Journal of Applied Mathematics https://www.hindawi.com The latest articles from Hindawi © 2017 , Hindawi Limited . All rights reserved. A Guide on Spectral Methods Applied to Discrete Data in One Dimension Mon, 24 Jul 2017 00:00:00 +0000 http://www.hindawi.com/journals/jam/2017/5108946/ This paper provides an overview about the usage of the Fourier transform and its related methods and focuses on the subtleties to which the users must pay attention. Typical questions, which are often addressed to the data, will be discussed. Such a problem can be the origin of frequency or band limitation of the signal or the source of artifacts, when a Fourier transform is carried out. Another topic is the processing of fragmented data. Here, the Lomb-Scargle method will be explained with an illustrative example to deal with this special type of signal. Furthermore, the time-dependent spectral analysis, with which one can evaluate the point in time when a certain frequency appears in the signal, is of interest. The goal of this paper is to collect the important information about the common methods to give the reader a guide on how to use these for application on one-dimensional data. The introduced methods are supported by the spectral package, which has been published for the statistical environment prior to this article. Martin Seilmayer and Matthias Ratajczak Copyright © 2017 Martin Seilmayer and Matthias Ratajczak. All rights reserved. Solvability of the Brinkman-Forchheimer-Darcy Equation Thu, 20 Jul 2017 06:16:13 +0000 http://www.hindawi.com/journals/jam/2017/7305230/ The nonlinear Brinkman-Forchheimer-Darcy equation is used to model some porous medium flow in chemical reactors of packed bed type. The results concerning the existence and uniqueness of a weak solution are presented for nonlinear convective flows in medium with variable porosity and for small data. Furthermore, the finite element approximations to the flow profiles in the fixed bed reactor are presented for several Reynolds numbers at the non-Darcy’s range. Piotr Skrzypacz and Dongming Wei Copyright © 2017 Piotr Skrzypacz and Dongming Wei. All rights reserved. Modelling of Rabies Transmission Dynamics Using Optimal Control Analysis Sun, 16 Jul 2017 08:56:27 +0000 http://www.hindawi.com/journals/jam/2017/2451237/ We examine an optimal way of eradicating rabies transmission from dogs into the human population, using preexposure prophylaxis (vaccination) and postexposure prophylaxis (treatment) due to public education. We obtain the disease-free equilibrium, the endemic equilibrium, the stability, and the sensitivity analysis of the optimal control model. Using the Latin hypercube sampling (LHS), the forward-backward sweep scheme and the fourth-order Range-Kutta numerical method predict that the global alliance for rabies control’s aim of working to eliminate deaths from canine rabies by 2030 is attainable through mass vaccination of susceptible dogs and continuous use of pre- and postexposure prophylaxis in humans. Joshua Kiddy K. Asamoah, Francis T. Oduro, Ebenezer Bonyah, and Baba Seidu Copyright © 2017 Joshua Kiddy K. Asamoah et al. All rights reserved. Nonlinear Waves in Rods and Beams of Power-Law Materials Thu, 13 Jul 2017 06:59:49 +0000 http://www.hindawi.com/journals/jam/2017/2095425/ Some novel traveling waves and special solutions to the 1D nonlinear dynamic equations of rod and beam of power-law materials are found in closed forms. The traveling solutions represent waves of high elevation that propagates without change of forms in time. These waves resemble the usual kink waves except that they do not possess bounded elevations. The special solutions satisfying certain boundary and initial conditions are presented to demonstrate the nonlinear behavior of the materials. This note demonstrates the apparent distinctions between linear elastic and nonlinear plastic waves. Dongming Wei, Piotr Skrzypacz, and Xijun Yu Copyright © 2017 Dongming Wei et al. All rights reserved. Improved Combinatorial Benders Decomposition for a Scheduling Problem with Unrelated Parallel Machines Mon, 03 Jul 2017 06:15:01 +0000 http://www.hindawi.com/journals/jam/2017/9452762/ This paper addresses the unrelated parallel machines scheduling problem with sequence and machine dependent setup times. Its goal is to minimize the makespan. The problem is