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   ISSN (Online) 2227-7390
   Published by MDPI Homepage  [156 journals]
  • Mathematics, Vol. 5, Pages 47: New Analytical Technique for Solving a
           System of Nonlinear Fractional Partial Differential Equations

    • Authors: Hayman Thabet, Subhash Kendre, Dimplekumar Chalishajar
      First page: 47
      Abstract: This paper introduces a new analytical technique (NAT) for solving a system of nonlinear fractional partial differential equations (NFPDEs) in full general set. Moreover, the convergence and error analysis of the proposed technique is shown. The approximate solutions for a system of NFPDEs are easily obtained by means of Caputo fractional partial derivatives based on the properties of fractional calculus. However, analytical and numerical traveling wave solutions for some systems of nonlinear wave equations are successfully obtained to confirm the accuracy and efficiency of the proposed technique. Several numerical results are presented in the format of tables and graphs to make a comparison with results previously obtained by other well-known methods.
      Citation: Mathematics
      PubDate: 2017-09-25
      DOI: 10.3390/math5040047
      Issue No: Vol. 5, No. 4 (2017)
  • Mathematics, Vol. 5, Pages 48: Least-Squares Solution of Linear
           Differential Equations

    • Authors: Daniele Mortari
      First page: 48
      Abstract: This study shows how to obtain least-squares solutions to initial value problems (IVPs), boundary value problems (BVPs), and multi-value problems (MVPs) for nonhomogeneous linear differential equations (DEs) with nonconstant coefficients of any order. However, without loss of generality, the approach has been applied to second-order DEs. The proposed method has two steps. The first step consists of writing a constrained expression, that has the DE constraints embedded. These kind of expressions are given in terms of a new unknown function, g ( t ) , and they satisfy the constraints, no matter what g ( t ) is. The second step consists of expressing g ( t ) as a linear combination of m independent known basis functions. Specifically, orthogonal polynomials are adopted for the basis functions. This choice requires rewriting the DE and the constraints in terms of a new independent variable, x ∈ [ − 1 , + 1 ] . The procedure leads to a set of linear equations in terms of the unknown coefficients of the basis functions that are then computed by least-squares. Numerical examples are provided to quantify the solutions’ accuracy for IVPs, BVPs and MVPs. In all the examples provided, the least-squares solution is obtained with machine error accuracy.
      Citation: Mathematics
      PubDate: 2017-10-08
      DOI: 10.3390/math5040048
      Issue No: Vol. 5, No. 4 (2017)
  • Mathematics, Vol. 5, Pages 49: An Optimal Control Approach for the
           Treatment of Solid Tumors with Angiogenesis Inhibitors

    • Authors: Adam Glick, Antonio Mastroberardino
      First page: 49
      Abstract: Cancer is a disease of unregulated cell growth that is estimated to kill over 600,000 people in the United States in 2017 according to the National Institute of Health. While there are several therapies to treat cancer, tumor resistance to these therapies is a concern. Drug therapies have been developed that attack proliferating endothelial cells instead of the tumor in an attempt to create a therapy that is resistant to resistance in contrast to other forms of treatment such as chemotherapy and radiation therapy. In this study, a two-compartment model in terms of differential equations is presented in order to determine the optimal protocol for the delivery of anti-angiogenesis therapy. Optimal control theory is applied to the model with a range of anti-angiogenesis doses to determine optimal doses to minimize tumor volume at the end of a two week treatment and minimize drug toxicity to the patient. Applying a continuous optimal control protocol to our model of angiogenesis and tumor cell growth shows promising results for tumor control while minimizing the toxicity to the patients. By investigating a variety of doses, we determine that the optimal angiogenesis inhibitor dose is in the range of 10–20 mg/kg. In this clinically useful range of doses, good tumor control is achieved for a two week treatment period. This work shows that varying the toxicity of the treatment to the patient will change the optimal dosing scheme but tumor control can still be achieved.
      Citation: Mathematics
      PubDate: 2017-10-10
      DOI: 10.3390/math5040049
      Issue No: Vol. 5, No. 4 (2017)
  • Mathematics, Vol. 5, Pages 50: The Stability of Parabolic Problems with
           Nonstandard p(x, t)-Growth

