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Mathematics, Volume 5, Issue 2 (June 2017) – 16 articles

Cover Story (view full-size image): The graphic reveals how the vertices of an icosahedral symmetry group can be represented as ordered pairs (couplets) in Dirichlet mathematical space. This representation encodes the doubling of physical dimensions and a possible route to understand higher dimensional symmetry groups that are relevant in several theories of modern physics to explain laws of nature. The special significance of aperiodic structures in this context is presented and discussed in this paper. View this paper
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Open AccessArticle
Lie Symmetries, Optimal System and Invariant Reductions to a Nonlinear Timoshenko System
Mathematics 2017, 5(2), 34; https://doi.org/10.3390/math5020034 - 17 Jun 2017
Cited by 2 | Viewed by 1617
Abstract
Lie symmetries and their Lie group transformations for a class of Timoshenko systems are presented. The class considered is the class of nonlinear Timoshenko systems of partial differential equations (PDEs). An optimal system of one-dimensional sub-algebras of the corresponding Lie algebra is derived. [...] Read more.
Lie symmetries and their Lie group transformations for a class of Timoshenko systems are presented. The class considered is the class of nonlinear Timoshenko systems of partial differential equations (PDEs). An optimal system of one-dimensional sub-algebras of the corresponding Lie algebra is derived. All possible invariant variables of the optimal system are obtained. The corresponding reduced systems of ordinary differential equations (ODEs) are also provided. All possible non-similar invariant conditions prescribed on invariant surfaces under symmetry transformations are given. As an application, explicit solutions of the system are given where the beam is hinged at one end and free at the other end. Full article
Open AccessArticle
An Analysis on the Fractional Asset Flow Differential Equations
Mathematics 2017, 5(2), 33; https://doi.org/10.3390/math5020033 - 16 Jun 2017
Cited by 2 | Viewed by 2254
Abstract
The asset flow differential equation (AFDE) is the mathematical model that plays an essential role for planning to predict the financial behavior in the market. In this paper, we introduce the fractional asset flow differential equations (FAFDEs) based on the Liouville-Caputo derivative. We [...] Read more.
The asset flow differential equation (AFDE) is the mathematical model that plays an essential role for planning to predict the financial behavior in the market. In this paper, we introduce the fractional asset flow differential equations (FAFDEs) based on the Liouville-Caputo derivative. We prove the existence and uniqueness of a solution for the FAFDEs. Furthermore, the stability analysis of the model is investigated and the numerical simulation is accordingly performed to support the proposed model. Full article
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Open AccessArticle
Metrization Theorem for Uniform Loops with the Invertibility Property
Mathematics 2017, 5(2), 32; https://doi.org/10.3390/math5020032 - 02 Jun 2017
Viewed by 1405
Abstract
In this paper, we have proved a metrization theorem that gives the sufficient conditions for a uniform IP-loop X to be metrizable by a left-invariant metric. It is shown that by consideration of topological IP-loop dual to X we obtain an analogical theorem [...] Read more.
In this paper, we have proved a metrization theorem that gives the sufficient conditions for a uniform IP-loop X to be metrizable by a left-invariant metric. It is shown that by consideration of topological IP-loop dual to X we obtain an analogical theorem for the case of the right-invariant metric. Full article
Open AccessArticle
Nonlinear Gronwall–Bellman Type Inequalities and Their Applications
Mathematics 2017, 5(2), 31; https://doi.org/10.3390/math5020031 - 31 May 2017
Cited by 4 | Viewed by 1880
Abstract
In this paper, some nonlinear Gronwall–Bellman type inequalities are established. Then, the obtained results are applied to study the Hyers–Ulam stability of a fractional differential equation and the boundedness of solutions to an integral equation, respectively. Full article
Open AccessArticle
Coincidence Points of a Sequence of Multivalued Mappings in Metric Space with a Graph
Mathematics 2017, 5(2), 30; https://doi.org/10.3390/math5020030 - 26 May 2017
Viewed by 1477
Abstract
In this article the coincidence points of a self map and a sequence of multivalued maps are found in the settings of complete metric space endowed with a graph. A novel result of Asrifa and Vetrivel is generalized and as an application we [...] Read more.
