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solution representation</title> <meta name="description" content="Search results for: discrete solution representation"> <meta name="keywords" content="discrete solution representation"> <meta name="viewport" content="width=device-width, initial-scale=1, minimum-scale=1, maximum-scale=1, user-scalable=no"> <meta charset="utf-8"> <link href="https://cdn.waset.org/favicon.ico" type="image/x-icon" rel="shortcut icon"> <link href="https://cdn.waset.org/static/plugins/bootstrap-4.2.1/css/bootstrap.min.css" rel="stylesheet"> <link href="https://cdn.waset.org/static/plugins/fontawesome/css/all.min.css" rel="stylesheet"> <link href="https://cdn.waset.org/static/css/site.css?v=150220211555" rel="stylesheet"> </head> <body> <header> <div class="container"> <nav class="navbar navbar-expand-lg navbar-light"> <a class="navbar-brand" href="https://waset.org"> <img src="https://cdn.waset.org/static/images/wasetc.png" alt="Open Science Research Excellence" title="Open Science Research Excellence" /> 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action="https://publications.waset.org/abstracts/search"> <div id="custom-search-input"> <div class="input-group"> <i class="fas fa-search"></i> <input type="text" class="search-query" name="q" placeholder="Author, Title, Abstract, Keywords" value="discrete solution representation"> <input type="submit" class="btn_search" value="Search"> </div> </div> </form> </div> </div> <div class="row mt-3"> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Commenced</strong> in January 2007</div> </div> </div> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Frequency:</strong> Monthly</div> </div> </div> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Edition:</strong> International</div> </div> </div> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Paper Count:</strong> 7405</div> </div> </div> </div> <h1 class="mt-3 mb-3 text-center" style="font-size:1.6rem;">Search results for: discrete solution representation</h1> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7405</span> Fundamental Solutions for Discrete Dynamical Systems Involving the Fractional Laplacian</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Jorge%20Gonzalez%20Camus">Jorge Gonzalez Camus</a>, <a href="https://publications.waset.org/abstracts/search?q=Valentin%20Keyantuo"> Valentin Keyantuo</a>, <a href="https://publications.waset.org/abstracts/search?q=Mahamadi%20Warma"> Mahamadi Warma</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this work, we obtain representation results for solutions of a time-fractional differential equation involving the discrete fractional Laplace operator in terms of generalized Wright functions. Such equations arise in the modeling of many physical systems, for example, chain processes in chemistry and radioactivity. The focus is on the linear problem of the simplified Moore - Gibson - Thompson equation, where the discrete fractional Laplacian and the Caputo fractional derivate of order on (0,2] are involved. As a particular case, we obtain the explicit solution for the discrete heat equation and discrete wave equation. Furthermore, we show the explicit solution for the equation involving the perturbed Laplacian by the identity operator. The main tool for obtaining the explicit solution are the Laplace and discrete Fourier transforms, and Stirling's formula. The methodology mainly is to apply both transforms in the equation, to find the inverse of each transform, and to prove that this solution is well defined, using Stirling´s formula. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=discrete%20fractional%20Laplacian" title="discrete fractional Laplacian">discrete fractional Laplacian</a>, <a href="https://publications.waset.org/abstracts/search?q=explicit%20representation%20of%20solutions" title=" explicit representation of solutions"> explicit representation of solutions</a>, <a href="https://publications.waset.org/abstracts/search?q=fractional%20heat%20and%20wave%20equations" title=" fractional heat and wave equations"> fractional heat and wave equations</a>, <a href="https://publications.waset.org/abstracts/search?q=fundamental" title=" fundamental"> fundamental</a> </p> <a href="https://publications.waset.org/abstracts/99922/fundamental-solutions-for-discrete-dynamical-systems-involving-the-fractional-laplacian" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/99922.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">209</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7404</span> An Improved Discrete Version of Teaching–Learning-Based ‎Optimization for Supply Chain Network Design</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Ehsan%20Yadegari">Ehsan Yadegari</a> </p> <p class="card-text"><strong>Abstract:</strong></p> While there are several metaheuristics and exact approaches to solving the Supply Chain Network Design (SCND) problem, there still remains an unfilled gap in using the Teaching-Learning-Based Optimization (TLBO) algorithm. The algorithm has demonstrated desirable results with problems with complicated combinational optimization. The present study introduces a Discrete Self-Study TLBO (DSS-TLBO) with priority-based solution representation that can solve a supply chain network configuration model to lower the total expenses of establishing facilities and the flow of materials. The network features four layers, namely suppliers, plants, distribution centers (DCs), and customer zones. It is designed to meet the customer’s demand through transporting the material between layers of network and providing facilities in the best economic Potential locations. To have a higher quality of the solution and increase the speed of TLBO, a distinct operator was introduced that ensures self-adaptation (self-study) in the algorithm based on the four types of local search. In addition, while TLBO is used in continuous solution representation and priority-based solution representation is discrete, a few modifications were added to the algorithm to remove the solutions that are infeasible. As shown by the results of experiments, the superiority of DSS-TLBO compared to pure TLBO, genetic algorithm (GA) and firefly Algorithm (FA) was established. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=supply%20chain%20network%20design" title="supply chain network design">supply chain network design</a>, <a href="https://publications.waset.org/abstracts/search?q=teaching%E2%80%93learning-based%20optimization" title=" teaching–learning-based optimization"> teaching–learning-based optimization</a>, <a href="https://publications.waset.org/abstracts/search?q=improved%20metaheuristics" title=" improved metaheuristics"> improved metaheuristics</a>, <a href="https://publications.waset.org/abstracts/search?q=discrete%20solution%20representation" title=" discrete solution representation"> discrete solution representation</a> </p> <a href="https://publications.waset.org/abstracts/184885/an-improved-discrete-version-of-teaching-learning-based-optimization-for-supply-chain-network-design" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/184885.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">52</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7403</span> Analytical Solution of Non–Autonomous Discrete Non-Linear Schrodinger Equation With Saturable Non-Linearity</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Mishu%20Gupta">Mishu Gupta</a>, <a href="https://publications.waset.org/abstracts/search?q=Rama%20Gupta"> Rama Gupta</a> </p> <p class="card-text"><strong>Abstract:</strong></p> It has been elucidated here that non- autonomous discrete non-linear Schrödinger equation is associated with saturable non-linearity through photo-refractive media. We have investigated the localized solution of non-autonomous saturable discrete non-linear Schrödinger equations. The similarity transformation has been involved in converting non-autonomous saturable discrete non-linear Schrödinger equation to constant-coefficient saturable discrete non-linear Schrödinger equation (SDNLSE), whose exact solution is already known. By back substitution, the solution of the non-autonomous version has been obtained. We have analysed our solution for the hyperbolic and periodic form of gain/loss term, and interesting results have been obtained. The most important characteristic role is that it helps us to analyse the propagation of electromagnetic waves in glass fibres and other optical wave mediums. Also, the usage of SDNLSE has been seen in tight binding for Bose-Einstein condensates in optical mediums. Even the solutions are interrelated, and its properties are prominently used in various physical aspects like optical waveguides, Bose-Einstein (B-E) condensates in optical mediums, Non-linear optics in photonic crystals, and non-linear kerr–type non-linearity effect and photo refracting medium. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=B-E-Bose-Einstein" title="B-E-Bose-Einstein">B-E-Bose-Einstein</a>, <a href="https://publications.waset.org/abstracts/search?q=DNLSE-Discrete%20non%20linear%20schrodinger%20equation" title=" DNLSE-Discrete non linear schrodinger equation"> DNLSE-Discrete non linear schrodinger equation</a>, <a href="https://publications.waset.org/abstracts/search?q=NLSE-non%20linear%20schrodinger%20equation" title=" NLSE-non linear schrodinger equation"> NLSE-non linear schrodinger equation</a>, <a href="https://publications.waset.org/abstracts/search?q=SDNLSE%20-%20saturable%20discrete%20non%20linear%20Schrodinger%20equation" title=" SDNLSE - saturable discrete non linear Schrodinger equation"> SDNLSE - saturable discrete non linear Schrodinger equation</a> </p> <a href="https://publications.waset.org/abstracts/121074/analytical-solution-of-non-autonomous-discrete-non-linear-schrodinger-equation-with-saturable-non-linearity" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/121074.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">155</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7402</span> Discrete Group Search Optimizer for the Travelling Salesman Problem</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Raed%20Alnajjar">Raed Alnajjar</a>, <a href="https://publications.waset.org/abstracts/search?q=Mohd%20Zakree"> Mohd Zakree</a>, <a href="https://publications.waset.org/abstracts/search?q=Ahmad%20Nazri"> Ahmad Nazri</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this study, we apply Discrete Group Search Optimizer (DGSO) for solving Traveling Salesman Problem (TSP). The DGSO is a nature inspired optimization algorithm that imitates the animal behavior, especially animal searching behavior. The proposed DGSO uses a vector representation and some discrete operators, such as destruction, construction, differential evolution, swap and insert. The TSP is a well-known hard combinatorial optimization problem, which seeks to find the shortest path among numbers of cities. The performance of the proposed DGSO is evaluated and tested on benchmark instances which listed in LIBTSP dataset. The experimental results show that the performance of the proposed DGSO is comparable with the other methods in the state of the art for some instances. The results show that DGSO outperform Ant Colony System (ACS) in some instances whilst outperform other metaheuristic in most instances. In addition to that, the new results obtained a number of optimal solutions and some best known results. DGSO was able to obtain feasible and good quality solution across all dataset. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=discrete%20group%20search%20optimizer%20%28DGSO%29%3B%20Travelling%20salesman%20problem%20%28TSP%29%3B%20Variable%20neighborhood%20search%28VNS%29" title="discrete group search optimizer (DGSO); Travelling salesman problem (TSP); Variable neighborhood search(VNS)">discrete group search optimizer (DGSO); Travelling salesman problem (TSP); Variable neighborhood search(VNS)</a> </p> <a href="https://publications.waset.org/abstracts/36200/discrete-group-search-optimizer-for-the-travelling-salesman-problem" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/36200.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">324</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7401</span> Multidimensional Integral and Discrete Opial–Type Inequalities</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Maja%20Andri%C4%87">Maja Andrić</a>, <a href="https://publications.waset.org/abstracts/search?q=Josip%20Pe%C4%8Dari%C4%87"> Josip Pečarić</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Over the last five decades, an enormous amount of work has been done on Opial’s integral inequality, dealing with new proofs, various generalizations, extensions and discrete analogs. The Opial inequality is recognized as a fundamental result in the analysis of qualitative properties of solution of differential equations. We use submultiplicative convex functions, appropriate representations of functions and inequalities involving means to obtain generalizations and extensions of certain known multidimensional integral and discrete Opial-type inequalities. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Opial%27s%20inequality" title="Opial's inequality">Opial's inequality</a>, <a href="https://publications.waset.org/abstracts/search?q=Jensen%27s%20inequality" title=" Jensen's inequality"> Jensen's inequality</a>, <a href="https://publications.waset.org/abstracts/search?q=integral%20inequality" title=" integral inequality"> integral inequality</a>, <a href="https://publications.waset.org/abstracts/search?q=discrete%20inequality" title=" discrete inequality"> discrete inequality</a> </p> <a href="https://publications.waset.org/abstracts/41583/multidimensional-integral-and-discrete-opial-type-inequalities" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/41583.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">439</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7400</span> Role of Discrete Event Simulation in the Assessment and Selection of the Potential Reconfigurable Manufacturing Solutions</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Mohsin%20Raza">Mohsin Raza</a>, <a href="https://publications.waset.org/abstracts/search?q=Arne%20Bilberg"> Arne Bilberg</a>, <a href="https://publications.waset.org/abstracts/search?q=Thomas%20Ditlev%20Brun%C3%B8"> Thomas Ditlev Brunø</a>, <a href="https://publications.waset.org/abstracts/search?q=Ann-Louise%20Andersen"> Ann-Louise Andersen</a>, <a href="https://publications.waset.org/abstracts/search?q=Filip%20SK%C3%A4rin"> Filip SKärin</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Shifting from a dedicated or flexible manufacturing system to a reconfigurable manufacturing system (RMS) requires a significant amount of time, money, and effort. Therefore, it is vital to verify beforehand that the potential reconfigurable solution will be able to achieve the organizational objectives. Discrete event simulation offers the opportunity of assessing several reconfigurable alternatives against the set objectives. This study signifies the importance of using discrete-event simulation as a tool to verify several reconfiguration options. Two different industrial cases have been presented in the study to elaborate on the role of discrete event simulation in the implementation methodology of RMSs. The study concluded that discrete event simulation is one of the important tools to consider in the RMS implementation methodology. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=reconfigurable%20manufacturing%20system" title="reconfigurable manufacturing system">reconfigurable manufacturing system</a>, <a href="https://publications.waset.org/abstracts/search?q=discrete%20event%20simulation" title=" discrete event simulation"> discrete event simulation</a>, <a href="https://publications.waset.org/abstracts/search?q=Tecnomatix%20plant%20simulation" title=" Tecnomatix plant simulation"> Tecnomatix plant simulation</a>, <a href="https://publications.waset.org/abstracts/search?q=RMS" title=" RMS"> RMS</a> </p> <a href="https://publications.waset.org/abstracts/150254/role-of-discrete-event-simulation-in-the-assessment-and-selection-of-the-potential-reconfigurable-manufacturing-solutions" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/150254.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">124</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7399</span> Numerical Solution of Integral Equations by Using Discrete GHM Multiwavelet</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Archit%20Yajnik">Archit Yajnik</a>, <a href="https://publications.waset.org/abstracts/search?q=Rustam%20Ali"> Rustam Ali</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, numerical method based on discrete GHM multiwavelets is presented for solving the Fredholm integral equations of second kind. There is hardly any article available in the literature in which the integral equations are numerically solved using discrete GHM multiwavelet. A number of examples are demonstrated to justify the applicability of the method. In GHM multiwavelets, the values of scaling and wavelet functions are calculated only at t = 0, 0.5 and 1. The numerical solution obtained by the present approach is compared with the traditional Quadrature method. It is observed that the present approach is more accurate and computationally efficient as compared to quadrature method. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=GHM%20multiwavelet" title="GHM multiwavelet">GHM multiwavelet</a>, <a href="https://publications.waset.org/abstracts/search?q=fredholm%20integral%20equations" title=" fredholm integral equations"> fredholm integral equations</a>, <a href="https://publications.waset.org/abstracts/search?q=quadrature%20method" title=" quadrature method"> quadrature method</a>, <a href="https://publications.waset.org/abstracts/search?q=function%20approximation" title=" function approximation"> function approximation</a> </p> <a href="https://publications.waset.org/abstracts/36311/numerical-solution-of-integral-equations-by-using-discrete-ghm-multiwavelet" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/36311.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">462</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7398</span> Equivalent Circuit Representation of Lossless and Lossy Power Transmission Systems Including Discrete Sampler</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Yuichi%20Kida">Yuichi Kida</a>, <a href="https://publications.waset.org/abstracts/search?q=Takuro%20Kida"> Takuro Kida</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In a new smart society supported by the recent development of 5G and 6G Communication systems, the im- portance of wireless power transmission is increasing. These systems contain discrete sampling systems in the middle of the transmission path and equivalent circuit representation of lossless or lossy power transmission through these systems is an important issue in circuit theory. In this paper, for the given weight function, we show that a lossless power transmission system with the given weight is expressed by an equivalent circuit representation of the Kida’s optimal signal prediction system followed by a reactance multi-port circuit behind it. Further, it is shown that, when the system is lossy, the system has an equivalent circuit in the form of connecting a multi-port positive-real circuit behind the Kida’s optimal signal prediction system. Also, for the convenience of the reader, in this paper, the equivalent circuit expression of the reactance multi-port circuit and the positive- real multi-port circuit by Cauer and Ohno, whose information is currently being lost even in the world of the Internet. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=signal%20prediction" title="signal prediction">signal prediction</a>, <a href="https://publications.waset.org/abstracts/search?q=pseudo%20inverse%20matrix" title=" pseudo inverse matrix"> pseudo inverse matrix</a>, <a href="https://publications.waset.org/abstracts/search?q=artificial%20%20intelligence" title=" artificial intelligence"> artificial intelligence</a>, <a href="https://publications.waset.org/abstracts/search?q=power%20transmission" title=" power transmission"> power transmission</a> </p> <a href="https://publications.waset.org/abstracts/144600/equivalent-circuit-representation-of-lossless-and-lossy-power-transmission-systems-including-discrete-sampler" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/144600.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">122</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7397</span> Numerical Treatment of Block Method for the Solution of Ordinary Differential Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=A.%20M.%20Sagir">A. M. Sagir</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Discrete linear multistep block method of uniform order for the solution of first order Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) is presented in this paper. The approach of interpolation and collocation approximation are adopted in the derivation of the method which is then applied to first order ordinary differential equations with associated initial conditions. The continuous hybrid formulations enable us to differentiate and evaluate at some grids and off – grid points to obtain four discrete schemes, which were used in block form for parallel or sequential solutions of the problems. Furthermore, a stability analysis and efficiency of the block method are tested on ordinary differential equations, and the results obtained compared favorably with the exact solution. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=block%20method" title="block method">block method</a>, <a href="https://publications.waset.org/abstracts/search?q=first%20order%20ordinary%20differential%20equations" title=" first order ordinary differential equations"> first order ordinary differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=hybrid" title=" hybrid"> hybrid</a>, <a href="https://publications.waset.org/abstracts/search?q=self-starting" title=" self-starting "> self-starting </a> </p> <a href="https://publications.waset.org/abstracts/3426/numerical-treatment-of-block-method-for-the-solution-of-ordinary-differential-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/3426.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">482</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7396</span> Optimization of Fourth Order Discrete-Approximation Inclusions</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Elimhan%20N.%20Mahmudov">Elimhan N. Mahmudov</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The paper concerns the necessary and sufficient conditions of optimality for Cauchy problem of fourth order discrete (PD) and discrete-approximate (PDA) inclusions. The main problem is formulation of the fourth order adjoint discrete and discrete-approximate inclusions and transversality conditions, which are peculiar to problems including fourth order derivatives and approximate derivatives. Thus the necessary and sufficient conditions of optimality are obtained incorporating the Euler-Lagrange and Hamiltonian forms of inclusions. Derivation of optimality conditions are based on the apparatus of locally adjoint mapping (LAM). Moreover in the application of these results we consider the fourth order linear discrete and discrete-approximate inclusions. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=difference" title="difference">difference</a>, <a href="https://publications.waset.org/abstracts/search?q=optimization" title=" optimization"> optimization</a>, <a href="https://publications.waset.org/abstracts/search?q=fourth" title=" fourth"> fourth</a>, <a href="https://publications.waset.org/abstracts/search?q=approximation" title=" approximation"> approximation</a>, <a href="https://publications.waset.org/abstracts/search?q=transversality" title=" transversality"> transversality</a> </p> <a href="https://publications.waset.org/abstracts/25199/optimization-of-fourth-order-discrete-approximation-inclusions" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/25199.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">374</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7395</span> Variational Evolutionary Splines for Solving a Model of Temporomandibular Disorders</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Alberto%20Hananel">Alberto Hananel</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The aim of this work is to modelize the occlusion of a person with temporomandibular disorders as an evolutionary equation and approach its solution by the construction and characterizing of discrete variational splines. To formulate the problem, certain boundary conditions have been considered. After showing the existence and the uniqueness of the solution of such a problem, a convergence result of a discrete variational evolutionary spline is shown. A stress analysis of the occlusion of a human jaw with temporomandibular disorders by finite elements is carried out in FreeFem++ in order to prove the validity of the presented method. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=approximation" title="approximation">approximation</a>, <a href="https://publications.waset.org/abstracts/search?q=evolutionary%20PDE" title=" evolutionary PDE"> evolutionary PDE</a>, <a href="https://publications.waset.org/abstracts/search?q=Finite%20Element%20Method" title=" Finite Element Method"> Finite Element Method</a>, <a href="https://publications.waset.org/abstracts/search?q=temporomandibular%20disorders" title=" temporomandibular disorders"> temporomandibular disorders</a>, <a href="https://publications.waset.org/abstracts/search?q=variational%20spline" title=" variational spline"> variational spline</a> </p> <a href="https://publications.waset.org/abstracts/51438/variational-evolutionary-splines-for-solving-a-model-of-temporomandibular-disorders" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/51438.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">378</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7394</span> Cubical Representation of Prime and Essential Prime Implicants of Boolean Functions</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Saurabh%20Rawat">Saurabh Rawat</a>, <a href="https://publications.waset.org/abstracts/search?q=Anushree%20Sah"> Anushree Sah</a> </p> <p class="card-text"><strong>Abstract:</strong></p> K Maps are generally and ideally, thought to be simplest form for obtaining solution of Boolean equations. Cubical Representation of Boolean equations is an alternate pick to incur a solution, otherwise to be meted out with Truth Tables, Boolean Laws, and different traits of Karnaugh Maps. Largest possible k- cubes that exist for a given function are equivalent to its prime implicants. A technique of minimization of Logic functions is tried to be achieved through cubical methods. The main purpose is to make aware and utilise the advantages of cubical techniques in minimization of Logic functions. All this is done with an aim to achieve minimal cost solution.r <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=K-maps" title="K-maps">K-maps</a>, <a href="https://publications.waset.org/abstracts/search?q=don%E2%80%99t%20care%20conditions" title=" don’t care conditions"> don’t care conditions</a>, <a href="https://publications.waset.org/abstracts/search?q=Boolean%20equations" title=" Boolean equations"> Boolean equations</a>, <a href="https://publications.waset.org/abstracts/search?q=cubes" title=" cubes"> cubes</a> </p> <a href="https://publications.waset.org/abstracts/6866/cubical-representation-of-prime-and-essential-prime-implicants-of-boolean-functions" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/6866.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">385</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7393</span> Discretization of Cuckoo Optimization Algorithm for Solving Quadratic Assignment Problems</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Elham%20Kazemi">Elham Kazemi</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Quadratic Assignment Problem (QAP) is one the combinatorial optimization problems about which research has been done in many companies for allocating some facilities to some locations. The issue of particular importance in this process is the costs of this allocation and the attempt in this problem is to minimize this group of costs. Since the QAP’s are from NP-hard problem, they cannot be solved by exact solution methods. Cuckoo Optimization Algorithm is a Meta-heuristicmethod which has higher capability to find the global optimal points. It is an algorithm which is basically raised to search a continuous space. The Quadratic Assignment Problem is the issue which can be solved in the discrete space, thus the standard arithmetic operators of Cuckoo Optimization Algorithm need to be redefined on the discrete space in order to apply the Cuckoo Optimization Algorithm on the discrete searching space. This paper represents the way of discretizing the Cuckoo optimization algorithm for solving the quadratic assignment problem. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Quadratic%20Assignment%20Problem%20%28QAP%29" title="Quadratic Assignment Problem (QAP)">Quadratic Assignment Problem (QAP)</a>, <a href="https://publications.waset.org/abstracts/search?q=Discrete%20Cuckoo%20Optimization%20Algorithm%20%28DCOA%29" title=" Discrete Cuckoo Optimization Algorithm (DCOA)"> Discrete Cuckoo Optimization Algorithm (DCOA)</a>, <a href="https://publications.waset.org/abstracts/search?q=meta-heuristic%20algorithms" title=" meta-heuristic algorithms"> meta-heuristic algorithms</a>, <a href="https://publications.waset.org/abstracts/search?q=optimization%20algorithms" title=" optimization algorithms"> optimization algorithms</a> </p> <a href="https://publications.waset.org/abstracts/25249/discretization-of-cuckoo-optimization-algorithm-for-solving-quadratic-assignment-problems" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/25249.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">517</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7392</span> Symbiotic Organism Search (SOS) for Solving the Capacitated Vehicle Routing Problem</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Eki%20Ruskartina">Eki Ruskartina</a>, <a href="https://publications.waset.org/abstracts/search?q=Vincent%20F.%20Yu"> Vincent F. Yu</a>, <a href="https://publications.waset.org/abstracts/search?q=Budi%20Santosa"> Budi Santosa</a>, <a href="https://publications.waset.org/abstracts/search?q=A.%20A.%20N.%20Perwira%20Redi"> A. A. N. Perwira Redi</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper introduces symbiotic organism search (SOS) for solving capacitated vehicle routing problem (CVRP). SOS is a new approach in metaheuristics fields and never been used to solve discrete problems. A sophisticated decoding method to deal with a discrete problem setting in CVRP is applied using the basic symbiotic organism search (SOS) framework. The performance of the algorithm was evaluated on a set of benchmark instances and compared results with best known solution. The computational results show that the proposed algorithm can produce good solution as a preliminary testing. These results indicated that the proposed SOS can be applied as an alternative to solve the capacitated vehicle routing problem. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=symbiotic%20organism%20search" title="symbiotic organism search">symbiotic organism search</a>, <a href="https://publications.waset.org/abstracts/search?q=capacitated%20vehicle%20routing%20problem" title=" capacitated vehicle routing problem"> capacitated vehicle routing problem</a>, <a href="https://publications.waset.org/abstracts/search?q=metaheuristic" title=" metaheuristic"> metaheuristic</a> </p> <a href="https://publications.waset.org/abstracts/27109/symbiotic-organism-search-sos-for-solving-the-capacitated-vehicle-routing-problem" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/27109.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">634</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7391</span> The Representation of J. D. Salinger’s Views on Changes in American Society in the 1940s in The Catcher in the Rye</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Jessadaporn%20Achariyopas">Jessadaporn Achariyopas</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The objectives of this study aim to analyze both the protagonist in The Catcher in the Rye in terms of ideological concepts and narrative techniques which influence the construction of the representation and the relationship between the representation and J. D. Salinger’s views on changes in American society in the 1940s. This area of study might concern two theories: namely, a theory of representation and narratology. In addition, this research is intended to answer the following three questions. Firstly, how is the production of meaning through language in The Catcher in the Rye constructed? Secondly, what are J. D. Salinger’s views on changes in American society in the 1940s? Lastly, how is the relationship between the representation and J. D. Salinger’s views? The findings showed that the protagonist’s views, J. D. Salinger’s views, and changes in American society in the 1940s are obviously interrelated. The production of meaning which is the representation of the protagonist’s views was constructed of narrative techniques. J. D. Salinger’s views on changes in American society in the 1940s were the same antisocial perspectives as Holden Caulfield’s which are phoniness, alienation and meltdown. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=representation" title="representation">representation</a>, <a href="https://publications.waset.org/abstracts/search?q=construction%20of%20the%20representation" title=" construction of the representation"> construction of the representation</a>, <a href="https://publications.waset.org/abstracts/search?q=systems%20of%20representation" title=" systems of representation"> systems of representation</a>, <a href="https://publications.waset.org/abstracts/search?q=phoniness" title=" phoniness"> phoniness</a>, <a href="https://publications.waset.org/abstracts/search?q=alienation" title=" alienation"> alienation</a>, <a href="https://publications.waset.org/abstracts/search?q=meltdown" title=" meltdown"> meltdown</a> </p> <a href="https://publications.waset.org/abstracts/9035/the-representation-of-j-d-salingers-views-on-changes-in-american-society-in-the-1940s-in-the-catcher-in-the-rye" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/9035.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">321</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7390</span> Starting Order Eight Method Accurately for the Solution of First Order Initial Value Problems of Ordinary Differential Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=James%20Adewale">James Adewale</a>, <a href="https://publications.waset.org/abstracts/search?q=Joshua%20Sunday"> Joshua Sunday</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, we developed a linear multistep method, which is implemented in predictor corrector-method. The corrector is developed by method of collocation and interpretation of power series approximate solutions at some selected grid points, to give a continuous linear multistep method, which is evaluated at some selected grid points to give a discrete linear multistep method. The predictors were also developed by method of collocation and interpolation of power series approximate solution, to give a continuous linear multistep method. The continuous linear multistep method is then solved for the independent solution to give a continuous block formula, which is evaluated at some selected grid point to give discrete block method. Basic properties of the corrector were investigated and found to be zero stable, consistent and convergent. The efficiency of the method was tested on some linear, non-learn, oscillatory and stiff problems of first order, initial value problems of ordinary differential equations. The results were found to be better in terms of computer time and error bound when compared with the existing methods. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=predictor" title="predictor">predictor</a>, <a href="https://publications.waset.org/abstracts/search?q=corrector" title=" corrector"> corrector</a>, <a href="https://publications.waset.org/abstracts/search?q=collocation" title=" collocation"> collocation</a>, <a href="https://publications.waset.org/abstracts/search?q=interpolation" title=" interpolation"> interpolation</a>, <a href="https://publications.waset.org/abstracts/search?q=approximate%20solution" title=" approximate solution"> approximate solution</a>, <a href="https://publications.waset.org/abstracts/search?q=independent%20solution" title=" independent solution"> independent solution</a>, <a href="https://publications.waset.org/abstracts/search?q=zero%20stable" title=" zero stable"> zero stable</a>, <a href="https://publications.waset.org/abstracts/search?q=consistent" title=" consistent"> consistent</a>, <a href="https://publications.waset.org/abstracts/search?q=convergent" title=" convergent"> convergent</a> </p> <a href="https://publications.waset.org/abstracts/17775/starting-order-eight-method-accurately-for-the-solution-of-first-order-initial-value-problems-of-ordinary-differential-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/17775.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">501</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7389</span> Numerical Modelling of Dry Stone Masonry Structures Based on Finite-Discrete Element Method</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=%C5%BD.%20Nikoli%C4%87">Ž. Nikolić</a>, <a href="https://publications.waset.org/abstracts/search?q=H.%20Smoljanovi%C4%87"> H. Smoljanović</a>, <a href="https://publications.waset.org/abstracts/search?q=N.%20%C5%BDivalji%C4%87"> N. Živaljić</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper presents numerical model based on finite-discrete element method for analysis of the structural response of dry stone masonry structures under static and dynamic loads. More precisely, each discrete stone block is discretized by finite elements. Material non-linearity including fracture and fragmentation of discrete elements as well as cyclic behavior during dynamic load are considered through contact elements which are implemented within a finite element mesh. The application of the model was conducted on several examples of these structures. The performed analysis shows high accuracy of the numerical results in comparison with the experimental ones and demonstrates the potential of the finite-discrete element method for modelling of the response of dry stone masonry structures. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=dry%20stone%20masonry%20structures" title="dry stone masonry structures">dry stone masonry structures</a>, <a href="https://publications.waset.org/abstracts/search?q=dynamic%20load" title=" dynamic load"> dynamic load</a>, <a href="https://publications.waset.org/abstracts/search?q=finite-discrete%20element%20method" title=" finite-discrete element method"> finite-discrete element method</a>, <a href="https://publications.waset.org/abstracts/search?q=static%20load" title=" static load"> static load</a> </p> <a href="https://publications.waset.org/abstracts/47740/numerical-modelling-of-dry-stone-masonry-structures-based-on-finite-discrete-element-method" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/47740.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">414</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7388</span> Discrete Swarm with Passive Congregation for Cost Minimization of the Multiple Vehicle Routing Problem</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Tarek%20Aboueldahab">Tarek Aboueldahab</a>, <a href="https://publications.waset.org/abstracts/search?q=Hanan%20Farag"> Hanan Farag</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Cost minimization of Multiple Vehicle Routing Problem becomes a critical issue in the field of transportation because it is NP-hard optimization problem and the search space is complex. Many researches use the hybridization of artificial intelligence (AI) models to solve this problem; however, it can not guarantee to reach the best solution due to the difficulty of searching the whole search space. To overcome this problem, we introduce the hybrid model of Discrete Particle Swarm Optimization (DPSO) with a passive congregation which enable searching the whole search space to compromise between both local and global search. The practical experiment shows that our model obviously outperforms other hybrid models in cost minimization. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=cost%20minimization" title="cost minimization">cost minimization</a>, <a href="https://publications.waset.org/abstracts/search?q=multi-vehicle%20routing%20problem" title=" multi-vehicle routing problem"> multi-vehicle routing problem</a>, <a href="https://publications.waset.org/abstracts/search?q=passive%20congregation" title=" passive congregation"> passive congregation</a>, <a href="https://publications.waset.org/abstracts/search?q=discrete%20swarm" title=" discrete swarm"> discrete swarm</a>, <a href="https://publications.waset.org/abstracts/search?q=passive%20congregation" title=" passive congregation"> passive congregation</a> </p> <a href="https://publications.waset.org/abstracts/157025/discrete-swarm-with-passive-congregation-for-cost-minimization-of-the-multiple-vehicle-routing-problem" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/157025.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">98</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7387</span> Discrete Breeding Swarm for Cost Minimization of Parallel Job Shop Scheduling Problem</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Tarek%20Aboueldahab">Tarek Aboueldahab</a>, <a href="https://publications.waset.org/abstracts/search?q=Hanan%20Farag"> Hanan Farag</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Parallel Job Shop Scheduling Problem (JSP) is a multi-objective and multi constrains NP- optimization problem. Traditional Artificial Intelligence techniques have been widely used; however, they could be trapped into the local minimum without reaching the optimum solution, so we propose a hybrid Artificial Intelligence model (AI) with Discrete Breeding Swarm (DBS) added to traditional Artificial Intelligence to avoid this trapping. This model is applied in the cost minimization of the Car Sequencing and Operator Allocation (CSOA) problem. The practical experiment shows that our model outperforms other techniques in cost minimization. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=parallel%20job%20shop%20scheduling%20problem" title="parallel job shop scheduling problem">parallel job shop scheduling problem</a>, <a href="https://publications.waset.org/abstracts/search?q=artificial%20intelligence" title=" artificial intelligence"> artificial intelligence</a>, <a href="https://publications.waset.org/abstracts/search?q=discrete%20breeding%20swarm" title=" discrete breeding swarm"> discrete breeding swarm</a>, <a href="https://publications.waset.org/abstracts/search?q=car%20sequencing%20and%20operator%20allocation" title=" car sequencing and operator allocation"> car sequencing and operator allocation</a>, <a href="https://publications.waset.org/abstracts/search?q=cost%20minimization" title=" cost minimization"> cost minimization</a> </p> <a href="https://publications.waset.org/abstracts/132701/discrete-breeding-swarm-for-cost-minimization-of-parallel-job-shop-scheduling-problem" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/132701.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">188</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7386</span> A Hybrid Watermarking Scheme Using Discrete and Discrete Stationary Wavelet Transformation For Color Images</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=B%C3%BClent%20Kantar">Bülent Kantar</a>, <a href="https://publications.waset.org/abstracts/search?q=Numan%20%C3%9Cnald%C4%B1"> Numan Ünaldı</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper presents a new method which includes robust and invisible digital watermarking on images that is colored. Colored images are used as watermark. Frequency region is used for digital watermarking. Discrete wavelet transform and discrete stationary wavelet transform are used for frequency region transformation. Low, medium and high frequency coefficients are obtained by applying the two-level discrete wavelet transform to the original image. Low frequency coefficients are obtained by applying one level discrete stationary wavelet transform separately to all frequency coefficient of the two-level discrete wavelet transformation of the original image. For every low frequency coefficient obtained from one level discrete stationary wavelet transformation, watermarks are added. Watermarks are added to all frequency coefficients of two-level discrete wavelet transform. Totally, four watermarks are added to original image. In order to get back the watermark, the original and watermarked images are applied with two-level discrete wavelet transform and one level discrete stationary wavelet transform. The watermark is obtained from difference of the discrete stationary wavelet transform of the low frequency coefficients. A total of four watermarks are obtained from all frequency of two-level discrete wavelet transform. Obtained watermark results are compared with real watermark results, and a similarity result is obtained. A watermark is obtained from the highest similarity values. Proposed methods of watermarking are tested against attacks of the geometric and image processing. The results show that proposed watermarking method is robust and invisible. All features of frequencies of two level discrete wavelet transform watermarking are combined to get back the watermark from the watermarked image. Watermarks have been added to the image by converting the binary image. These operations provide us with better results in getting back the watermark from watermarked image by attacking of the geometric and image processing. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=watermarking" title="watermarking">watermarking</a>, <a href="https://publications.waset.org/abstracts/search?q=DWT" title=" DWT"> DWT</a>, <a href="https://publications.waset.org/abstracts/search?q=DSWT" title=" DSWT"> DSWT</a>, <a href="https://publications.waset.org/abstracts/search?q=copy%20right%20protection" title=" copy right protection"> copy right protection</a>, <a href="https://publications.waset.org/abstracts/search?q=RGB" title=" RGB "> RGB </a> </p> <a href="https://publications.waset.org/abstracts/16927/a-hybrid-watermarking-scheme-using-discrete-and-discrete-stationary-wavelet-transformation-for-color-images" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/16927.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">535</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7385</span> Superconvergence of the Iterated Discrete Legendre Galerkin Method for Fredholm-Hammerstein Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Payel%20Das">Payel Das</a>, <a href="https://publications.waset.org/abstracts/search?q=Gnaneshwar%20Nelakanti"> Gnaneshwar Nelakanti</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper we analyse the iterated discrete Legendre Galerkin method for Fredholm-Hammerstein integral equations with smooth kernel. Using sufficiently accurate numerical quadrature rule, we obtain superconvergence rates for the iterated discrete Legendre Galerkin solutions in both infinity and $L^2$-norm. Numerical examples are given to illustrate the theoretical results. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=hammerstein%20integral%20equations" title="hammerstein integral equations">hammerstein integral equations</a>, <a href="https://publications.waset.org/abstracts/search?q=spectral%20method" title=" spectral method"> spectral method</a>, <a href="https://publications.waset.org/abstracts/search?q=discrete%20galerkin" title=" discrete galerkin"> discrete galerkin</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical%0D%0Aquadrature" title=" numerical quadrature"> numerical quadrature</a>, <a href="https://publications.waset.org/abstracts/search?q=superconvergence" title=" superconvergence"> superconvergence</a> </p> <a href="https://publications.waset.org/abstracts/22260/superconvergence-of-the-iterated-discrete-legendre-galerkin-method-for-fredholm-hammerstein-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/22260.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">469</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7384</span> A Look at the Quantum Theory of Atoms in Molecules from the Discrete Morse Theory</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Dairo%20Jose%20Hernandez%20Paez">Dairo Jose Hernandez Paez</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The quantum theory of atoms in molecules (QTAIM) allows us to obtain topological information on electronic density in quantum mechanical systems. The QTAIM starts by considering the electron density as a continuous mathematical object. On the other hand, the discretization of electron density is also a mathematical object, which, from discrete mathematics, would allow a new approach to its topological study. From this point of view, it is necessary to develop a series of steps that provide the theoretical support that guarantees its application. Some of the steps that we consider most important are mentioned below: (1) obtain good representations of the electron density through computational calculations, (2) design a methodology for the discretization of electron density, and construct the simplicial complex. (3) Make an analysis of the discrete vector field associating the simplicial complex. (4) Finally, in this research, we propose to use the discrete Morse theory as a mathematical tool to carry out studies of electron density topology. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=discrete%20mathematics" title="discrete mathematics">discrete mathematics</a>, <a href="https://publications.waset.org/abstracts/search?q=Discrete%20Morse%20theory" title=" Discrete Morse theory"> Discrete Morse theory</a>, <a href="https://publications.waset.org/abstracts/search?q=electronic%20density" title=" electronic density"> electronic density</a>, <a href="https://publications.waset.org/abstracts/search?q=computational%20calculations" title=" computational calculations"> computational calculations</a> </p> <a href="https://publications.waset.org/abstracts/163354/a-look-at-the-quantum-theory-of-atoms-in-molecules-from-the-discrete-morse-theory" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/163354.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">104</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7383</span> Innovative Pictogram Chinese Characters Representation</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=J.