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

[...] Read more.
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.
Full article

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.

[...] 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

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

►
Figures
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

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

[...] 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

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.
Full article

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 **

►
Figures
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 $SU\left(5\right)$ , ${E}_{6}$ , ${E}_{8}$ 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.
Full article

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

[...] 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

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 **

►
Figures
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 ${\mathbf{f}}_{\mathrm{TV}}^{*}$ . 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.
Full article

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 **

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 $\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.
Full article

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 **

►
Figures
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

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 **

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

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

►
Figures
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 $\alpha $ .
Full article

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

[...] Read more.
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.
Full article

On Optimal Fuzzy Best Proximity Coincidence Points of Proximal Contractions Involving Cyclic Mappings in Non-Archimedean Fuzzy Metric Spaces*Mathematics* **2017**, *5*(2), 22; doi:10.3390/math5020022 - 1 April 2017**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 $\left(g\text{},\text{}T\right)$ 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 $\left(g\text{},\text{}T\right)$ 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

On Some Extended Block Krylov Based Methods for Large Scale Nonsymmetric Stein Matrix Equations*Mathematics* **2017**, *5*(2), 21; doi:10.3390/math5020021 - 27 March 2017**Abstract **

►
Figures
In the present paper, we consider the large scale Stein matrix equation with a low-rank constant term $AXB-X+E{F}^{T}=0$ . 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 $AXB-X+E{F}^{T}=0$ . 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

In this paper, some properties of $\mathcal{F}$ -harmonic and conformal $\mathcal{F}$ -harmonic maps between doubly warped product manifolds are studied and new examples of non-harmonic $\mathcal{F}$ -harmonic maps are constructed.

[...] Read more.
In this paper, some properties of $\mathcal{F}$ -harmonic and conformal $\mathcal{F}$ -harmonic maps between doubly warped product manifolds are studied and new examples of non-harmonic $\mathcal{F}$ -harmonic maps are constructed.
Full article

A Generalization of *b*-Metric Space and Some Fixed Point Theorems*Mathematics* **2017**, *5*(2), 19; doi:10.3390/math5020019 - 23 March 2017**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

Characterization of the Minimizing Graph of the Connected Graphs Whose Complements Are Bicyclic*Mathematics* **2017**, *5*(1), 18; doi:10.3390/math5010018 - 11 March 2017**Abstract **

►
Figures
In a certain class of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum. A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to

[...] Read more.
In a certain class of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum. A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. Let ${\mathcal{G}}_{1,n}^{c}$ and ${\mathcal{G}}_{2,n}^{c}$ be the classes of the connected graphs of order *n* whose complements are bicyclic with exactly two and three cycles, respectively. In this paper, we characterize the unique minimizing graph among all the graphs which belong to ${\mathcal{G}}_{n}^{c}={\mathcal{G}}_{1,n}^{c}\cup {\mathcal{G}}_{2,n}^{c}$ , a class of the connected graphs of order *n* whose complements are bicyclic.
Full article

On the Additively Weighted Harary Index of Some Composite Graphs*Mathematics* **2017**, *5*(1), 16; doi:10.3390/math5010016 - 7 March 2017**Abstract **

►
Figures
The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The additively weighted Harary index ${H}_{A}\left(G\right)$ is a modification of the Harary index in which the contributions of

[...] Read more.
The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The additively weighted Harary index ${H}_{A}\left(G\right)$ is a modification of the Harary index in which the contributions of vertex pairs are weighted by the sum of their degrees. This new invariant was introduced in (Alizadeh, Iranmanesh and Došlić. *Additively weighted Harary index of some composite graphs*, Discrete Math, 2013) and they posed the following question: *What is the behavior of *${H}_{A}\left(G\right)$ when G is a composite graph resulting for example by: splice, link, corona and rooted product? We investigate the additively weighted Harary index for these standard graph products. Then we obtain lower and upper bounds for some of them.
Full article

Certain Concepts of Bipolar Fuzzy Directed Hypergraphs*Mathematics* **2017**, *5*(1), 17; doi:10.3390/math5010017 - 4 March 2017**Abstract **

►
Figures
A hypergraph is the most developed tool for modeling various practical problems in different fields, including computer sciences, biological sciences, social networks and psychology. Sometimes, given data in a network model are based on bipolar information rather than one sided. To deal with

[...] Read more.
A hypergraph is the most developed tool for modeling various practical problems in different fields, including computer sciences, biological sciences, social networks and psychology. Sometimes, given data in a network model are based on bipolar information rather than one sided. To deal with such types of problems, we use mathematical models that are based on bipolar fuzzy (BF) sets. In this research paper, we introduce the concept of BF directed hypergraphs. We describe certain operations on BF directed hypergraphs, including addition, multiplication, vertex-wise multiplication and structural subtraction. We introduce the concept of $B=({m}^{+},{m}^{-})$ -tempered BF directed hypergraphs and investigate some of their properties. We also present an algorithm to compute the minimum arc length of a BF directed hyperpath.
Full article