In this article the coincidence points of a self map and a sequence of multivalued maps are found in the settings of complete metric space endowed with a graph. A novel result of Asrifa and Vetrivel is generalized and as an application we obtain an existence theorem for a special type of fractional integral equation. Moreover, we establish a result on the convergence of successive approximation of a system of Bernstein operators on a Banach space. Full article
Open AccessArticle
Emergence of an Aperiodic Dirichlet Space from the Tetrahedral Units of an Icosahedral Internal Space
Mathematics 2017, 5(2), 29; https://doi.org/10.3390/math5020029 - 26 May 2017
Cited by 2 | Viewed by 3036
Abstract
We present the emergence of a root system in six dimensions from the tetrahedra of an icosahedral core known as the 20-group (20G) within the framework of Clifford’s geometric algebra. Consequently, we establish a connection between a three-dimensional icosahedral seed, a six-dimensional (6D) [...] Read more.
We present the emergence of a root system in six dimensions from the tetrahedra of an icosahedral core known as the 20-group (20G) within the framework of Clifford’s geometric algebra. Consequently, we establish a connection between a three-dimensional icosahedral seed, a six-dimensional (6D) Dirichlet quantized host and a higher dimensional lattice structure. The 20G, owing to its icosahedral symmetry, bears the signature of a 6D lattice that manifests in the Dirichlet integer representation. We present an interpretation whereby the three-dimensional 20G can be regarded as the core substratum from which the higher dimensional lattices emerge. This emergent geometry is based on an induction principle supported by the Clifford multi-vector formalism of three-dimensional (3D) Euclidean space. This lays a geometric framework for understanding several physics theories related to S U ( 5 ) , E 6 , E 8 Lie algebras and their composition with the algebra associated with the even unimodular lattice in R 3 , 1 . The construction presented here is inspired by Penrose’s three world model. Full article
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Open AccessArticle
A Two-Stage Method for Piecewise-Constant Solution for Fredholm Integral Equations of the First Kind
Mathematics 2017, 5(2), 28; https://doi.org/10.3390/math5020028 - 22 May 2017
Cited by 1 | Viewed by 1840
Abstract
A numerical method is proposed for estimating piecewise-constant solutions for Fredholm integral equations of the first kind. Two functionals, namely the weighted total variation (WTV) functional and the simplified Modica-Mortola (MM) functional, are introduced. The solution procedure consists of two stages. In the [...] Read more.
A numerical method is proposed for estimating piecewise-constant solutions for Fredholm integral equations of the first kind. Two functionals, namely the weighted total variation (WTV) functional and the simplified Modica-Mortola (MM) functional, are introduced. The solution procedure consists of two stages. In the first stage, the WTV functional is minimized to obtain an approximate solution f TV * . In the second stage, the simplified MM functional is minimized to obtain the final result by using the damped Newton (DN) method with f TV * as the initial guess. The numerical implementation is given in detail, and numerical results of two examples are presented to illustrate the efficiency of the proposed approach. Full article
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Open AccessArticle
Analysis of Magneto-hydrodynamics Flow and Heat Transfer of a Viscoelastic Fluid through Porous Medium in Wire Coating Analysis
Mathematics 2017, 5(2), 27; https://doi.org/10.3390/math5020027 - 16 May 2017
Cited by 12 | Viewed by 2119
Abstract
Wire coating process is a continuous extrusion process for primary insulation of conducting wires with molten polymers for mechanical strength and protection in aggressive environments. Nylon, polysulfide, low/high density polyethylene (LDPE/HDPE) and plastic polyvinyl chloride (PVC) are the common and important plastic resin [...] Read more.
