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Mathematics, Volume 6, Issue 7 (July 2018)

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Cover Story (view full-size image) Diversity is fundamental to ecology, and its measurement is essential for many ecosystem studies. [...] Read more.
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Open AccessArticle The Relation between the Probability of Collision-Free Broadcast Transmission in a Wireless Network and the Stirling Number of the Second Kind
Mathematics 2018, 6(7), 127; https://doi.org/10.3390/math6070127
Received: 5 June 2018 / Revised: 12 July 2018 / Accepted: 18 July 2018 / Published: 21 July 2018
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Abstract
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
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The broadcast performance of the 802.11 wireless protocol depends on several factors. One of the important factor is the number of nodes simultaneously contending for the shared channel. The Medium Access Control (MAC) technique of 802.11 is called the Distributed Coordination Function (DCF). DCF is a Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) scheme with binary slotted exponential backoff. A collision is the result of two or more stations transmitting simultaneously. Given the simplicity of the DCF scheme, it was adapted for Dedicated Short Range Communication (DSRC) based vehicular communication. A broadcast mechanism is used to disseminate emergency and safety related messages in a vehicular network. Emergency and safety related messages have a strict end-to-end latency of 100 ms and a Packet Delivery Ratio (PDR) of 90% and above. The PDR can be evaluated through the packet loss probability. The packet loss probability PL is given by, PL = 1 − (1Pe)(1PC), 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. Full article
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Open AccessArticle Eccentricity Based Topological Indices of an Oxide Network
Mathematics 2018, 6(7), 126; https://doi.org/10.3390/math6070126
Received: 5 June 2018 / Revised: 8 July 2018 / Accepted: 11 July 2018 / Published: 18 July 2018
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Abstract
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
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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
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
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Open AccessArticle Soft Rough Neutrosophic Influence Graphs with Application
Mathematics 2018, 6(7), 125; https://doi.org/10.3390/math6070125
Received: 28 June 2018 / Revised: 13 July 2018 / Accepted: 15 July 2018 / Published: 18 July 2018
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Abstract
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
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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
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Open AccessArticle Decomposition of Dynamical Signals into Jumps, Oscillatory Patterns, and Possible Outliers
Mathematics 2018, 6(7), 124; https://doi.org/10.3390/math6070124
Received: 31 January 2018 / Revised: 30 May 2018 / Accepted: 19 June 2018 / Published: 16 July 2018
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Abstract
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
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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
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Open AccessArticle On the Most Extended Modal Operator of First Type over Interval-Valued Intuitionistic Fuzzy Sets
Mathematics 2018, 6(7), 123; https://doi.org/10.3390/math6070123
Received: 30 May 2018 / Revised: 4 July 2018 / Accepted: 4 July 2018 / Published: 13 July 2018
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Abstract
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. Full article
(This article belongs to the Special Issue Fuzzy Mathematics)
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Open AccessArticle A Generalized Fejér–Hadamard Inequality for Harmonically Convex Functions via Generalized Fractional Integral Operator and Related Results
Mathematics 2018, 6(7), 122; https://doi.org/10.3390/math6070122
Received: 5 June 2018 / Revised: 7 July 2018 / Accepted: 9 July 2018 / Published: 11 July 2018
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Abstract
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.
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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
Open AccessArticle Some Generalized Dice Measures for Double-Valued Neutrosophic Sets and Their Applications
Mathematics 2018, 6(7), 121; https://doi.org/10.3390/math6070121
Received: 10 June 2018 / Revised: 7 July 2018 / Accepted: 9 July 2018 / Published: 10 July 2018
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Abstract
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
[...] Read more.
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
Open AccessArticle A Characterization of Projective Special Unitary Group PSU(3,3) and Projective Special Linear Group PSL(3,3) by NSE
Mathematics 2018, 6(7), 120; https://doi.org/10.3390/math6070120
Received: 17 May 2018 / Revised: 27 June 2018 / Accepted: 29 June 2018 / Published: 10 July 2018
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Abstract
Let G be a finite group and ω(G) be the set of element orders of G. Let kω(G) and mk be the number of elements of order k in G. Let n
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Let G be a finite group and ω(G) be the set of element orders of G. Let kω(G) and mk be the number of elements of order k in G. Let nse(G)={mk|kω(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 GH. Full article
(This article belongs to the Special Issue Computer Algebra in Scientific Computing)
Open AccessReview Ecological Diversity: Measuring the Unmeasurable
Mathematics 2018, 6(7), 119; https://doi.org/10.3390/math6070119
Received: 1 May 2018 / Revised: 4 July 2018 / Accepted: 5 July 2018 / Published: 10 July 2018
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Abstract
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
[...] Read more.
