On the Uniqueness Results and Value Distribution of Meromorphic Mappings*Mathematics* **2017**, *5*(3), 42; doi:10.3390/math5030042 (registering DOI) - 17 August 2017**Abstract **

This research concentrates on the analysis of meromorphic mappings. We derived several important results for value distribution of specific difference polynomials of meromorphic mappings, which generalize the work of Laine and Yang. In addition, we proved uniqueness theorems of meromorphic mappings. The difference

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This research concentrates on the analysis of meromorphic mappings. We derived several important results for value distribution of specific difference polynomials of meromorphic mappings, which generalize the work of Laine and Yang. In addition, we proved uniqueness theorems of meromorphic mappings. The difference polynomials of these functions have the same fixed points or share a nonzero value. This extends the research work of Qi, Yang and Liu, where they used the finite ordered meromorphic mappings.
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On the Duality of Regular and Local Functions*Mathematics* **2017**, *5*(3), 41; doi:10.3390/math5030041 - 9 August 2017**Abstract **

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In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty principle. They both express Fourier dualities within the space of tempered distributions and these dualities are also inverse of each other. While Poisson’s summation formula expresses a duality between discretization and periodization,

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In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty principle. They both express Fourier dualities within the space of tempered distributions and these dualities are also inverse of each other. While Poisson’s summation formula expresses a duality between discretization and periodization, Heisenberg’s uncertainty principle expresses a duality between regularization and localization. We define regularization and localization on generalized functions and show that the Fourier transform of regular functions are local functions and, vice versa, the Fourier transform of local functions are regular functions.
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Integral Representations of the Catalan Numbers and Their Applications*Mathematics* **2017**, *5*(3), 40; doi:10.3390/math5030040 - 3 August 2017**Abstract **

In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations, collect applications of some integral representations, and present sums of several

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In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations, collect applications of some integral representations, and present sums of several power series whose coefficients involve the Catalan numbers.
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Confidence Intervals for Mean and Difference between Means of Normal Distributions with Unknown Coefficients of Variation*Mathematics* **2017**, *5*(3), 39; doi:10.3390/math5030039 - 28 July 2017**Abstract **

This paper proposes confidence intervals for a single mean and difference of two means of normal distributions with unknown coefficients of variation (CVs). The generalized confidence interval (GCI) approach and large sample (LS) approach were proposed to construct confidence intervals for the single

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This paper proposes confidence intervals for a single mean and difference of two means of normal distributions with unknown coefficients of variation (CVs). The generalized confidence interval (GCI) approach and large sample (LS) approach were proposed to construct confidence intervals for the single normal mean with unknown CV. These confidence intervals were compared with existing confidence interval for the single normal mean based on the Student’s t-distribution (small sample size case) and the z-distribution (large sample size case). Furthermore, the confidence intervals for the difference between two normal means with unknown CVs were constructed based on the GCI approach, the method of variance estimates recovery (MOVER) approach and the LS approach and then compared with the Welch–Satterthwaite (WS) approach. The coverage probability and average length of the proposed confidence intervals were evaluated via Monte Carlo simulation. The results indicated that the GCIs for the single normal mean and the difference of two normal means with unknown CVs are better than the other confidence intervals. Finally, three datasets are given to illustrate the proposed confidence intervals.
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Variable Shape Parameter Strategy in Local Radial Basis Functions Collocation Method for Solving the 2D Nonlinear Coupled Burgers’ Equations*Mathematics* **2017**, *5*(3), 38; doi:10.3390/math5030038 - 21 July 2017**Abstract **

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This study aimed at investigating a local radial basis function collocation method (LRBFCM) in the reproducing kernel Hilbert space. This method was, in fact, a meshless one which applied the local sub-clusters of domain nodes for the approximation of the arbitrary field. For

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This study aimed at investigating a local radial basis function collocation method (LRBFCM) in the reproducing kernel Hilbert space. This method was, in fact, a meshless one which applied the local sub-clusters of domain nodes for the approximation of the arbitrary field. For time-dependent partial differential equations (PDEs), it would be changed to a system of ordinary differential equations (ODEs). Here, we intended to decrease the error through utilizing variable shape parameter (VSP) strategies. This method was an appropriate way to solve the two-dimensional nonlinear coupled Burgers’ equations comprised of Dirichlet and mixed boundary conditions. Numerical examples indicated that the variable shape parameter strategies were more efficient than constant ones for various values of the Reynolds number.
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Elimination of Quotients in Various Localisations of Premodels into Models*Mathematics* **2017**, *5*(3), 37; doi:10.3390/math5030037 - 9 July 2017**Abstract **

The contribution of this article is quadruple. It (1) unifies various schemes of premodels/models including situations such as presheaves/sheaves, sheaves/flabby sheaves, prespectra/$\Omega $ -spectra, simplicial topological spaces/(complete) Segal spaces, pre-localised rings/localised rings, functors in categories/strong stacks and, to some extent, functors from a

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The contribution of this article is quadruple. It (1) unifies various schemes of premodels/models including situations such as presheaves/sheaves, sheaves/flabby sheaves, prespectra/$\Omega $ -spectra, simplicial topological spaces/(complete) Segal spaces, pre-localised rings/localised rings, functors in categories/strong stacks and, to some extent, functors from a limit sketch to a model category versus the homotopical models for the limit sketch; (2) provides a general construction from the premodels to the models; (3) proposes technics that allow one to assess the nature of the universal properties associated with this construction; (4) shows that the obtained localisation admits a particular presentation, which organises the structural and relational information into bundles of data. This presentation is obtained via a process called an *elimination of quotients* and its aim is to facilitate the handling of the relational information appearing in the construction of higher dimensional objects such as weak $(\omega ,n)$ -categories, weak $\omega $ -groupoids and higher moduli stacks.
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Lattices and Rational Points*Mathematics* **2017**, *5*(3), 36; doi:10.3390/math5030036 - 9 July 2017**Abstract **