Wire coating process is a continuous extrusion process for primary insulation of conducting wires with molten polymers for mechanical strength and protection in aggressive environments. Nylon, polysulfide, low/high density polyethylene (LDPE/HDPE) and plastic polyvinyl chloride (PVC) are the common and important plastic resin used for wire coating. In the current study, wire coating is performed using viscoelastic third grade fluid in the presence of applied magnetic field and porous medium. The governing equations are first modeled and then solved analytically by utilizing the homotopy analysis method (HAM). The convergence of the series solution is established. A numerical technique called ND-solve method is used for comparison and found good agreement. The effect of pertinent parameters on the velocity field and temperature profile is shown with the help of graphs. It is observed that the velocity profiles increase as the value of viscoelastic third grade parameter β increase and decrease as the magnetic parameter M and permeability parameter K increase. It is also observed that the temperature profiles increases as the Brinkman number B r , permeability parameter K , magnetic parameter M and viscoelastic third grade parameter (non-Newtonian parameter) β increase. Full article
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Open AccessArticle
A New Variational Iteration Method for a Class of Fractional Convection-Diffusion Equations in Large Domains
Mathematics 2017, 5(2), 26; https://doi.org/10.3390/math5020026 - 11 May 2017
Cited by 6 | Viewed by 1964
Abstract
In this paper, we introduced a new generalization method to solve fractional convection–diffusion equations based on the well-known variational iteration method (VIM) improved by an auxiliary parameter. The suggested method was highly effective in controlling the convergence region of the approximate solution. By [...] Read more.
In this paper, we introduced a new generalization method to solve fractional convection–diffusion equations based on the well-known variational iteration method (VIM) improved by an auxiliary parameter. The suggested method was highly effective in controlling the convergence region of the approximate solution. By solving some fractional convection–diffusion equations with a propounded method and comparing it with standard VIM, it was concluded that complete reliability, efficiency, and accuracy of this method are guaranteed. Additionally, we studied and investigated the convergence of the proposed method, namely the VIM with an auxiliary parameter. We also offered the optimal choice of the auxiliary parameter in the proposed method. It was noticed that the approach could be applied to other models of physics. Full article
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Open AccessArticle
Discrete-Time Fractional Optimal Control
Mathematics 2017, 5(2), 25; https://doi.org/10.3390/math5020025 - 19 Apr 2017
Cited by 10 | Viewed by 2350
Abstract
A formulation and solution of the discrete-time fractional optimal control problem in terms of the Caputo fractional derivative is presented in this paper. The performance index (PI) is considered in a quadratic form. The necessary and transversality conditions are obtained using a Hamiltonian [...] Read more.
A formulation and solution of the discrete-time fractional optimal control problem in terms of the Caputo fractional derivative is presented in this paper. The performance index (PI) is considered in a quadratic form. The necessary and transversality conditions are obtained using a Hamiltonian approach. Both the free and fixed final state cases have been considered. Numerical examples are taken up and their solution technique is presented. Results are produced for different values of α . Full article
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Open AccessArticle
Fixed Points of Set Valued Mappings in Terms of Start Point on a Metric Space Endowed with a Directed Graph
Mathematics 2017, 5(2), 24; https://doi.org/10.3390/math5020024 - 19 Apr 2017
Cited by 3 | Viewed by 1720
Abstract
In the present article, we introduce the new concept of start point in a directed graph and provide the characterizations required for a directed graph to have a start point. We also define the notion of a self path set valued map and [...] Read more.
In the present article, we introduce the new concept of start point in a directed graph and provide the characterizations required for a directed graph to have a start point. We also define the notion of a self path set valued map and establish its relation with start point in the setting of a metric space endowed with a directed graph. Further, some fixed point theorems for set valued maps have been proven in this context. A version of the Knaster–Tarski theorem has also been established using our results. Full article
(This article belongs to the Special Issue Fixed Point Theorems and Applications)
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Open AccessArticle
Best Proximity Point Results in Non-Archimedean Modular Metric Space
Mathematics 2017, 5(2), 23; https://doi.org/10.3390/math5020023 - 05 Apr 2017
Cited by 2 | Viewed by 1309
Abstract
In this paper, we introduce the new notion of Suzuki-type (α,β,θ,γ)-contractive mapping and investigate the existence and uniqueness of the best proximity point for such mappings in non-Archimedean modular metric space using the weak [...] Read more.