solved by a combinatorial Benders decomposition. This method can be slow to converge. Therefore, three procedures are introduced to accelerate its convergence. The first procedure is a new method that consists of terminating the execution of the master problem when a repeated optimal solution is found. The second procedure is based on the multicut technique. The third procedure is based on the warm-start. The improved Benders decomposition scheme is compared to a mathematical formulation and a standard implementation of Benders decomposition algorithm. In the experiments, two test sets from the literature are used, with 240 and 600 instances with up to 60 jobs and 5 machines. For the first set the proposed method performs 21.85% on average faster than the standard implementation of the Benders algorithm. For the second set the proposed method failed to find an optimal solution in only 31 in 600 instances, obtained an average gap of 0.07%, and took an average computational time of 377.86 s, while the best results of the other methods were 57, 0.17%, and 573.89 s, respectively. Francisco Regis Abreu Gomes and Geraldo Robson Mateus Copyright © 2017 Francisco Regis Abreu Gomes and Geraldo Robson Mateus. All rights reserved. Finite Element Solution of an Unsteady MHD Flow through Porous Medium between Two Parallel Flat Plates Wed, 14 Jun 2017 10:56:28 +0000 http://www.hindawi.com/journals/jam/2017/6856470/ Finite element solution of unsteady magnetohydrodynamics (MHD) flow of an electrically conducting, incompressible viscous fluid past through porous medium between two parallel plates is presented in the presence of a transverse magnetic field and Hall effect. The results obtained from some test cases are then compared with previous published work using the finite difference method (FDM). Numerical examples show that the finite element method (FEM) gives more accurate results in comparison with the finite difference method (FDM). AbdelLatif Sa’adAldin and Naji Qatanani Copyright © 2017 AbdelLatif Sa’adAldin and Naji Qatanani. All rights reserved. Urban Lead: Modeling Its Distribution and Effects on Children Sun, 11 Jun 2017 09:26:55 +0000 http://www.hindawi.com/journals/jam/2017/6107430/ We model the transportation of lead from the atmosphere and from the surface of the soil simultaneously at the macroscale and mesoscale to study its health effects on children in Jersey City, NJ. We conceptualize Jersey City as an open system where lead is continuously emitted from a local smelting plant and a local power plant, deposited onto the surface soil of playgrounds, and ingested by children. The model is constructed using the diffusion-advection partial differential equation in three spatial dimensions and one temporal dimension with an initial condition and boundary conditions. The model is solved using the Crank-Nicolson numerical method at the macroscale to determine the deposition of lead from the smelting plant and the local power plant and at the mesoscale to refine the amount of lead deposition for the areas considered. We then determine the health consequences for the average child using the bioaccessibility of lead from soil to children, the bioavailability of ingested lead to the circulatory system, and the biological half-life of lead isotopes in the blood. The health effects on children from lead are directly proportional to the blood lead concentration. Zhixiong Chen, Yi Ding, Andrew Getz, and Bernard Lipat Copyright © 2017 Zhixiong Chen et al. All rights reserved. The Implied Risk Neutral Density Dynamics: Evidence from the S&P TSX 60 Index Sun, 11 Jun 2017 09:09:37 +0000 http://www.hindawi.com/journals/jam/2017/3156250/ The risk neutral density is an important tool for analyzing the dynamics of financial markets and traders’ attitudes and reactions to already experienced shocks by financial markets as well as the potential ones. In this paper, we present a new method for the extraction information content from option prices. By eliminating bias caused by daily variation of contract maturity through a completely nonparametric technique based on kernel regression, we allow comparing evolution of risk neutral density and extracting from time continuous indicators that detect evolution of traders’ attitudes, risk perception, and belief homogeneity. This method is useful to develop trading