Mathematics, Volume 7, Issue 6 (June 2019) – 83 articles

Humans breathe air into the respiratory system through the trachea, but with all the pollutants in our environment (both outside and inside), the air we breathe may not be clean. When that is so, the respiratory system secretes mucus to trap dirt that is inhaled through the nostrils. The respiratory tract contains hair-like structures in the epithelial tissue, called cilia: These wave back and forth to help expel particles of dust, dirt, mucus, and contaminants from the body. Cilia are found in this layer (a porous medium) and the fluid in this layer is called the periciliary layer (PCL). This study aims to determine the velocity of the PCL fluid flow in motile cilia. Usually, fluids move due to pressure changes, but in this study, the velocity of solids or of the cilia moves the PCL fluid. Stokes-Brinkman equations are used to determine the velocity of PCL fluid flow when cilia form an angle with the horizontal plane. The Beavers and Joseph boundary condition is applied in this study. The asymptotic expansion method is adapted in order to determine the velocity of PCL from the movement of the cilia. Full article

In this paper, we introduce the notions of $(\in ,\in )$ -fuzzy positive implicative filters and $(\in ,\in \vee q)$ -fuzzy positive implicative filters in hoops and investigate their properties. We also define some equivalent definitions of them, and then we use the congruence relation on hoop defined in blue[Aaly Kologani, M.; Mohseni Takallo, M.; Kim, H.S. Fuzzy filters of hoops based on fuzzy points. Mathematics. 2019, 7, 430; doi:10.3390/math7050430] by using an $(\in ,\in )$ -fuzzy filter in hoop. We show that the quotient structure of this relation is a Brouwerian semilattice. Full article

The basic concepts of exterior calculus for space–time multivectors are presented: Interior and exterior products, interior and exterior derivatives, oriented integrals over hypersurfaces, circulation and flux of multivector fields. Two Stokes theorems relating the exterior and interior derivatives with circulation and flux, respectively, are derived. As an application, it is shown how the exterior-calculus space–time formulation of the electromagnetic Maxwell equations and Lorentz force recovers the standard vector-calculus formulations, in both differential and integral forms. Full article

The neutrosophic triplets in neutrosophic rings $⟨Q\cup I⟩$ and $⟨R\cup I⟩$ are investigated in this paper. However, non-trivial neutrosophic triplets are not found in $⟨Z\cup I⟩$ . In the neutrosophic ring of integers $Z\backslash \{0,1\}$ , no element has inverse in Z. It is proved that these rings can contain only three types of neutrosophic triplets, these collections are distinct, and these collections form a torsion free abelian group as triplets under component wise product. However, these collections are not even closed under component wise addition. Full article

A generic family of optimal sixteenth-order multiple-root finders are theoretically developed from general settings of weight functions under the known multiplicity. Special cases of rational weight functions are considered and relevant coefficient relations are derived in such a way that all the extraneous fixed points are purely imaginary. A number of schemes are constructed based on the selection of desired free parameters among the coefficient relations. Numerical and dynamical aspects on the convergence of such schemes are explored with tabulated computational results and illustrated attractor basins. Overall conclusion is drawn along with future work on a different family of optimal root-finders. Full article

Nowadays, resource utilization and management are very important for businesses. They try to make a profit by providing high levels of efficiency from available sources. Their labor force is one of these sources. Therefore, scheduling based on personnel satisfaction has become an important problem in recent years. In this study, a case study was carried out in a glass factory in Ankara which has 7 department and 80 personnel. The aim of the study is to provide better service by distributing personnel to shifts in a fair and balanced manner. Assignment points are different for the departments where the personnel will work. Every personnel member is assigned to the department as best as possible. A goal programming method was used, and the results were better than those obtained using other methods. Full article

Based on the very recent work by Shehu and Agbebaku in Comput. Appl. Math. 2017, we introduce an extension of their iterative algorithm by combining it with inertial extrapolation for solving split inclusion problems and fixed point problems. Under suitable conditions, we prove that the proposed algorithm converges strongly to common elements of the solution set of the split inclusion problems and fixed point problems. Full article