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
(This article belongs to the Special Issue Progress in Mathematical Ecology) Printed Edition available
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Open AccessArticle Global Stability of Within-Host Virus Dynamics Models with Multitarget Cells
Mathematics 2018, 6(7), 118; https://doi.org/10.3390/math6070118
Received: 26 April 2018 / Revised: 29 June 2018 / Accepted: 2 July 2018 / Published: 10 July 2018
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Abstract
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
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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. Full article
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Open AccessArticle Some Simultaneous Generalizations of Well-Known Fixed Point Theorems and Their Applications to Fixed Point Theory
Mathematics 2018, 6(7), 117; https://doi.org/10.3390/math6070117
Received: 10 June 2018 / Revised: 5 July 2018 / Accepted: 6 July 2018 / Published: 9 July 2018
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Abstract
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
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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
(This article belongs to the Special Issue Fixed Point Theory)
Open AccessArticle On p-Cyclic Orbital M-K Contractions in a Partial Metric Space
Mathematics 2018, 6(7), 116; https://doi.org/10.3390/math6070116
Received: 15 June 2018 / Revised: 29 June 2018 / Accepted: 6 July 2018 / Published: 9 July 2018
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Abstract
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
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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
Open AccessArticle Boundary Value Problem of the Operator ⊕k Related to the Biharmonic Operator and the Diamond Operator
Mathematics 2018, 6(7), 115; https://doi.org/10.3390/math6070115
Received: 26 May 2018 / Revised: 1 July 2018 / Accepted: 3 July 2018 / Published: 5 July 2018
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Abstract
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 k, where k=kk,
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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 k, where k=kk, 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 k, in which their derivations are also provided. To illustrate our findings, the example with prescribed boundary conditions is exhibited. Full article
Open AccessArticle Complex Symmetric Formulation of Maxwell Equations for Fields and Potentials
Mathematics 2018, 6(7), 114; https://doi.org/10.3390/math6070114
Received: 14 May 2018 / Revised: 19 June 2018 / Accepted: 19 June 2018 / Published: 3 July 2018
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Abstract
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
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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. Full article
Open AccessFeature PaperArticle Analysis of the Incidence of Poxvirus on the Dynamics between Red and Grey Squirrels
Mathematics 2018, 6(7), 113; https://doi.org/10.3390/math6070113
Received: 23 March 2018 / Revised: 29 May 2018 / Accepted: 18 June 2018 / Published: 1 July 2018
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Abstract
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
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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
(This article belongs to the Special Issue Progress in Mathematical Ecology) Printed Edition available
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Open AccessArticle On Generalized Roughness in LA-Semigroups
Mathematics 2018, 6(7), 112; https://doi.org/10.3390/math6070112
Received: 30 May 2018 / Revised: 16 June 2018 / Accepted: 25 June 2018 / Published: 27 June 2018
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Abstract
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
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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
(This article belongs to the Special Issue Fuzzy Mathematics)
Open AccessFeature PaperArticle Symmetries and Invariants for Non-Hermitian Hamiltonians
Mathematics 2018, 6(7), 111; https://doi.org/10.3390/math6070111
Received: 1 June 2018 / Revised: 22 June 2018 / Accepted: 24 June 2018 / Published: 27 June 2018
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Abstract
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 [
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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=HA. 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
(This article belongs to the Special Issue Time and Time Dependence in Quantum Mechanics)
Open AccessArticle The Emergence of Fuzzy Sets in the Decade of the Perceptron—Lotfi A. Zadeh’s and Frank Rosenblatt’s Research Work on Pattern Classification
Mathematics 2018, 6(7), 110; https://doi.org/10.3390/math6070110
Received: 25 May 2018 / Revised: 16 June 2018 / Accepted: 19 June 2018 / Published: 26 June 2018
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Abstract
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,
[...] Read more.
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
(This article belongs to the Special Issue Fuzzy Mathematics)
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Open AccessArticle An Implicit Hybrid Method for Solving Fractional Bagley-Torvik Boundary Value Problem
Mathematics 2018, 6(7), 109; https://doi.org/10.3390/math6070109
Received: 31 May 2018 / Revised: 13 June 2018 / Accepted: 15 June 2018 / Published: 25 June 2018
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Abstract
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
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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
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications)
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Open AccessArticle Near Fixed Point Theorems in the Space of Fuzzy Numbers
Mathematics 2018, 6(7), 108; https://doi.org/10.3390/math6070108
Received: 24 May 2018 / Revised: 19 June 2018 / Accepted: 21 June 2018 / Published: 25 June 2018
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Abstract
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
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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
(This article belongs to the Special Issue Nonlinear Analysis Using Fuzzy Mathematics)
Open AccessArticle Recognition of M × M by Its Complex Group Algebra Where M Is a Simple K3-Group
Mathematics 2018, 6(7), 107; https://doi.org/10.3390/math6070107
Received: 4 April 2018 / Revised: 1 May 2018 / Accepted: 9 May 2018 / Published: 25 June 2018
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Abstract
In this paper we prove that if M is a simple K3-group, then M×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×
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In this paper we prove that if M is a simple K3-group, then M×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×M is uniquely determined by the structure of its complex group algebra. Full article
Open AccessArticle Fuzzy Semi-Metric Spaces
Mathematics 2018, 6(7), 106; https://doi.org/10.3390/math6070106
Received: 27 May 2018 / Revised: 15 June 2018 / Accepted: 19 June 2018 / Published: 22 June 2018
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Abstract
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. Full article
(This article belongs to the Special Issue Fuzzy Mathematics)
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