strategies and monetary policies. Nessim Souissi Copyright © 2017 Nessim Souissi. All rights reserved. A Mathematical Model of Malaria Transmission with Structured Vector Population and Seasonality Sun, 04 Jun 2017 09:02:23 +0000 http://www.hindawi.com/journals/jam/2017/6754097/ In this paper, we formulate a mathematical model of nonautonomous ordinary differential equations describing the dynamics of malaria transmission with age structure for the vector population. The biting rate of mosquitoes is considered as a positive periodic function which depends on climatic factors. The basic reproduction ratio of the model is obtained and we show that it is the threshold parameter between the extinction and the persistence of the disease. Thus, by applying the theorem of comparison and the theory of uniform persistence, we prove that if the basic reproduction ratio is less than , then the disease-free equilibrium is globally asymptotically stable and if it is greater than , then there exists at least one positive periodic solution. Finally, numerical simulations are carried out to illustrate our analytical results. Bakary Traoré, Boureima Sangaré, and Sado Traoré Copyright © 2017 Bakary Traoré et al. All rights reserved. Per-Spectral Characterizations of Bicyclic Networks Wed, 31 May 2017 09:03:09 +0000 http://www.hindawi.com/journals/jam/2017/7541312/ Spectral techniques are used for the study of several network properties: community detection, bipartition, clustering, design of highly synchronizable networks, and so forth. In this paper, we investigate which kinds of bicyclic networks are determined by their per-spectra. We find that the permanental spectra cannot determine sandglass graphs in general. When we restrict our consideration to connected graphs or quadrangle-free graphs, sandglass graphs are determined by their permanental spectra. Furthermore, we construct countless pairs of per-cospectra bicyclic networks. Tingzeng Wu and Huazhong Lü Copyright © 2017 Tingzeng Wu and Huazhong Lü. All rights reserved. Comparing an Approximate Queuing Approach with Simulation for the Solution of a Cross-Docking Problem Sun, 28 May 2017 00:00:00 +0000 http://www.hindawi.com/journals/jam/2017/4987127/ Cross-docking is a logistics management concept in which products are temporarily unloaded at intermediate facilities and loaded onto output trucks to be sent to their final destination. In this paper, we propose an approximate nonstationary queuing model to size the number of docks to receive the trucks, so that their unloading will be as short as possible at the receiving dock, thus making the cross-docking process more efficient. It is observed that the stochastic queuing process may not reach the steady equilibrium state. A type of modeling that does not depend on the stationary characteristics of the process developed is applied. In order to measure the efficiency, performance, and possible adjustments of the parameters of the algorithm, an alternative simulation model is proposed using the Arena® software. The simulation uses analytic tools to make the problem more detailed, which is not allowed in the theoretical model. The computational analysis compares the results of the simulated model with the ones obtained with the theoretical algorithm, considering the queue length and the average waiting time of the trucks. Based on the results obtained, the simulation represented very well the proposed problem and possible changes can be easily detected with small adjustments in the simulated model. Roberta Briesemeister and Antônio G. N. Novaes Copyright © 2017 Roberta Briesemeister and Antônio G. N. Novaes. All rights reserved. Computational Methods for Solving Linear Fuzzy Volterra Integral Equation Sun, 28 May 2017 00:00:00 +0000 http://www.hindawi.com/journals/jam/2017/2417195/ Two numerical schemes, namely, the Taylor expansion and the variational iteration methods, have been implemented to give an approximate solution of the fuzzy linear Volterra integral equation of the second kind. To display the validity and applicability of the numerical methods, one illustrative example with known exact solution is presented. Numerical results show that the convergence and accuracy of these methods were in a good agreement with the exact solution. However, according to comparison of these methods, we conclude