    • Authors: André Erhardt
      First page: 50
      Abstract: In this paper, we study weak solutions to the following nonlinear parabolic partial differential equation ∂ t u − div a ( x , t , ∇ u ) + λ ( u p ( x , t ) − 2 u ) = 0 in Ω T , where λ ≥ 0 and ∂ t u denote the partial derivative of u with respect to the time variable t, while ∇ u denotes the one with respect to the space variable x. Moreover, the vector-field a ( x , t , · ) satisfies certain nonstandard p ( x , t ) -growth and monotonicity conditions. In this manuscript, we establish the existence of a unique weak solution to the corresponding Dirichlet problem. Furthermore, we prove the stability of this solution, i.e., we show that two weak solutions with different initial values are controlled by these initial values.
      Citation: Mathematics
      PubDate: 2017-10-12
      DOI: 10.3390/math5040050
      Issue No: Vol. 5, No. 4 (2017)
  • Mathematics, Vol. 5, Pages 35: Banach Subspaces of Continuous Functions
           Possessing Schauder Bases

    • Authors: Sergey Ludkowski
      First page: 35
      Abstract: In this article, Müntz spaces M Λ , C of continuous functions supplied with the absolute maximum norm are considered. An existence of Schauder bases in Müntz spaces M Λ , C is investigated. Moreover, Fourier series approximation of functions in Müntz spaces M Λ , C is studied.
      PubDate: 2017-06-24
      DOI: 10.3390/math5030035
      Issue No: Vol. 5, No. 3 (2017)
  • Mathematics, Vol. 5, Pages 36: Lattices and Rational Points

    • Authors: Evelina Viada
      First page: 36
      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 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 Q -rank ≤ 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.
      Citation: Mathematics
      PubDate: 2017-07-09
      DOI: 10.3390/math5030036
      Issue No: Vol. 5, No. 3 (2017)
  • Mathematics, Vol. 5, Pages 37: Elimination of Quotients in Various
           Localisations of Premodels into Models

    • Authors: Rémy Tuyéras
      First page: 37
      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/ Ω -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 ( ω , n ) -categories, weak ω -groupoids and higher moduli stacks.
      Citation: Mathematics
      PubDate: 2017-07-09
      DOI: 10.3390/math5030037
      Issue No: Vol. 5, No. 3 (2017)
  • Mathematics, Vol. 5, Pages 38: Variable Shape Parameter Strategy in Local
           Radial Basis Functions Collocation Method for Solving the 2D Nonlinear
           Coupled Burgers’ Equations

    • Authors: Hananeh Nojavan, Saeid Abbasbandy, Tofigh Allahviranloo
      First page: 38
      Abstract: 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.
      Citation: Mathematics
      PubDate: 2017-07-21
      DOI: 10.3390/math5030038
      Issue No: Vol. 5, No. 3 (2017)
  • Mathematics, Vol. 5, Pages 39: Confidence Intervals for Mean and
           Difference between Means of Normal Distributions with Unknown Coefficients
           of Variation

    • Authors: Warisa Thangjai, Suparat Niwitpong, Sa-Aat Niwitpong
      First page: 39
      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 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.
      Citation: Mathematics
      PubDate: 2017-07-28
      DOI: 10.3390/math5030039
      Issue No: Vol. 5, No. 3 (2017)
  • Mathematics, Vol. 5, Pages 40: Integral Representations of the Catalan
           Numbers and Their Applications

    • Authors: Feng Qi, Bai-Ni Guo
      First page: 40
      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 power series whose coefficients involve the Catalan numbers.
      Citation: Mathematics
      PubDate: 2017-08-03
      DOI: 10.3390/math5030040
      Issue No: Vol. 5, No. 3 (2017)
  • Mathematics, Vol. 5, Pages 41: On the Duality of Regular and Local

    • Authors: Jens Fischer
      First page: 41
      Abstract: 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.
      Citation: Mathematics
      PubDate: 2017-08-09
      DOI: 10.3390/math5030041
      Issue No: Vol. 5, No. 3 (2017)
  • Mathematics, Vol. 5, Pages 42: On the Uniqueness Results and Value
           Distribution of Meromorphic Mappings