The paper proves a unified analysis for finite-time anti-synchronization of a class of integer-order and fractional-order chaotic systems. We establish an effective controller to ensure that the chaotic system with unknown parameters achieves anti-synchronization in finite time under our controller. Then, we apply our results to the integer-order and fractional-order Lorenz system, respectively. Finally, numerical simulations are presented to show the feasibility of the proposed control scheme. At the same time, through the numerical simulation results, it is show that for the Lorenz chaotic system, when the order is greater, the more quickly is anti-synchronization achieved. Full article

This paper studies the parameter identification problems for multivariable output-error-like systems with colored noises. Based on the hierarchical identification principle, the original system is decomposed into several subsystems. However, each subsystem contains the same parameter vector, which leads to redundant computation. By taking the average of the parameter estimation vectors of each subsystem, a partially-coupled subsystem recursive generalized extended least squares (PC-S-RGELS) algorithm is presented to cut down the redundant parameter estimates. Furthermore, a partially-coupled recursive generalized extended least squares (PC-RGELS) algorithm is presented to further reduce the computational cost and the redundant estimates by using the coupling identification concept. Finally, an example indicates the effectiveness of the derived algorithms. Full article

Let $\mathsf{\Gamma }$ be a numerical semigroup. We associate an undirected graph $G\left(\mathsf{\Gamma }\right)$ with a numerical semigroup $\mathsf{\Gamma }$ with vertex set $\{v_i:i\in \mathbb{N}\backslash \mathsf{\Gamma }\}$ and edge set $\{v_i{v}_j\iff i+j\in \mathsf{\Gamma }\}$ . In this article, we discuss the connectedness, diameter, girth, and some other related properties of the graph $G\left(\mathsf{\Gamma }\right)$ . Full article

The extending structures problem for strict Lie 2-algebras is studied. To provide the theoretical answer to this problem, this paper introduces the unified product of a given strict Lie 2-algebra $\mathfrak{g}$ and 2-vector space V. The unified product includes some interesting products such as semi-direct product, crossed product, and bicrossed product. The paper focuses on crossed and bicrossed products, which give the answer to the extension problem and factorization problem, respectively. Full article

We consider two different time-inhomogeneous diffusion processes useful to model the evolution of a population in a random environment. The first is a Gompertz-type diffusion process with time-dependent growth intensity, carrying capacity and noise intensity, whose conditional median coincides with the deterministic solution. The second is a shifted-restricted Gompertz-type diffusion process with a reflecting condition in zero state and with time-dependent regulation functions. For both processes, we analyze the transient and the asymptotic behavior of the transition probability density functions and their conditional moments. Particular attention is dedicated to the first-passage time, by deriving some closed form for its density through special boundaries. Finally, special cases of periodic regulation functions are discussed. Full article

This article is a review of problems and difficulties arising in the construction of fractional-dynamic analogs of standard models by using fractional calculus. These fractional generalizations allow us to take into account the effects of memory and non-locality, distributed lag, and scaling. We formulate rules (principles) for constructing fractional generalizations of standard models, which were described by differential equations of integer order. Important requirements to building fractional generalization of dynamical models (the rules for “fractional-dynamic generalizers”) are represented as the derivability principle, the multiplicity principle, the solvability and correspondence principles, and the interpretability principle. The characteristic properties of fractional derivatives of non-integer order are the violation of standard rules and properties that are fulfilled for derivatives of integer order. These non-standard mathematical properties allow us to describe non-standard processes and phenomena associated with non-locality and memory. However, these non-standard properties lead to restrictions in the sequential and self-consistent construction of fractional generalizations of standard models. In this article, we give examples of problems arising due to the non-standard properties of fractional derivatives in construction of fractional generalizations of standard dynamic models in economics. Full article

