Mathematics, Volume 6, Issue 7 (July 2018) – 22 articles

The broadcast performance of the 802.11 wireless protocol depends on several factors. One of the important factor is the number of nodes simultaneously contending for the shared channel. The Medium Access Control (MAC) technique of 802.11 is called the Distributed Coordination Function (DCF). DCF is a Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) scheme with binary slotted exponential backoff. A collision is the result of two or more stations transmitting simultaneously. Given the simplicity of the DCF scheme, it was adapted for Dedicated Short Range Communication (DSRC) based vehicular communication. A broadcast mechanism is used to disseminate emergency and safety related messages in a vehicular network. Emergency and safety related messages have a strict end-to-end latency of 100 ms and a Packet Delivery Ratio (PDR) of 90% and above. The PDR can be evaluated through the packet loss probability. The packet loss probability $P_L$ is given by, $P_L$ = 1 − ($1-P_e$)($1-P_C$), where $P_e$ is the probability of channel error and $P_C$ is the probability of collision. $P_e$ depends on several environmental and operating factors and thus cannot be improved. The only way to reduce $P_L$ is by reducing $P_C$. Currently, expensive radio hardware are used to measure $P_L$. Several adaptive algorithms are available to reduce $P_C$. In this paper, we establish a closed relation between $P_C$ 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 $P_C$ and $P_L$ are known, one can precisely determine the distribution of $P_e$. Full article

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. Full article

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. Full article

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. Full article

In this paper, we obtain a version of the Fejér–Hadamard inequality for harmonically convex functions via generalized fractional integral operator. In addition, we establish an integral identity and some Fejér–Hadamard type integral inequalities for harmonically convex functions via a generalized fractional integral operator. Being generalizations, our results reproduce some known results. Full article

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. Full article

Let G be a finite group and $\omega \left(G\right)$ be the set of element orders of G. Let $k\in \omega \left(G\right)$ and $m_k$ be the number of elements of order k in G. Let $nse\left(G\right)=\{m_k|k\in \omega \left(G\right)\}$. 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 $G\cong H$. Full article

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’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. Full article

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 ${\mathcal{R}}_0$. 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. Full article

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’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. Full article

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. Full article

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’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. Full article

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) L^p-spaces in nonlinear Maxwell equations. Full article

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. Full article

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. Full article

We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation $AH{A}^{†}$ 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&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. Full article

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. Full article

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. Full article

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. Full article

In this paper we prove that if M is a simple $K_3$-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. Full article