In this article, we show how to use the first and second Minkowski Theorems and some Diophantine geometry to bound explicitly the height of the points of rank $N-1$ on transverse curves in ${E}^{N}$ , where *E* is an elliptic

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In this article, we show how to use the first and second Minkowski Theorems and some Diophantine geometry to bound explicitly the height of the points of rank $N-1$ on transverse curves in ${E}^{N}$ , where *E* is an elliptic curve without Complex Multiplication (CM). We then apply our result to give a method for finding the rational points on such curves, when *E* has $\mathbb{Q}$ -rank $\le N-1$ . We also give some explicit examples. This result generalises from rank 1 to rank $N-1$ previous results of S. Checcoli, F. Veneziano and the author.
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Banach Subspaces of Continuous Functions Possessing Schauder Bases*Mathematics* **2017**, *5*(3), 35; doi:10.3390/math5030035 - 24 June 2017**Abstract **

In this article, Müntz spaces ${M}_{\Lambda ,C}$ of continuous functions supplied with the absolute maximum norm are considered. An existence of Schauder bases in Müntz spaces ${M}_{\Lambda ,C}$ is investigated. Moreover, Fourier series approximation of functions in Müntz spaces

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In this article, Müntz spaces ${M}_{\Lambda ,C}$ of continuous functions supplied with the absolute maximum norm are considered. An existence of Schauder bases in Müntz spaces ${M}_{\Lambda ,C}$ is investigated. Moreover, Fourier series approximation of functions in Müntz spaces ${M}_{\Lambda ,C}$ is studied.
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Lie Symmetries, Optimal System and Invariant Reductions to a Nonlinear Timoshenko System*Mathematics* **2017**, *5*(2), 34; doi:10.3390/math5020034 - 17 June 2017**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.

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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.
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An Analysis on the Fractional Asset Flow Differential Equations*Mathematics* **2017**, *5*(2), 33; doi:10.3390/math5020033 - 16 June 2017**Abstract **

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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

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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.
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Metrization Theorem for Uniform Loops with the Invertibility Property*Mathematics* **2017**, *5*(2), 32; doi:10.3390/math5020032 - 2 June 2017**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

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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.
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Nonlinear Gronwall–Bellman Type Inequalities and Their Applications*Mathematics* **2017**, *5*(2), 31; doi:10.3390/math5020031 - 31 May 2017**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.
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Emergence of an Aperiodic Dirichlet Space from the Tetrahedral Units of an Icosahedral Internal Space*Mathematics* **2017**, *5*(2), 29; doi:10.3390/math5020029 - 26 May 2017**Abstract **

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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)

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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 $SU\left(5\right)$ , ${E}_{6}$ , ${E}_{8}$ Lie algebras and their composition with the algebra associated with the even unimodular lattice in ${\mathbb{R}}^{3,1}$ . The construction presented here is inspired by Penrose’s three world model.
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Coincidence Points of a Sequence of Multivalued Mappings in Metric Space with a Graph*Mathematics* **2017**, *5*(2), 30; doi:10.3390/math5020030 - 26 May 2017**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

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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.
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A Two-Stage Method for Piecewise-Constant Solution for Fredholm Integral Equations of the First Kind*Mathematics* **2017**, *5*(2), 28; doi:10.3390/math5020028 - 22 May 2017**Abstract **

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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

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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 ${\mathbf{f}}_{\mathrm{TV}}^{*}$ . In the second stage, the simplified MM functional is minimized to obtain the final result by using the damped Newton (DN) method with ${\mathbf{f}}_{\mathrm{TV}}^{*}$ 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.
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Analysis of Magneto-hydrodynamics Flow and Heat Transfer of a Viscoelastic Fluid through Porous Medium in Wire Coating Analysis*Mathematics* **2017**, *5*(2), 27; doi:10.3390/math5020027 - 16 May 2017**Abstract **

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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

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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 $\beta $ 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 $Br$ , permeability parameter $K$ , magnetic parameter $M$ and viscoelastic third grade parameter (non-Newtonian parameter) $\beta $ increase.
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A New Variational Iteration Method for a Class of Fractional Convection-Diffusion Equations in Large Domains*Mathematics* **2017**, *5*(2), 26; doi:10.3390/math5020026 - 11 May 2017**Abstract **

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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

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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.
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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; doi:10.3390/math5020024 - 19 April 2017**Abstract **

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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

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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.
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Discrete-Time Fractional Optimal Control*Mathematics* **2017**, *5*(2), 25; doi:10.3390/math5020025 - 19 April 2017**Abstract **

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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

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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 $\alpha $ .
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Best Proximity Point Results in Non-Archimedean Modular Metric Space*Mathematics* **2017**, *5*(2), 23; doi:10.3390/math5020023 - 5 April 2017**Abstract **

In this paper, we introduce the new notion of Suzuki-type $(\alpha ,\beta ,\theta ,\gamma )$ -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

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In this paper, we introduce the new notion of Suzuki-type $(\alpha ,\beta ,\theta ,\gamma )$ -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 ${P}_{\lambda}$ -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.
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