that the variational iteration method provides more accurate results. Jihan Hamaydi and Naji Qatanani Copyright © 2017 Jihan Hamaydi and Naji Qatanani. All rights reserved. On a Bivariate Spectral Homotopy Analysis Method for Unsteady Mixed Convection Boundary Layer Flow, Heat, and Mass Transfer due to a Stretching Surface in a Rotating Fluid Tue, 09 May 2017 10:04:17 +0000 http://www.hindawi.com/journals/jam/2017/5962073/ A bivariate spectral homotopy analysis method (BSHAM) is extended to solutions of systems of nonlinear coupled partial differential equations (PDEs). The method has been used successfully to solve a nonlinear PDE and is now tested with systems. The method is based on a new idea of finding solutions that obey a rule of solution expression that is defined in terms of the bivariate Lagrange interpolation polynomials. The BSHAM is used to solve a system of coupled nonlinear partial differential equations modeling the unsteady mixed convection boundary layer flow, heat, and mass transfer due to a stretching surface in a rotating fluid, taking into consideration the effect of buoyancy forces. Convergence of the numerical solutions was monitored using the residual error of the PDEs. The effects of the flow parameters on the local skin-friction coefficient, the Nusselt number, and the Sherwood number were presented in graphs. Sandile S. Motsa and Zodwa G. Makukula Copyright © 2017 Sandile S. Motsa and Zodwa G. Makukula. All rights reserved. Bound for the 2-Page Fixed Linear Crossing Number of Hypercube Graph via SDP Relaxation Mon, 08 May 2017 00:00:00 +0000 http://www.hindawi.com/journals/jam/2017/7640347/ The crossing number of graph is the minimum number of edges crossing in any drawing of in a plane. In this paper we describe a method of finding the bound of 2-page fixed linear crossing number of . We consider a conflict graph of . Then, instead of minimizing the crossing number of , we show that it is equivalent to maximize the weight of a cut of . We formulate the original problem into the MAXCUT problem. We consider a semidefinite relaxation of the MAXCUT problem. An example of a case where is hypercube is explicitly shown to obtain an upper bound. The numerical results confirm the effectiveness of the approximation. A. Suebsriwichai and T. Mouktonglang Copyright © 2017 A. Suebsriwichai and T. Mouktonglang. All rights reserved. Generation Expansion Models including Technical Constraints and Demand Uncertainty Thu, 06 Apr 2017 00:00:00 +0000 http://www.hindawi.com/journals/jam/2017/3424129/ This article presents a Generation Expansion Model of the power system taking into account the operational constraints and the uncertainty of long-term electricity demand projections. The model is based on a discretization of the load duration curve and explicitly considers that power plant ramping capabilities must meet demand variations. A model predictive control method is used to improve the long-term planning decisions while considering the uncertainty of demand projections. The model presented in this paper allows integrating technical constraints and uncertainty in the simulations, improving the accuracy of the results, while maintaining feasible computational time. Results are tested over three scenarios based on load data of an energy retailer in Colombia. P. Deossa, K. De Vos, G. Deconinck, and J. Espinosa Copyright © 2017 P. Deossa et al. All rights reserved. Solutions of First-Order Volterra Type Linear Integrodifferential Equations by Collocation Method Mon, 20 Mar 2017 00:00:00 +0000 http://www.hindawi.com/journals/jam/2017/1510267/ The numerical solutions of linear integrodifferential equations of Volterra type have been considered. Power series is used as the basis polynomial to approximate the solution of the problem. Furthermore, standard and Chebyshev-Gauss-Lobatto collocation points were, respectively, chosen to collocate the approximate solution. Numerical experiments are performed on some sample problems already solved by homotopy analysis method and finite difference methods. Comparison of the absolute error is obtained from the present method and those from aforementioned methods. It is also observed that the absolute errors obtained are very low establishing convergence and computational efficiency. Olumuyiwa A. Agbolade and Timothy A. Anake Copyright © 2017 