting results that cannot be realized in the classical sense or by the Laplace transform.

]]>Mathematics doi: 10.3390/math6030031

Authors: Choonkil Park

Using the fixed point method, we prove the Hyers-Ulam stability of quasi-∗-multipliers on Banach ∗-algebras and unital C*-algebras, associated to bi-additive s-functional inequalities.

]]>Mathematics doi: 10.3390/math6030030

Authors: Choonkil Park

In this paper, we investigate C * -ternary biderivations and C * -ternary bihomomorphism in C * -ternary algebras, associated with bi-additive s-functional inequalities.

]]>Mathematics doi: 10.3390/math6020029

Authors: Hsien-Chung Wu

We propose the so-called fuzzy semi-metric space in which the symmetric condition is not assumed to be satisfied. In this case, there are four kinds of triangle inequalities that should be considered. The purpose of this paper is to study the common coincidence points and common fixed points in the newly proposed fuzzy semi-metric spaces endowed with the so-called ⋈-triangle inequality. The other three different kinds of triangle inequalities will be the future research, since they cannot be similarly investigated as the case of ⋈-triangle inequality.

]]>Mathematics doi: 10.3390/math6020028

Authors: Muhammad Akram Gulfam Shahzadi

Fuzzy graph theory is a conceptual framework to study and analyze the units that are intensely or frequently connected in a network. It is used to study the mathematical structures of pairwise relations among objects. An m-polar fuzzy (mF, for short) set is a useful notion in practice, which is used by researchers or modelings on real world problems that sometimes involve multi-agents, multi-attributes, multi-objects, multi-indexes and multi-polar information. In this paper, we apply the concept of mF sets to hypergraphs, and present the notions of regular mF hypergraphs and totally regular mF hypergraphs. We describe the certain properties of regular mF hypergraphs and totally regular mF hypergraphs. We discuss the novel applications of mF hypergraphs in decision-making problems. We also develop efficient algorithms to solve decision-making problems.

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