Fuzzy graph theory is a useful and well-known tool to model and solve many real-life optimization problems. Since real-life problems are often uncertain due to inconsistent and indeterminate information, it is very hard for an expert to model those problems using a fuzzy graph. A neutrosophic graph can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problem, where fuzzy graphs may fail to reveal satisfactory results. The concepts of the regularity and degree of a node play a significant role in both the theory and application of graph theory in the neutrosophic environment. In this work, we describe the utility of the regular neutrosophic graph and bipartite neutrosophic graph to model an assignment problem, a road transport network, and a social network. For this purpose, we introduce the definitions of the regular neutrosophic graph, star neutrosophic graph, regular complete neutrosophic graph, complete bipartite neutrosophic graph, and regular strong neutrosophic graph. We define the $d_m$ - and $t{d}_m$ -degrees of a node in a regular neutrosophic graph. Depending on the degree of the node, this paper classifies the regularity of a neutrosophic graph into three types, namely $d_m$ -regular, $t{d}_m$ -regular, and m-highly irregular neutrosophic graphs. We present some theorems and properties of those regular neutrosophic graphs. The concept of an m-highly irregular neutrosophic graph on cycle and path graphs is also investigated in this paper. The definition of busy and free nodes in a regular neutrosophic graph is presented here. We introduce the idea of the $\mu$ -complement and h-morphism of a regular neutrosophic graph. Some properties of complement and isomorphic regular neutrosophic graphs are presented here. Full article

This work deals with a new modified version of the Adomian-Rach decomposition method (MDM). The MDM is based on combining a series solution and decomposition method for solving nonlinear differential equations with Adomian polynomials for nonlinearities. With application to a class of nonlinear oscillators known as the Lienard-type equations, convergence and error analysis are discussed. Several physical problems modeled by Lienard-type equations are considered to illustrate the effectiveness, performance and reliability of the method. In comparison to the 4th Runge-Kutta method (RK4), highly accurate solutions on a large domain are obtained. Full article

Complex fuzzy set (CFS) is a recent development in the field of fuzzy set (FS) theory. The significance of CFS lies in the fact that CFS assigned membership grades from a unit circle in the complex plane, i.e., in the form of a complex number whose amplitude term belongs to a $[0,1]$ interval. The interval-valued complex fuzzy set (IVCFS) is one of the extensions of the CFS in which the amplitude term is extended from the real numbers to the interval-valued numbers. The novelty of IVCFS lies in its larger range comparative to CFS. We often use fuzzy distance measures to solve some problems in our daily life. Hence, this paper develops some series of distance measures between IVCFSs by using Hamming and Euclidean metrics. The boundaries of these distance measures for IVCFSs are obtained. Finally, we study two geometric properties include rotational invariance and reflectional invariance of these distance measures. Full article

In the paper, we introduce some subclasses of harmonic mapping, the analytic part of which is related to general starlike (or convex) functions with a symmetric conjecture point defined by subordination. Using the conditions satisfied by the analytic part, we obtain the integral expressions, the coefficient estimates, distortion estimates and the growth estimates of the co-analytic part g, and Jacobian estimates, the growth estimates and covering theorem of the harmonic function f. Through the above research, the geometric properties of the classes are obtained. In particular, we draw figures of extremum functions to better reflect the geometric properties of the classes. For the first time, we introduce and obtain the properties of harmonic univalent functions with respect to symmetric conjugate points. The conclusion of this paper extends the original research. Full article

We introduce log-preinvex and log-invex functions on a Riemannian manifold. Some properties and relationships of these functions are discussed. A characterization for the existence of a global minimum point of a mathematical programming problem is presented. Moreover, a mean value inequality under geodesic log-preinvexity is extended to Cartan-Hadamard manifolds. Full article