%20H.%20Low">J. H. Low</a>, <a href="https://publications.waset.org/abstracts/search?q=S.%20H.%20Hew"> S. H. Hew</a>, <a href="https://publications.waset.org/abstracts/search?q=C.%20O.%20Wong"> C. O. Wong</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper proposes an innovative approach to represent the pictogram Chinese characters. The advantage of this representation is using an extraordinary to represent the pictogram Chinese character. This extraordinary representation is created accordingly to the original pictogram Chinese characters revolution. The purpose of this innovative creation is to assistant the learner learning Chinese as second language (SCL) in Chinese language learning specifically on memorize Chinese characters. Commonly, the SCL will give up and frustrate easily while memorize the Chinese characters by rote. So, our innovative representation is able to help on memorize the Chinese character by the help of visually storytelling. This innovative representation enhances the Chinese language learning experience of SCL. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Chinese%20e-learning" title="Chinese e-learning">Chinese e-learning</a>, <a href="https://publications.waset.org/abstracts/search?q=innovative%20Chinese%20character%20representation" title=" innovative Chinese character representation"> innovative Chinese character representation</a>, <a href="https://publications.waset.org/abstracts/search?q=knowledge%20management" title=" knowledge management"> knowledge management</a>, <a href="https://publications.waset.org/abstracts/search?q=language%20learning" title=" language learning"> language learning</a> </p> <a href="https://publications.waset.org/abstracts/4346/innovative-pictogram-chinese-characters-representation" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/4346.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">487</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7382</span> Discrete Tracking Control of Nonholonomic Mobile Robots: Backstepping Design Approach</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Alexander%20S.%20Andreev">Alexander S. Andreev</a>, <a href="https://publications.waset.org/abstracts/search?q=Olga%20A.%20Peregudova"> Olga A. Peregudova</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, we propose a discrete tracking control of nonholonomic mobile robots with two degrees of freedom. The electro-mechanical model of a mobile robot moving on a horizontal surface without slipping, with two rear wheels controlled by two independent DC electric, and one front roal wheel is considered. We present back-stepping design based on the Euler approximate discrete-time model of a continuous-time plant. Theoretical considerations are verified by numerical simulation. The work was supported by RFFI (15-01-08482). <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=actuator%20dynamics" title="actuator dynamics">actuator dynamics</a>, <a href="https://publications.waset.org/abstracts/search?q=back%20stepping" title=" back stepping"> back stepping</a>, <a href="https://publications.waset.org/abstracts/search?q=discrete-time%20controller" title=" discrete-time controller"> discrete-time controller</a>, <a href="https://publications.waset.org/abstracts/search?q=Lyapunov%20function" title=" Lyapunov function"> Lyapunov function</a>, <a href="https://publications.waset.org/abstracts/search?q=wheeled%20mobile%20robot" title=" wheeled mobile robot "> wheeled mobile robot </a> </p> <a href="https://publications.waset.org/abstracts/15632/discrete-tracking-control-of-nonholonomic-mobile-robots-backstepping-design-approach" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/15632.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">415</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7381</span> Implementation of Invisible Digital Watermarking</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=V.%20Monisha">V. Monisha</a>, <a href="https://publications.waset.org/abstracts/search?q=D.%20Sindhuja"> D. Sindhuja</a>, <a href="https://publications.waset.org/abstracts/search?q=M.%20Sowmiya"> M. Sowmiya</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Over the decade, the applications about multimedia have been developed rapidly. The advancement in the communication field at the faster pace, it is necessary to protect the data during transmission. Thus, security of multimedia contents becomes a vital issue, and it is a need for protecting the digital content against malfunctions. Digital watermarking becomes the solution for the copyright protection and authentication of data in the network. In multimedia applications, embedded watermarks should be robust, and imperceptible. For improving robustness, the discrete wavelet transform is used. Both encoding and extraction algorithm can be done using MATLAB R2012a. In this Discrete wavelet transform (DWT) domain of digital image, watermarking algorithm is used, and hardware implementation can be done on Xilinx based FPGA. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=digital%20watermarking" title="digital watermarking">digital watermarking</a>, <a href="https://publications.waset.org/abstracts/search?q=DWT" title=" DWT"> DWT</a>, <a href="https://publications.waset.org/abstracts/search?q=robustness" title=" robustness"> robustness</a>, <a href="https://publications.waset.org/abstracts/search?q=FPGA" title=" FPGA"> FPGA</a> </p> <a href="https://publications.waset.org/abstracts/47802/implementation-of-invisible-digital-watermarking" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/47802.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">413</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7380</span> An Efficient Motion Recognition System Based on LMA Technique and a Discrete Hidden Markov Model</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Insaf%20Ajili">Insaf Ajili</a>, <a href="https://publications.waset.org/abstracts/search?q=Malik%20Mallem"> Malik Mallem</a>, <a href="https://publications.waset.org/abstracts/search?q=Jean-Yves%20Didier"> Jean-Yves Didier</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Human motion recognition has been extensively increased in recent years due to its importance in a wide range of applications, such as human-computer interaction, intelligent surveillance, augmented reality, content-based video compression and retrieval, etc. However, it is still regarded as a challenging task especially in realistic scenarios. It can be seen as a general machine learning problem which requires an effective human motion representation and an efficient learning method. In this work, we introduce a descriptor based on Laban Movement Analysis technique, a formal and universal language for human movement, to capture both quantitative and qualitative aspects of movement. We use Discrete Hidden Markov Model (DHMM) for training and classification motions. We improve the classification algorithm by proposing two DHMMs for each motion class to process the motion sequence in two different directions, forward and backward. Such modification allows avoiding the misclassification that can happen when recognizing similar motions. Two experiments are conducted. In the first one, we evaluate our method on a public dataset, the Microsoft Research Cambridge-12 Kinect gesture data set (MSRC-12) which is a widely used dataset for evaluating action/gesture recognition methods. In the second experiment, we build a dataset composed of 10 gestures(Introduce yourself, waving, Dance, move, turn left, turn right, stop, sit down, increase velocity, decrease velocity) performed by 20 persons. The evaluation of the system includes testing the efficiency of our descriptor vector based on LMA with basic DHMM method and comparing the recognition results of the modified DHMM with the original one. Experiment results demonstrate that our method outperforms most of existing methods that used the MSRC-12 dataset, and a near perfect classification rate in our dataset. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=human%20motion%20recognition" title="human motion recognition">human motion recognition</a>, <a href="https://publications.waset.org/abstracts/search?q=motion%20representation" title=" motion representation"> motion representation</a>, <a href="https://publications.waset.org/abstracts/search?q=Laban%20Movement%20Analysis" title=" Laban Movement Analysis"> Laban Movement Analysis</a>, <a href="https://publications.waset.org/abstracts/search?q=Discrete%20Hidden%20Markov%20Model" title=" Discrete Hidden Markov Model"> Discrete Hidden Markov Model</a> </p> <a href="https://publications.waset.org/abstracts/87469/an-efficient-motion-recognition-system-based-on-lma-technique-and-a-discrete-hidden-markov-model" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/87469.