    • Authors: Rahman Ullah, Xiao-Min Li, Faiz Faizullah, Hong-Xun Yi, Riaz Khan
      First page: 42
      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 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.
      Citation: Mathematics
      PubDate: 2017-08-17
      DOI: 10.3390/math5030042
      Issue No: Vol. 5, No. 3 (2017)
  • Mathematics, Vol. 5, Pages 43: On Minimal Covolume Hyperbolic Lattices

    • Authors: Ruth Kellerhals
      First page: 43
      Abstract: We study lattices with a non-compact fundamental domain of small volume in hyperbolic space H n . First, we identify the arithmetic lattices in Isom + H n of minimal covolume for even n up to 18. Then, we discuss the related problem in higher odd dimensions and provide solutions for n = 11 and n = 13 in terms of the rotation subgroup of certain Coxeter pyramid groups found by Tumarkin. The results depend on the work of Belolipetsky and Emery, as well as on the Euler characteristic computation for hyperbolic Coxeter polyhedra with few facets by means of the program CoxIter developed by Guglielmetti. This work complements the survey about hyperbolic orbifolds of minimal volume.
      Citation: Mathematics
      PubDate: 2017-08-22
      DOI: 10.3390/math5030043
      Issue No: Vol. 5, No. 3 (2017)
  • Mathematics, Vol. 5, Pages 44: Topics of Measure Theory on Infinite
           Dimensional Spaces

    • Authors: José Velhinho
      First page: 44
      Abstract: This short review is devoted to measures on infinite dimensional spaces. We start by discussing product measures and projective techniques. Special attention is paid to measures on linear spaces, and in particular to Gaussian measures. Transformation properties of measures are considered, as well as fundamental results concerning the support of the measure.
      Citation: Mathematics
      PubDate: 2017-08-29
      DOI: 10.3390/math5030044
      Issue No: Vol. 5, No. 3 (2017)
  • Mathematics, Vol. 5, Pages 45: Fusion Estimation from Multisensor
           Observations with Multiplicative Noises and Correlated Random Delays in

    • Authors: Raquel Caballero-Águila, Aurora Hermoso-Carazo, Josefa Linares-Pérez
      First page: 45
      Abstract: In this paper, the information fusion estimation problem is investigated for a class of multisensor linear systems affected by different kinds of stochastic uncertainties, using both the distributed and the centralized fusion methodologies. It is assumed that the measured outputs are perturbed by one-step autocorrelated and cross-correlated additive noises, and also stochastic uncertainties caused by multiplicative noises and randomly missing measurements in the sensor outputs are considered. At each sampling time, every sensor output is sent to a local processor and, due to some kind of transmission failures, one-step correlated random delays may occur. Using only covariance information, without requiring the evolution model of the signal process, a local least-squares (LS) filter based on the measurements received from each sensor is designed by an innovation approach. All these local filters are then fused to generate an optimal distributed fusion filter by a matrix-weighted linear combination, using the LS optimality criterion. Moreover, a recursive algorithm for the centralized fusion filter is also proposed and the accuracy of the proposed estimators, which is measured by the estimation error covariances, is analyzed by a simulation example.
      Citation: Mathematics
      PubDate: 2017-09-04
      DOI: 10.3390/math5030045
      Issue No: Vol. 5, No. 3 (2017)
  • Mathematics, Vol. 5, Pages 46: The Catastrophe of Electric Vehicle Sales

    • Authors: Timothy Sands
      First page: 46
      Abstract: Electric vehicles have undergone a recent faddy trend in the United States and Europe, and several recent publications trumpet the continued rise of electric vehicles citing steadily-climbing monthly vehicle sales. The broad purpose of this study is to examine this optimism with some degree of mathematical rigor. Specifically, the methodology will use catastrophe theory to explore the possibility of a sudden, seemingly-unexplainable crash in vehicle sales. The study begins by defining optimal system equations that well-model the available sales data. Next, these optimal models are used to investigate the potential response to a slow dynamic acting on the relatively faster dynamic of the optimal system equations. Catastrophe theory indicates a potential sudden crash in sales when a slow dynamic is at-work. It is noteworthy that the prediction can be made even while sales are increasing.
      Citation: Mathematics
      PubDate: 2017-09-17
      DOI: 10.3390/math5030046
      Issue No: Vol. 5, No. 3 (2017)
  • Mathematics, Vol. 5, Pages 19: A Generalization of b-Metric Space and Some
           Fixed Point Theorems

