Journal of Applied Mathematics
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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.

Comparing an Approximate Queuing Approach with Simulation for the Solution of a CrossDocking Problem
Sun, 28 May 2017 00:00:00 +0000
http://www.hindawi.com/journals/jam/2017/4987127/
Crossdocking 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 crossdocking 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.

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 skinfriction 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 2Page 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 2page 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 longterm 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 longterm 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 FirstOrder 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 ChebyshevGaussLobatto 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 secondorder 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. ArizaHernandez, Jorge SanchezOrtiz, Martin P. ArcigaAlejandre, and Luis X. VivasCruz
Copyright © 2017 Francisco J. ArizaHernandez et al. All rights reserved.

Implicit OneStep Block Hybrid ThirdDerivative Method for the Direct Solution of Initial Value Problems of SecondOrder Ordinary Differential Equations
Wed, 18 Jan 2017 09:11:45 +0000
http://www.hindawi.com/journals/jam/2017/8510948/
A new onestep block method with generalized three hybrid points for solving initial value problems of secondorder 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 secondorder 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 SecondOrder Fredholm Integrodifferential Equations with Boundary Conditions by QuadratureDifference Method
Wed, 11 Jan 2017 00:00:00 +0000
http://www.hindawi.com/journals/jam/2017/2645097/
In this research, the quadraturedifference method with Gauss Elimination (GE) method is applied for solving the secondorder of linear Fredholm integrodifferential equations (LFIDEs). In order to derive an approximation equation, the combinations of Composite Simpson’s 1/3 rule and secondorder finitedifference method are used to discretize the secondorder 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 quadraturedifference 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 wellstudied 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. GarciaMartinez, F. R. McMorris, O. Ortega, and R. C. Powers
Copyright © 2017 C. GarciaMartinez 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 nonNewtonian 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 temperaturedependent plastic dynamic viscosity is considered. An attempt has been made to focus on the case of twodimensional Casson fluid flow over a horizontal melting surface embedded in a thermally stratified medium. Since the viscosity of the nonNewtonian 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 spacedependent heat source; plastic dynamic viscosity and thermal conductivity of the nonNewtonian 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 RungeKutta 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 illintentioned 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 AlexandrovFenchel inequality, we establish the general BrunnMinkowski 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 inputoutput 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 backoff 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. Poomsaard
Copyright © 2016 A. Panpa and T. Poomsaard. All rights reserved.

A Note on the vec Operator Applied to Unbalanced BlockStructured 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 blockstructured 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 2Generator 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 2generator groups of class 2. The results obtained here are supported by Groups, Algorithm and Programming (GAP).
Mahmoud Bashir Alhasanat, Bilal AlHasanat, and Eman AlSarairah
Copyright © 2016 Mahmoud Bashir Alhasanat et al. All rights reserved.

Bounds on the Spectral Radius of a Nonnegative Matrix and Its Applications
Sun, 06 Nov 2016 10:36:59 +0000
http://www.hindawi.com/journals/jam/2016/3812736/
We obtain the sharp bounds for the spectral radius of a nonnegative matrix and then obtain some known results or new results by applying these bounds to a graph or a digraph and revise and improve two known results.
Danping Huang and Lihua You
Copyright © 2016 Danping Huang and Lihua You. All rights reserved.

A ThreeState MarkovModulated Switching Model for Exchange Rates
Mon, 31 Oct 2016 06:04:37 +0000
http://www.hindawi.com/journals/jam/2016/5061749/
Several authors have examined the long swings hypothesis in exchange rates using a twostate Markov switching model. This study developed a model to investigate long swings hypothesis in currencies which may exhibit a state pattern. The proposed model was then applied to euros, British pounds, Japanese yen, and Nigerian naira. Specification measures such as AIC, BIC, and HIC favoured a threestate pattern in Nigerian naira but a twostate one in the other three currencies. For the period January 2004 to May 2016, empirical results suggested the presence of asymmetric swings in naira and yen and long swings in euros and pounds. In addition, taking as the benchmark for smoothing probabilities, choice models provided a clear reading of the cycle in a manner that is consistent with the realities of the movements in corresponding exchange rate series.
Idowu Oluwasayo Ayodeji
Copyright © 2016 Idowu Oluwasayo Ayodeji. All rights reserved.

An Analytically Tractable Model for Pricing Multiasset Options with Correlated JumpDiffusion Equity Processes and a TwoFactor Stochastic Yield Curve
Thu, 27 Oct 2016 07:16:46 +0000
http://www.hindawi.com/journals/jam/2016/8029750/
This paper shows how to value multiasset options analytically in a modeling framework that combines both continuous and discontinuous variations in the underlying equity or foreign exchange processes and a stochastic, twofactor yield curve. All correlations are taken into account, between the factors driving the yield curve, between fixed income and equity as asset classes, and between the individual equity assets themselves. The valuation method is applied to three of the most popular twoasset options.
Tristan Guillaume
Copyright © 2016 Tristan Guillaume. All rights reserved.

A New Algorithm for Positive Semidefinite Matrix Completion
Tue, 25 Oct 2016 08:34:35 +0000
http://www.hindawi.com/journals/jam/2016/1659019/
Positive semidefinite matrix completion (PSDMC) aims to recover positive semidefinite and lowrank matrices from a subset of entries of a matrix. It is widely applicable in many fields, such as statistic analysis and system control. This task can be conducted by solving the nuclear norm regularized linear least squares model with positive semidefinite constraints. We apply the widely used alternating direction method of multipliers to solve the model and get a novel algorithm. The applicability and efficiency of the new algorithm are demonstrated in numerical experiments. Recovery results show that our algorithm is helpful.
Fangfang Xu and Peng Pan
Copyright © 2016 Fangfang Xu and Peng Pan. All rights reserved.