Olumuyiwa A. Agbolade and Timothy A. Anake. All rights reserved. Sparse Optimization of Vibration Signal by ADMM Wed, 15 Feb 2017 00:00:00 +0000 http://www.hindawi.com/journals/jam/2017/4612853/ In this paper, the alternating direction method of multipliers (ADMM) algorithm is applied to the compressed sensing theory to realize the sparse optimization of vibration signal. Solving the basis pursuit problem for minimizing the norm minimization under the equality constraints, the sparse matrix obtained by the ADMM algorithm can be reconstructed by inverse sparse orthogonal matrix inversion. This paper analyzes common sparse orthogonal basis on the reconstruction results, that is, discrete Fourier orthogonal basis, discrete cosine orthogonal basis, and discrete wavelet orthogonal basis. In particular, we will show that, from the point of view of central tendency, the discrete cosine orthogonal basis is more suitable, for instance, at the vibration signal data because its error is close to zero. Moreover, using the discrete wavelet transform in signal reconstruction there still are some outliers but the error is unstable. We also use the time complex degree and validity, for the analysis of the advantages and disadvantages of the ADMM algorithm applied to sparse signal optimization. The advantage of this method is that these abnormal values are limited in the control range. Song Wanqing Copyright © 2017 Song Wanqing. All rights reserved. First Integrals and Hamiltonians of Some Classes of ODEs of Maximal Symmetry Tue, 14 Feb 2017 06:33:31 +0000 http://www.hindawi.com/journals/jam/2017/7302081/ Complete sets of linearly independent first integrals are found for the most general form of linear equations of maximal symmetry algebra of order ranging from two to eight. The corresponding Hamiltonian systems are constructed and it is shown that their general solutions can also be found by a simple superposition formula from the solutions of a scalar second-order source equation. J. C. Ndogmo Copyright © 2017 J. C. Ndogmo. All rights reserved. Bayesian Analysis for a Fractional Population Growth Model Mon, 23 Jan 2017 00:00:00 +0000 http://www.hindawi.com/journals/jam/2017/9654506/ We implement the Bayesian statistical inversion theory to obtain the solution for an inverse problem of growth data, using a fractional population growth model. We estimate the parameters in the model and we make a comparison between this model and an exponential one, based on an approximation of Bayes factor. A simulation study is carried out to show the performance of the estimators and the Bayes factor. Finally, we present a real data example to illustrate the effectiveness of the method proposed here and the pertinence of using a fractional model. Francisco J. Ariza-Hernandez, Jorge Sanchez-Ortiz, Martin P. Arciga-Alejandre, and Luis X. Vivas-Cruz Copyright © 2017 Francisco J. Ariza-Hernandez et al. All rights reserved. Implicit One-Step Block Hybrid Third-Derivative Method for the Direct Solution of Initial Value Problems of Second-Order Ordinary Differential Equations Wed, 18 Jan 2017 09:11:45 +0000 http://www.hindawi.com/journals/jam/2017/8510948/ A new one-step block method with generalized three hybrid points for solving initial value problems of second-order ordinary differential equations directly is proposed. In deriving this method, a power series approximate function is interpolated at while its second and third derivatives are collocated at all points in the given interval. The proposed method is then tested on initial value problems of second-order ordinary differential equations solved by other methods previously. The numerical results confirm the superiority of the new method to the existing methods in terms of accuracy. Mohammad Alkasassbeh and Zurni Omar Copyright © 2017 Mohammad Alkasassbeh and Zurni Omar. All rights reserved. Numerical Solution of Second-Order Fredholm Integrodifferential Equations with Boundary Conditions by Quadrature-Difference Method Wed, 11 Jan 2017 00:00:00 +0000 http://www.hindawi.com/journals/jam/2017/2645097/ In this research, the quadrature-difference method with Gauss Elimination (GE) method is applied for solving the second-order of linear Fredholm integrodifferential equations (LFIDEs). In order to derive an approximation equation, the combinations of Composite Simpson’s 