    • Authors: Tayyab Kamran, Maria Samreen, Qurat UL Ain
      First page: 19
      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.
      PubDate: 2017-03-23
      DOI: 10.3390/math5020019
      Issue No: Vol. 5, No. 2 (2017)
  • Mathematics, Vol. 5, Pages 20: F-Harmonic Maps between Doubly Warped
           Product Manifolds

    • Authors: Seyed Torbaghan, Morteza Rezaii
      First page: 20
      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.
      PubDate: 2017-03-23
      DOI: 10.3390/math5020020
      Issue No: Vol. 5, No. 2 (2017)
  • Mathematics, Vol. 5, Pages 21: On Some Extended Block Krylov Based Methods
           for Large Scale Nonsymmetric Stein Matrix Equations

    • Authors: Abdeslem Bentbib, Khalide Jbilou, EL Sadek
      First page: 21
      Abstract: 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.
      PubDate: 2017-03-27
      DOI: 10.3390/math5020021
      Issue No: Vol. 5, No. 2 (2017)
  • Mathematics, Vol. 5, Pages 22: On Optimal Fuzzy Best Proximity Coincidence
           Points of Proximal Contractions Involving Cyclic Mappings in
           Non-Archimedean Fuzzy Metric Spaces

    • Authors: Manuel Sen, Mujahid Abbas, Naeem Saleem
      First page: 22
      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 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.
      PubDate: 2017-04-01
      DOI: 10.3390/math5020022
      Issue No: Vol. 5, No. 2 (2017)
  • Mathematics, Vol. 5, Pages 23: Best Proximity Point Results in
           Non-Archimedean Modular Metric Space

    • Authors: Mohadeshe Paknazar, Manuel Sen
      First page: 23
      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 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.
      PubDate: 2017-04-05
      DOI: 10.3390/math5020023
      Issue No: Vol. 5, No. 2 (2017)
  • Mathematics, Vol. 5, Pages 24: Fixed Points of Set Valued Mappings in
           Terms of Start Point on a Metric Space Endowed with a Directed Graph

    • Authors: Murchana Neog, Pradip Debnath
      First page: 24
      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 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.
      PubDate: 2017-04-19
      DOI: 10.3390/math5020024
      Issue No: Vol. 5, No. 2 (2017)
  • Mathematics, Vol. 5, Pages 25: Discrete-Time Fractional Optimal Control

    • Authors: Tirumalasetty Chiranjeevi, Raj Biswas
      First page: 25
      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 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 .
      PubDate: 2017-04-19
      DOI: 10.3390/math5020025
      Issue No: Vol. 5, No. 2 (2017)
  • Mathematics, Vol. 5, Pages 26: A New Variational Iteration Method for a
           Class of Fractional Convection-Diffusion Equations in Large Domains

    • Authors: Mohammad Abolhasani, Saeid Abbasbandy, Tofigh Allahviranloo
      First page: 26
      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 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.
      PubDate: 2017-05-11
      DOI: 10.3390/math5020026
      Issue No: Vol. 5, No. 2 (2017)
  • Mathematics, Vol. 5, Pages 27: Analysis of Magneto-hydrodynamics Flow and
           Heat Transfer of a Viscoelastic Fluid through Porous Medium in Wire
           Coating Analysis

    • Authors: Zeeshan Khan, Muhammad Khan, Saeed Islam, Bilal Jan, Fawad Hussain, Haroon Ur Rasheed, Waris Khan
      First page: 27
      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 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.
      PubDate: 2017-05-16
      DOI: 10.3390/math5020027
      Issue No: Vol. 5, No. 2 (2017)
  • Mathematics, Vol. 5, Pages 28: A Two-Stage Method for Piecewise-Constant
           Solution for Fredholm Integral Equations of the First Kind

    • Authors: Fu-Rong Lin, Shi-Wei Yang
      First page: 28
      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 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.
      PubDate: 2017-05-22
      DOI: 10.3390/math5020028
      Issue No: Vol. 5, No. 2 (2017)
  • Mathematics, Vol. 5, Pages 29: Emergence of an Aperiodic Dirichlet Space
           from the Tetrahedral Units of an Icosahedral Internal Space