ILS Heuristics for the SingleMachine Scheduling Problem with SequenceDependent Family Setup Times to Minimize Total Tardiness
Thu, 20 Oct 2016 07:34:39 +0000
http://www.hindawi.com/journals/jam/2016/9598041/
This paper addresses a singlemachine scheduling problem with sequencedependent family setup times. In this problem the jobs are classified into families according to their similarity characteristics. Setup times are required on each occasion when the machine switches from processing jobs in one family to jobs in another family. The performance measure to be minimized is the total tardiness with respect to the given due dates of the jobs. The problem is classified as hard in the ordinary sense. Since the computational complexity associated with the mathematical formulation of the problem makes it difficult for optimization solvers to deal with largesized instances in reasonable solution time, efficient heuristic algorithms are needed to obtain nearoptimal solutions. In this work we propose three heuristics based on the Iterated Local Search (ILS) metaheuristic. The first heuristic is a basic ILS, the second uses a dynamic perturbation size, and the third uses a Path Relinking (PR) technique as an intensification strategy. We carry out comprehensive computational and statistical experiments in order to analyze the performance of the proposed heuristics. The computational experiments show that the ILS heuristics outperform a genetic algorithm proposed in the literature. The ILS heuristic with dynamic perturbation size and PR intensification has a superior performance compared to other heuristics.
Vinícius Vilar Jacob and José Elias C. Arroyo
Copyright © 2016 Vinícius Vilar Jacob and José Elias C. Arroyo. All rights reserved.

Pricing Basket Options by Polynomial Approximations
Wed, 05 Oct 2016 08:02:27 +0000
http://www.hindawi.com/journals/jam/2016/9747394/
We propose a closedform approximation for the price of basket options under a multivariate BlackScholes model. The method is based on Taylor and Chebyshev expansions and involves mixed exponentialpower moments of a Gaussian distribution. Our numerical results show that both approaches are comparable in accuracy to a standard Monte Carlo method, with a lesser computational effort.
Pablo Olivares and Alexander Alvarez
Copyright © 2016 Pablo Olivares and Alexander Alvarez. All rights reserved.

A Formula for the Energy of Circulant Graphs with Two Generators
Thu, 22 Sep 2016 08:34:43 +0000
http://www.hindawi.com/journals/jam/2016/1793978/
We derive closed formulas for the energy of circulant graphs generated by and , where is an integer. We also find a formula for the energy of the complete graph without a Hamilton cycle.
Justine Louis
Copyright © 2016 Justine Louis. All rights reserved.

Convergence Analysis on Unstructured Meshes of a DDFV Method for Flow Problems with Full Neumann Boundary Conditions
Sun, 18 Sep 2016 14:00:47 +0000
http://www.hindawi.com/journals/jam/2016/5891064/
A Discrete Duality Finite Volume (DDFV) method to solve on unstructured meshes the flow problems in anisotropic nonhomogeneous porous media with full Neumann boundary conditions is proposed in the present work. We start with the derivation of the discrete problem. A result of existence and uniqueness of a solution for that problem is given thanks to the properties of its associated matrix combined with adequate assumptions on data. Their theoretical properties, namely, stability and error estimates (in discrete energy norms and norm), are investigated. Numerical test is provided.
A. Kinfack Jeutsa, A. Njifenjou, and J. Nganhou
Copyright © 2016 A. Kinfack Jeutsa et al. All rights reserved.

Multiple Solutions of Mixed Convective MHD Casson Fluid Flow in a Channel
Sun, 04 Sep 2016 11:17:09 +0000
http://www.hindawi.com/journals/jam/2016/7535793/
A numerical investigation is made to determine the occurrence of the multiple solutions of MHD Casson fluid in a porous channel. Governing partial differential equation of the proposed problem converted into nonlinear ordinary differential equations by using similarity transformation. Numerical technique known as shooting method is used to investigate the existence of the multiple solutions for the variations of different parameters. Effects of physical parameters on velocity profile, temperature, concentration, and skin friction are presented in pictorial and tabulation representation.
Jawad Raza, Azizah Mohd Rohni, and Zurni Omar
Copyright © 2016 Jawad Raza et al. All rights reserved.

Bounding Regions to Plane Steepest Descent Curves of Quasiconvex Families
Wed, 27 Jul 2016 08:35:57 +0000
http://www.hindawi.com/journals/jam/2016/4873276/
Twodimensional steepest descent curves (SDC) for a quasiconvex family are considered; the problem of their extensions (with constraints) outside of a convex body is studied. It is shown that possible extensions are constrained to lie inside of suitable bounding regions depending on . These regions are bounded by arcs of involutes of and satisfy many inclusions properties. The involutes of the boundary of an arbitrary plane convex body are defined and written by their support function. Extensions SDC of minimal length are constructed. Selfcontracting sets (with opposite orientation) are considered: necessary and/or sufficient conditions for them to be subsets of SDC are proved.
Marco Longinetti, Paolo Manselli, and Adriana Venturi
Copyright © 2016 Marco Longinetti et al. All rights reserved.