1/3 rule and second-order finite-difference method are used to discretize the second-order of LFIDEs. This approximation equation will be used to generate a system of linear algebraic equations and will be solved by using Gauss Elimination. In addition, the formulation and the implementation of the quadrature-difference method are explained in detail. Finally, some numerical experiments were carried out to examine the accuracy of the proposed method. Chriscella Jalius and Zanariah Abdul Majid Copyright © 2017 Chriscella Jalius and Zanariah Abdul Majid. All rights reserved. Axioms for Consensus Functions on the -Cube Mon, 09 Jan 2017 10:01:11 +0000 http://www.hindawi.com/journals/jam/2017/8025616/ A value of a sequence of elements of a finite metric space is an element for which is minimum. The –function with domain the set of all finite sequences on and defined by is a value of is called the –function on . The and functions are the well-studied median and mean functions, respectively. In this note, simple characterizations of the –functions on the -cube are given. In addition, the center function (using the minimax criterion) is characterized as well as new results proved for the median and antimedian functions. C. Garcia-Martinez, F. R. McMorris, O. Ortega, and R. C. Powers Copyright © 2017 C. Garcia-Martinez et al. All rights reserved. Viscous Dissipation Effects on the Motion of Casson Fluid over an Upper Horizontal Thermally Stratified Melting Surface of a Paraboloid of Revolution: Boundary Layer Analysis Wed, 04 Jan 2017 09:05:32 +0000 http://www.hindawi.com/journals/jam/2017/1697135/ The problem of a non-Newtonian fluid flow past an upper surface of an object that is neither a perfect horizontal/vertical nor inclined/cone in which dissipation of energy is associated with temperature-dependent plastic dynamic viscosity is considered. An attempt has been made to focus on the case of two-dimensional Casson fluid flow over a horizontal melting surface embedded in a thermally stratified medium. Since the viscosity of the non-Newtonian fluid tends to take energy from the motion (kinetic energy) and transform it into internal energy, the viscous dissipation term is accommodated in the energy equation. Due to the existence of internal space-dependent heat source; plastic dynamic viscosity and thermal conductivity of the non-Newtonian fluid are assumed to vary linearly with temperature. Based on the boundary layer assumptions, suitable similarity variables are applied to nondimensionalized, parameterized and reduce the governing partial differential equations into a coupled ordinary differential equations. These equations along with the boundary conditions are solved numerically using the shooting method together with the Runge-Kutta technique. The effects of pertinent parameters are established. A significant increases in is guaranteed with when magnitude of is large. decreases with and . T. M. Ajayi, A. J. Omowaye, and I. L. Animasaun Copyright © 2017 T. M. Ajayi et al. All rights reserved. On the Usefulness of Cooperation in Person Games Tue, 13 Dec 2016 13:22:53 +0000 http://www.hindawi.com/journals/jam/2016/9734615/ The person games in which each player maximizes his payoff function are considered. We have studied an interesting question for the cooperative game theory about the usefulness of uniting the players in a union. The aim of such cooperation is for each player to get a positive increase to his guaranteed payoff. We have obtained some effective sufficient conditions under which the joining of the players in union is useful for each player. The linear case, specially, is being considered. In the second part of the paper, we have studied the question about the usefulness of cooperation of the players in the presence of the th player, an ill-intentioned destructive player, whose whole aim is not to win but to harm each player individually, and also the union of these players, for example, global terrorism. It should be noted that the considered situation in the second part is related to A. V. Kryazhimskiy’s talk delivered in the summer of 2014. We obtain constructive conditions under which the union of the players is beneficial in this situation as well. Mikhail Sergeevich Nikolskii and Aboubacar Moussa Copyright © 2016 Mikhail Sergeevich Nikolskii and Aboubacar Moussa. All rights