    • Authors: Amrik Sen, Raymond Aschheim, Klee Irwin
      First page: 29
      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) 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 mode.
      PubDate: 2017-05-26
      DOI: 10.3390/math5020029
      Issue No: Vol. 5, No. 2 (2017)
  • Mathematics, Vol. 5, Pages 30: Coincidence Points of a Sequence of
           Multivalued Mappings in Metric Space with a Graph

    • Authors: Muhammad Khan, Akbar Azam, Nayyar Mehmood
      First page: 30
      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 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.
      PubDate: 2017-05-26
      DOI: 10.3390/math5020030
      Issue No: Vol. 5, No. 2 (2017)
  • Mathematics, Vol. 5, Pages 31: Nonlinear Gronwall–Bellman Type
           Inequalities and Their Applications

    • Authors: Weimin Wang, Yuqiang Feng, Yuanyuan Wang
      First page: 31
      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.
      PubDate: 2017-05-31
      DOI: 10.3390/math5020031
      Issue No: Vol. 5, No. 2 (2017)
  • Mathematics, Vol. 5, Pages 32: Metrization Theorem for Uniform Loops with
           the Invertibility Property

    • Authors: Dagmar Markechová, Peter Vrábel, Beáta Stehlíková
      First page: 32
      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 for the case of the right-invariant metric.
      PubDate: 2017-06-02
      DOI: 10.3390/math5020032
      Issue No: Vol. 5, No. 2 (2017)
  • Mathematics, Vol. 5, Pages 33: An Analysis on the Fractional Asset Flow
           Differential Equations

    • Authors: Din Prathumwan, Wannika Sawangtong, Panumart Sawangtong
      First page: 33
      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 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.
      PubDate: 2017-06-16
      DOI: 10.3390/math5020033
      Issue No: Vol. 5, No. 2 (2017)
  • Mathematics, Vol. 5, Pages 34: Lie Symmetries, Optimal System and
           Invariant Reductions to a Nonlinear Timoshenko System

    • Authors: Shadi Al-Omari, Fiazuddin Zaman, Hassan Azad
      First page: 34
      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. 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.
      PubDate: 2017-06-17
      DOI: 10.3390/math5020034
      Issue No: Vol. 5, No. 2 (2017)
  • Mathematics, Vol. 5, Pages 2: On Autonomy Imposition in Zero Interval
           Limit Perturbation Expansion for the Spectral Entities of
           Hilbert–Schmidt Integral Operators

    • Authors: Süha Tuna, Metin Demiralp
      First page: 2
      Abstract: In this work, we deal with the autonomy issue in the perturbation expansion for the eigenfunctions of a compact Hilbert–Schmidt integral operator. Here, the autonomy points to the perturbation expansion coefficients of the relevant eigenfunction not depending on the perturbation parameter explicitly, but the dependence on this parameter arises from the coordinate change at the zero interval limit. Moreover, the related half interval length is utilized as the perturbation parameter in the perturbative analyses. Thus, the zero interval limit perturbation for solving the eigenproblem under consideration is developed. The aim of this work is to show that the autonomy imposition brings an important restriction on the kernel of the corresponding integral operator, and the constructed perturbation series is not capable of expressing the exact solution approximately unless a specific type of kernel is considered. The general structure for the encountered constraints is revealed, and the specific class of kernels is identified to this end. Error analysis of the developed method is given. These analyses are also supported by certain illustrative implementations involving the kernels, which are and are not in the specific class addressed above. Thus, the efficiency of the developed method is shown, and the relevant analyses are confirmed.
      PubDate: 2017-01-06
      DOI: 10.3390/math5010002
      Issue No: Vol. 5, No. 1 (2017)
  • Mathematics, Vol. 5, Pages 3: From the Underdamped Generalized Elastic
           Model to the Single Particle Langevin Description