Connectivity is a critical parameter that can measure the reliability of networks. Let $Q\subseteq V\left(G\right)$ be a vertex set. If $G-Q$ is disconnected and every component of $G-Q$ contains at least $k+1$ vertices, then Q is an extra-cut. The number of vertices in the smallest extra-cut is the extraconnectivity ${\kappa }_k\left(G\right)$ . Suppose $\omega \left(G\right)$ is the number of components of G and $W\subseteq V\left(G\right)$ ; if $\omega (G-W)\ge t$ , then w is a t-component cut of G. The number of vertices in the least t-component cut is the t-component connectivity $c{\kappa }_t\left(G\right)$ of G. The t-component edge connectivity $c{\lambda }_t\left(G\right)$ is defined similarly. In this note, we study the BC networks and obtain the t-component (edge) connectivity of bijective connection networks for some t. Full article

In this paper, a new collocation method based on Haar wavelet is developed for numerical solution of Riccati type differential equations with non-integer order. The fractional derivatives are considered in the Caputo sense. The method is applied to one test problem. The maximum absolute estimated error functions are calculated, and the performance of the process is demonstrated by calculating the maximum absolute estimated error functions for a distinct number of nodal points. The results show that the method is applicable and efficient. Full article

In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm–Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively. Full article

In this paper, we are concerned with the non-commutativity measure of quantum discord. We first present an explicit expression of the non-commutativity measure of quantum discord in the two-qubit case. Then we compare the geometric quantum discords for two dynamic models with their non-commutativity measure of quantum discords. Furthermore, we show that the results conducted by the non-commutativity measure of quantum discord are different from those conducted by both or one of the Hilbert-Schmidt distance discord and trace distance discord. These intrinsic differences indicate that the non-commutativity measure of quantum discord is incompatible with at least one of the well-known geometric quantum discords in the quantitative and qualitative representation of quantum correlations. Full article

In this paper, the problems of basic fuzzy event space and of probability fuzzy space are studied. Firstly, the concepts of basic fuzzy event, fuzzy event and basic fuzzy event space are defined, related properties are investigated, and some results that will be used in the next study of probability fuzzy space are obtained. Then, the definitions of the probability function for fuzzy events and probability fuzzy space are given, some properties of the defined probability function are obtained. In addition, some models of probability distribution of probability fuzzy space based on a known probability space are proposed, and some examples are given to show the usability of the proposed models of probability distribution. Full article

The behaviour of many dynamic real phenomena shows different phases, with each one following a sigmoidal type pattern. This requires studying sigmoidal curves with more than one inflection point. In this work, a diffusion process is introduced whose mean function is a curve of this type, concretely a transformation of the well-known Gompertz model after introducing in its expression a polynomial term. The maximum likelihood estimation of the parameters of the model is studied, and various criteria are provided for the selection of the degree of the polynomial when real situations are addressed. Finally, some simulated examples are presented. Full article

In this paper, we study the semilocal convergence of the multi-point variant of Jarratt method under two different mild situations. The first one is the assumption that just a second-order Fréchet derivative is bounded instead of third-order. In addition, in the next one, the bound of the norm of the third order Fréchet derivative is assumed at initial iterate rather than supposing it on the domain of the nonlinear operator and it also satisfies the local $\omega$ -continuity condition in order to prove the convergence, existence-uniqueness followed by a priori error bound. During the study, it is noted that some norms and functions have to recalculate and its significance can be also seen in the numerical section. Full article

We mainly study the transcendental entire solutions of the differential equation $f^n(z)+P(f)=p_1{e}^{{\alpha }_{1}z}+p_2{e}^{{\alpha }_{2}z}$ , where $p_1$ , $p_2$ , ${\alpha }_1$ and ${\alpha }_2$ are nonzero constants satisfying ${\alpha }_1\ne {\alpha }_2$ and $P(f)$ is a differential polynomial in f of degree $n-1$ . We improve Chen and Gao’s results and partially answer a question proposed by Li (J. Math. Anal. Appl. 375 (2011), pp. 310–319). Full article

In this manuscript, we introduce a new type of contraction, bilateral contraction which merges two significant approaches in the fixed point theory: Caristi type contractions and Jaggi type contractions. The principal aim of the main result is to enrich the literature by combining the techniques of the mentioned two celebrated results that belong to Jaggi and Caristi. We consider an example to indicate the validity and genuine nature of the main result. Full article