reserved. Uniqueness of Solutions to a Nonlinear Elliptic Hessian Equation Thu, 01 Dec 2016 11:29:47 +0000 http://www.hindawi.com/journals/jam/2016/4649150/ Through an Alexandrov-Fenchel inequality, we establish the general Brunn-Minkowski inequality. Then we obtain the uniqueness of solutions to a nonlinear elliptic Hessian equation on . Siyuan Li Copyright © 2016 Siyuan Li. All rights reserved. Theoretical Analysis of the Noise Power Ratio of Nonlinear Power Amplifiers Thu, 01 Dec 2016 06:51:10 +0000 http://www.hindawi.com/journals/jam/2016/8710860/ This paper presents a theoretical analysis and derives the amplifier output noise power spectral density result in a closed form when the input to the amplifier is a band limited Gaussian noise. From the computed power spectral density the NPR is evaluated by a simple subtraction. The method can be applied to any amplifier with known input-output characteristics. The method may be applied to analyze various other important characteristics of the nonlinear amplifier such as spectral regrowth that refers to the spreading of the signal bandwidth when a band limited signal is inputted to the nonlinear amplifier. The paper presents numerical results on the NPR as a function of the noise bandwidth, depth level of the notch, and the output power back-off obtained from the analysis presented in the paper. Rajendra Kumar Copyright © 2016 Rajendra Kumar. All rights reserved. A New Double Color Image Watermarking Algorithm Based on the SVD and Arnold Scrambling Thu, 24 Nov 2016 13:59:50 +0000 http://www.hindawi.com/journals/jam/2016/2497379/ We propose a new image watermarking scheme based on the real SVD and Arnold scrambling to embed a color watermarking image into a color host image. Before embedding watermark, the color watermark image with size of is scrambled by Arnold transformation to obtain a meaningless image . Then, the color host image with size of is divided into nonoverlapping pixel blocks. In each pixel block , we form a real matrix with the red, green, and blue components of and perform the SVD of . We then replace the three smallest singular values of by the red, green, and blue values of with scaling factor, to form a new watermarked host image . With the reserve procedure, we can extract the watermark from the watermarked host image. In the process of the algorithm, we only need to perform real number algebra operations, which have very low computational complexity and are more effective than the one using the quaternion SVD of color image. Ying Li, Musheng Wei, Fengxia Zhang, and Jianli Zhao Copyright © 2016 Ying Li et al. All rights reserved. On Graceful Spider Graphs with at Most Four Legs of Lengths Greater than One Wed, 23 Nov 2016 09:01:45 +0000 http://www.hindawi.com/journals/jam/2016/5327026/ A graceful labeling of a tree with edges is a bijection such that equal to . A spider graph is a tree with at most one vertex of degree greater than . We show that all spider graphs with at most four legs of lengths greater than one admit graceful labeling. A. Panpa and T. Poomsa-ard Copyright © 2016 A. Panpa and T. Poomsa-ard. All rights reserved. A Note on the vec Operator Applied to Unbalanced Block-Structured Matrices Wed, 09 Nov 2016 09:07:35 +0000 http://www.hindawi.com/journals/jam/2016/4590817/ The vec operator transforms a matrix to a column vector by stacking each column on top of the next. It is useful to write the vec of a block-structured matrix in terms of the vec operator applied to each of its component blocks. We derive a simple formula for doing so, which applies regardless of whether the blocks are of the same or of different sizes. Hal Caswell and Silke F. van Daalen Copyright © 2016 Hal Caswell and Silke F. van Daalen. All rights reserved. The Order Classes of 2-Generator -Groups Tue, 08 Nov 2016 07:17:44 +0000 http://www.hindawi.com/journals/jam/2016/8435768/ In order to classify a finite group using its elements orders, the order classes are defined. This partition determines the number of elements for each order. The aim of this paper is to find the order classes of 2-generator -groups of class 2. The results obtained here are supported by Groups, Algorithm and Programming (GAP). Mahmoud Bashir Alhasanat, Bilal Al-Hasanat, and Eman Al-Sarairah Copyright © 2016 Mahmoud Bashir Alhasanat et al. All rights reserved.