    • Authors: Alessandro Taloni
      First page: 3
      Abstract: The generalized elastic model encompasses several linear stochastic models describing the dynamics of polymers, membranes, rough surfaces, and fluctuating interfaces. While usually defined in the overdamped case, in this paper we formally include the inertial term to account for the initial diffusive stages of the stochastic dynamics. We derive the generalized Langevin equation for a probe particle and we show that this equation reduces to the usual Langevin equation for Brownian motion, and to the fractional Langevin equation on the long-time limit.
      PubDate: 2017-01-06
      DOI: 10.3390/math5010003
      Issue No: Vol. 5, No. 1 (2017)
  • Mathematics, Vol. 5, Pages 4: Logical Entropy of Dynamical Systems—A
           General Model

    • Authors: Abolfazl Ebrahimzadeh, Zahra Giski, Dagmar Markechová
      First page: 4
      Abstract: In the paper by Riečan and Markechová (Fuzzy Sets Syst. 96, 1998), some fuzzy modifications of Shannon’s and Kolmogorov-Sinai’s entropy were studied and the general scheme involving the presented models was introduced. Our aim in this contribution is to provide analogies of these results for the case of the logical entropy. We define the logical entropy and logical mutual information of finite partitions on the appropriate algebraic structure and prove basic properties of these measures. It is shown that, as a special case, we obtain the logical entropy of fuzzy partitions studied by Markechová and Riečan (Entropy 18, 2016). Finally, using the suggested concept of entropy of partitions we define the logical entropy of a dynamical system and prove that it is the same for two dynamical systems that are isomorphic.
      PubDate: 2017-01-06
      DOI: 10.3390/math5010004
      Issue No: Vol. 5, No. 1 (2017)
  • Mathematics, Vol. 5, Pages 5: Data Clustering with Quantum Mechanics

    • Authors: Tony Scott, Madhusudan Therani, Xing Wang
      First page: 5
      Abstract: Data clustering is a vital tool for data analysis. This work shows that some existing useful methods in data clustering are actually based on quantum mechanics and can be assembled into a powerful and accurate data clustering method where the efficiency of computational quantum chemistry eigenvalue methods is therefore applicable. These methods can be applied to scientific data, engineering data and even text.
      PubDate: 2017-01-06
      DOI: 10.3390/math5010005
      Issue No: Vol. 5, No. 1 (2017)
  • Mathematics, Vol. 5, Pages 6: Zoology of Atlas-Groups: Dessins
           D’enfants, Finite Geometries and Quantum Commutation

    • Authors: Michel Planat, Hishamuddin Zainuddin
      First page: 6
      Abstract: Every finite simple group P can be generated by two of its elements. Pairs of generators for P are available in the Atlas of finite group representations as (not necessarily minimal) permutation representations P . It is unusual, but significant to recognize that a P is a Grothendieck’s “dessin d’enfant” D and that a wealth of standard graphs and finite geometries G —such as near polygons and their generalizations—are stabilized by a D . In our paper, tripods P − D − G of rank larger than two, corresponding to simple groups, are organized into classes, e.g., symplectic, unitary, sporadic, etc. (as in the Atlas). An exhaustive search and characterization of non-trivial point-line configurations defined from small index representations of simple groups is performed, with the goal to recognize their quantum physical significance. All of the defined geometries G ′ s have a contextuality parameter close to its maximal value of one.
      PubDate: 2017-01-14
      DOI: 10.3390/math5010006
      Issue No: Vol. 5, No. 1 (2017)
  • Mathematics, Vol. 5, Pages 7: Deterministic Seirs Epidemic Model for
           Modeling Vital Dynamics, Vaccinations, and Temporary Immunity