In this paper, we introduce the new notion of Suzuki-type ( α , β , θ , γ ) -contractive mapping and investigate the existence and uniqueness of the best proximity point for such mappings in non-Archimedean modular metric space using the weak P λ -property. Meanwhile, we present an illustrative example to emphasize the realized improvements. These obtained results extend and improve certain well-known results in the literature. Full article
Open AccessArticle
On Optimal Fuzzy Best Proximity Coincidence Points of Proximal Contractions Involving Cyclic Mappings in Non-Archimedean Fuzzy Metric Spaces
Mathematics 2017, 5(2), 22; https://doi.org/10.3390/math5020022 - 01 Apr 2017
Cited by 8 | Viewed by 1738
Abstract
The main objective of this paper is to deal with some properties of interest in two types of fuzzy ordered proximal contractions of cyclic self-mappings T integrated in a pair (g,T) of mappings. In particular, g is a non-contractive [...] Read more.
The main objective of this paper is to deal with some properties of interest in two types of fuzzy ordered proximal contractions of cyclic self-mappings T integrated in a pair ( g , T ) of mappings. In particular, g is a non-contractive fuzzy self-mapping, in the framework of non-Archimedean ordered fuzzy complete metric spaces and T is a p -cyclic proximal contraction. Two types of such contractions (so called of type I and of type II) are dealt with. In particular, the existence, uniqueness and limit properties for sequences to optimal fuzzy best proximity coincidence points are investigated for such pairs of mappings. Full article
(This article belongs to the Special Issue Fixed Point Theorems and Applications)
Open AccessArticle
On Some Extended Block Krylov Based Methods for Large Scale Nonsymmetric Stein Matrix Equations
Mathematics 2017, 5(2), 21; https://doi.org/10.3390/math5020021 - 27 Mar 2017
Cited by 4 | Viewed by 1542
Abstract
In the present paper, we consider the large scale Stein matrix equation with a low-rank constant term AXBX+EFT=0. These matrix equations appear in many applications in discrete-time control problems, filtering and image [...] Read more.
In the present paper, we consider the large scale Stein matrix equation with a low-rank constant term A X B X + E F T = 0 . These matrix equations appear in many applications in discrete-time control problems, filtering and image restoration and others. The proposed methods are based on projection onto the extended block Krylov subspace with a Galerkin approach (GA) or with the minimization of the norm of the residual. We give some results on the residual and error norms and report some numerical experiments. Full article
(This article belongs to the Special Issue Numerical Linear Algebra with Applications)
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Open AccessArticle
F-Harmonic Maps between Doubly Warped Product Manifolds
Mathematics 2017, 5(2), 20; https://doi.org/10.3390/math5020020 - 23 Mar 2017
Cited by 3 | Viewed by 1349
Abstract
In this paper, some properties of F-harmonic and conformal F-harmonic maps between doubly warped product manifolds are studied and new examples of non-harmonic F-harmonic maps are constructed. [...] Read more.
In this paper, some properties of F -harmonic and conformal F -harmonic maps between doubly warped product manifolds are studied and new examples of non-harmonic F -harmonic maps are constructed. Full article
Open AccessArticle
A Generalization of b-Metric Space and Some Fixed Point Theorems
Mathematics 2017, 5(2), 19; https://doi.org/10.3390/math5020019 - 23 Mar 2017
Cited by 66 | Viewed by 2750
Abstract
In this paper, inspired by the concept of b-metric space, we introduce the concept of extended b-metric space. We also establish some fixed point theorems for self-mappings defined on such spaces. Our results extend/generalize many pre-existing results in literature. Full article
(This article belongs to the Special Issue Fixed Point Theorems and Applications)
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