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">207</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7379</span> Real-Time Image Encryption Using a 3D Discrete Dual Chaotic Cipher</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=M.%20F.%20Haroun">M. F. Haroun</a>, <a href="https://publications.waset.org/abstracts/search?q=T.%20A.%20Gulliver"> T. A. Gulliver</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, an encryption algorithm is proposed for real-time image encryption. The scheme employs a dual chaotic generator based on a three dimensional (3D) discrete Lorenz attractor. Encryption is achieved using non-autonomous modulation where the data is injected into the dynamics of the master chaotic generator. The second generator is used to permute the dynamics of the master generator using the same approach. Since the data stream can be regarded as a random source, the resulting permutations of the generator dynamics greatly increase the security of the transmitted signal. In addition, a technique is proposed to mitigate the error propagation due to the finite precision arithmetic of digital hardware. In particular, truncation and rounding errors are eliminated by employing an integer representation of the data which can easily be implemented. The simple hardware architecture of the algorithm makes it suitable for secure real-time applications. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=chaotic%20systems" title="chaotic systems">chaotic systems</a>, <a href="https://publications.waset.org/abstracts/search?q=image%20encryption" title=" image encryption"> image encryption</a>, <a href="https://publications.waset.org/abstracts/search?q=non-autonomous%20modulation" title=" non-autonomous modulation"> non-autonomous modulation</a>, <a href="https://publications.waset.org/abstracts/search?q=FPGA" title=" FPGA"> FPGA</a> </p> <a href="https://publications.waset.org/abstracts/25228/real-time-image-encryption-using-a-3d-discrete-dual-chaotic-cipher" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/25228.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">506</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7378</span> Existence of Positive Solutions for Second-Order Difference Equation with Discrete Boundary Value Problem</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Thanin%20Sitthiwirattham">Thanin Sitthiwirattham</a>, <a href="https://publications.waset.org/abstracts/search?q=Jiraporn%20Reunsumrit">Jiraporn Reunsumrit</a> </p> <p class="card-text"><strong>Abstract:</strong></p> We study the existence of positive solutions to the three points difference summation boundary value problem. We show the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed point theorem due to Krasnoselskii in cones. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=positive%20solution" title="positive solution">positive solution</a>, <a href="https://publications.waset.org/abstracts/search?q=boundary%20value%20problem" title=" boundary value problem"> boundary value problem</a>, <a href="https://publications.waset.org/abstracts/search?q=fixed%20point%20theorem" title=" fixed point theorem"> fixed point theorem</a>, <a href="https://publications.waset.org/abstracts/search?q=cone" title=" cone"> cone</a> </p> <a href="https://publications.waset.org/abstracts/5029/existence-of-positive-solutions-for-second-order-difference-equation-with-discrete-boundary-value-problem" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/5029.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">439</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7377</span> A Particle Filter-Based Data Assimilation Method for Discrete Event Simulation</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Zhi%20Zhu">Zhi Zhu</a>, <a href="https://publications.waset.org/abstracts/search?q=Boquan%20Zhang"> Boquan Zhang</a>, <a href="https://publications.waset.org/abstracts/search?q=Tian%20Jing"> Tian Jing</a>, <a href="https://publications.waset.org/abstracts/search?q=Jingjing%20Li"> Jingjing Li</a>, <a href="https://publications.waset.org/abstracts/search?q=Tao%20Wang"> Tao Wang</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Data assimilation is a model and data hybrid-driven method that dynamically fuses new observation data with a numerical model to iteratively approach the real system state. It is widely used in state prediction and parameter inference of continuous systems. Because of the discrete event system’s non-linearity and non-Gaussianity, traditional Kalman Filter based on linear and Gaussian assumptions cannot perform data assimilation for such systems, so particle filter has gradually become a technical approach for discrete event simulation data assimilation. Hence, we proposed a particle filter-based discrete event simulation data assimilation method and took the unmanned aerial vehicle (UAV) maintenance service system as a proof of concept to conduct simulation experiments. The experimental results showed that the filtered state data is closer to the real state of the system, which verifies the effectiveness of the proposed method. This research can provide a reference framework for the data assimilation process of other complex nonlinear systems, such as discrete-time and agent simulation. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=discrete%20event%20simulation" title="discrete event simulation">discrete event simulation</a>, <a href="https://publications.waset.org/abstracts/search?q=data%20assimilation" title=" data assimilation"> data assimilation</a>, <a href="https://publications.waset.org/abstracts/search?q=particle%20filter" title=" particle filter"> particle filter</a>, <a href="https://publications.waset.org/abstracts/search?q=model%20and%20data-driven" title=" model and data-driven"> model and data-driven</a> </p> <a href="https://publications.waset.org/abstracts/192559/a-particle-filter-based-data-assimilation-method-for-discrete-event-simulation" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/192559.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">13</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">7376</span> Convergence of Sinc Methods Applied to Kuramoto-Sivashinsky Equation</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Kamel%20Al-Khaled">Kamel Al-Khaled</a> </p> <p class="card-text"><strong>Abstract:</strong></p> A comparative study of the Sinc-Galerkin and Sinc-Collocation methods for solving the Kuramoto-Sivashinsky equation is given. Both approaches depend on using Sinc basis functions. Firstly, a numerical scheme using Sinc-Galerkin method is developed to approximate the solution of Kuramoto-Sivashinsky equation. Sinc approximations to both derivatives and indefinite integrals reduces the solution to an explicit system of algebraic equations. The error in the solution is shown to converge to the exact solution at an exponential. The convergence proof of the solution for the discrete system is given using fixed-point iteration. Secondly, a combination of a Crank-Nicolson formula in the time direction, with the Sinc-collocation in the space direction is presented, where the derivatives in the space variable are replaced by the necessary matrices to produce a system of algebraic equations. The methods are tested on two examples. The demonstrated results show that both of the presented methods more or less have the same accuracy. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Sinc-Collocation" title="Sinc-Collocation">Sinc-Collocation</a>, <a href="https://publications.waset.org/abstracts/search?q=nonlinear%20PDEs" title=" nonlinear PDEs"> nonlinear PDEs</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical%20methods" title=" numerical methods"> numerical methods</a>, <a href="https://publications.waset.org/abstracts/search?q=fixed-point" title=" fixed-point"> fixed-point</a> </p> <a href="https://publications.waset.org/abstracts/9717/convergence-of-sinc-methods-applied-to-kuramoto-sivashinsky-equation" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/9717.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads 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