    • Authors: Marek Trawicki
      First page: 7
      Abstract: In this paper, the author proposes a new SEIRS model that generalizes several classical deterministic epidemic models (e.g., SIR and SIS and SEIR and SEIRS) involving the relationships between the susceptible S, exposed E, infected I, and recovered R individuals for understanding the proliferation of infectious diseases. As a way to incorporate the most important features of the previous models under the assumption of homogeneous mixing (mass-action principle) of the individuals in the population N, the SEIRS model utilizes vital dynamics with unequal birth and death rates, vaccinations for newborns and non-newborns, and temporary immunity. In order to determine the equilibrium points, namely the disease-free and endemic equilibrium points, and study their local stability behaviors, the SEIRS model is rescaled with the total time-varying population and analyzed according to its epidemic condition R0 for two cases of no epidemic (R0 ≤ 1) and epidemic (R0 > 1) using the time-series and phase portraits of the susceptible s, exposed e, infected i, and recovered r individuals. Based on the experimental results using a set of arbitrarily-defined parameters for horizontal transmission of the infectious diseases, the proportional population of the SEIRS model consisted primarily of the recovered r (0.7–0.9) individuals and susceptible s (0.0–0.1) individuals (epidemic) and recovered r (0.9) individuals with only a small proportional population for the susceptible s (0.1) individuals (no epidemic). Overall, the initial conditions for the susceptible s, exposed e, infected i, and recovered r individuals reached the corresponding equilibrium point for local stability: no epidemic (DFE X ¯ D F E ) and epidemic (EE X ¯ E E ).
      PubDate: 2017-01-17
      DOI: 10.3390/math5010007
      Issue No: Vol. 5, No. 1 (2017)
  • Mathematics, Vol. 5, Pages 8: An Analysis of the Influence of Graph Theory
           When Preparing for Programming Contests

    • Authors: Cristina Jordán, Jon Gómez, J. Conejero
      First page: 8
      Abstract: The subject known as Programming Contests in the Bachelor’s Degree in Computer Engineering course focuses on solving programming problems frequently met within contests such as the Southwest Europe Regional Contest (SWERC). In order to solve these problems one first needs to model the problem correctly, find the ideal solution, and then be able to program it without making any mistakes in a very short period of time. Leading multinationals such as Google, Apple, IBM, Facebook and Microsoft place a very high value on these abilities when selecting candidates for posts in their companies. In this communication we present some preliminary results of an analysis of the interaction between two optional subjects in the Computer Science Degree course: Programming Contests (PC) and Graphs, Models and Applications (GMA). The results of this analysis enabled us to make changes to some of the contents in GMA in order to better prepare the students to deal with the challenges they have to face in programming contests.
      PubDate: 2017-01-20
      DOI: 10.3390/math5010008
      Issue No: Vol. 5, No. 1 (2017)
  • Mathematics, Vol. 5, Pages 9: Existence of Mild Solutions for Impulsive
           Fractional Integro-Differential Inclusions with State-Dependent Delay

    • Authors: Selvaraj Suganya, Mani Mallika Arjunan
      First page: 9
      Abstract: In this manuscript, we implement Bohnenblust–Karlin’s fixed point theorem to demonstrate the existence of mild solutions for a class of impulsive fractional integro-differential inclusions (IFIDI) with state-dependent delay (SDD) in Banach spaces. An example is provided to illustrate the obtained abstract results.
      PubDate: 2017-01-25
      DOI: 10.3390/math5010009
      Issue No: Vol. 5, No. 1 (2017)
  • Mathematics, Vol. 5, Pages 10: Approximation in Müntz Spaces MΛ,p of Lp
           Functions for 1 < p < ∞ and Bases

    • Authors: Sergey Ludkowski
      First page: 10
      Abstract: Müntz spaces satisfying the Müntz and gap conditions are considered. A Fourier approximation of functions in the Müntz spaces MΛ,p of Lp functions is studied, where 1 &lt; p &lt; ∞. It is proven that up to an isomorphism and a change of variables, these spaces are contained in Weil–Nagy’s class. Moreover, the existence of Schauder bases in the Müntz spaces MΛ,p is investigated.
      PubDate: 2017-01-25
      DOI: 10.3390/math5010010
      Issue No: Vol. 5, No. 1 (2017)
  • Mathematics, Vol. 5, Pages 11: The Split Common Fixed Point Problem for a
           Family of Multivalued Quasinonexpansive Mappings and Totally
           Asymptotically Strictly Pseudocontractive Mappings in Banach Spaces

    • Authors: Ali Abkar, Elahe Shahrosvand, Azizollah Azizi
      First page: 11
      Abstract: In this paper, we introduce an iterative algorithm for solving the split common fixed point problem for a family of multi-valued quasinonexpansive mappings and totally asymptotically strictly pseudocontractive mappings, as well as for a family of totally quasi-ϕ-asymptotically nonexpansive mappings and k-quasi-strictly pseudocontractive mappings in the setting of Banach spaces. Our results improve and extend the results of Tang et al., Takahashi, Moudafi, Censor et al., and Byrne et al.
      PubDate: 2017-02-11
      DOI: 10.3390/math5010011
      Issue No: Vol. 5, No. 1 (2017)
  • Mathematics, Vol. 5, Pages 12: Fractional Fokker-Planck Equation

    • Authors: Gerd Baumann, Frank Stenger
      First page: 12
      Abstract: We shall discuss the numerical solution of the Cauchy problem for the fully fractional Fokker-Planck (fFP) equation in connection with Sinc convolution methods. The numerical approximation is based on Caputo and Riesz-Feller fractional derivatives. The use of the transfer function in Laplace and Fourier spaces in connection with Sinc convolutions allow to find exponentially converging computing schemes. Examples using different initial conditions demonstrate the effective computations with a small number of grid points on an infinite spatial domain.
      PubDate: 2017-02-11
      DOI: 10.3390/math5010012
      Issue No: Vol. 5, No. 1 (2017)
  • Mathematics, Vol. 5, Pages 13: A Few Finite Trigonometric Sums

    • Authors: Chandan Datta, Pankaj Agrawal
      First page: 13
      Abstract: Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known; however, sums with products of trigonometric functions can become complicated, and may not have a simple expression in a number of cases. Some of these sums have interesting properties, and can have amazingly simple values. However, only some of them are available in the literature. We obtain a number of such sums using the method of residues.
      PubDate: 2017-02-18
      DOI: 10.3390/math5010013
      Issue No: Vol. 5, No. 1 (2017)
  • Mathematics, Vol. 5, Pages 14: A Novel Iterative Algorithm Applied to
           Totally Asymptotically Nonexpansive Mappings in CAT(0) Spaces

    • Authors: Ali Abkar, Mohsen Shekarbaigi
      First page: 14
      Abstract: In this paper we introduce a new iterative algorithm for approximating fixed points of totally asymptotically quasi-nonexpansive mappings on CAT(0) spaces. We prove a strong convergence theorem under suitable conditions. The result we obtain improves and extends several recent results stated by many others; they also complement many known recent results in the literature. We then provide some numerical examples to illustrate our main result and to display the efficiency of the proposed algorithm.
      PubDate: 2017-02-22
      DOI: 10.3390/math5010014
      Issue No: Vol. 5, No. 1 (2017)
  • Mathematics, Vol. 5, Pages 15: Dialectical Multivalued Logic and
           Probabilistic Theory

    • Authors: José Usó Doménech, Josué Nescolarde-Selva, Lorena Segura-Abad
      First page: 15
      Abstract: There are two probabilistic algebras: one for classical probability and the other for quantum mechanics. Naturally, it is the relation to the object that decides, as in the case of logic, which algebra is to be used. From a paraconsistent multivalued logic therefore, one can derive a probability theory, adding the correspondence between truth value and fortuity.
      PubDate: 2017-02-23
      DOI: 10.3390/math5010015
      Issue No: Vol. 5, No. 1 (2017)
  • Mathematics, Vol. 5, Pages 16: On the Additively Weighted Harary Index of
           Some Composite Graphs

    • Authors: Behrooz Khosravi, Elnaz Ramezani
      First page: 16
      Abstract: 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 ( G ) 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 ( G ) 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.
      PubDate: 2017-03-07
      DOI: 10.3390/math5010016
      Issue No: Vol. 5, No. 1 (2017)
  • Mathematics, Vol. 5, Pages 17: Certain Concepts of Bipolar Fuzzy Directed

    • Authors: Muhammad Akram, Anam Luqman
      First page: 17
      Abstract: 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.
      PubDate: 2017-03-04
      DOI: 10.3390/math5010017
      Issue No: Vol. 5, No. 1 (2017)
  • Mathematics, Vol. 5, Pages 18: Characterization of the Minimizing Graph of
           the Connected Graphs Whose Complements Are Bicyclic

    • Authors: Muhammad Javaid
      First page: 18
      Abstract: 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 G 1 , n c and 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 G n c = G 1 , n c ∪ G 2 , n c , a class of the connected graphs of order n whose complements are bicyclic.
      PubDate: 2017-03-11
      DOI: 10.3390/math5010018
      Issue No: Vol. 5, No. 1 (2017)
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