Search results for: Resource constrained project scheduling problem

Authors: A. M. Sagir

Abstract:

In this paper, self-starting block hybrid method of order (5,5,5,5)T is proposed for the solution of the special second order ordinary differential equations with associated initial or boundary 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. The computational burden and computer time wastage involved in the usual reduction of second order problem into system of first order equations are avoided by this approach. Furthermore, a stability analysis and efficiency of the block method are tested on stiff ordinary differential equations, and the results obtained compared favorably with the exact solution.

Keywords: Block Method, Hybrid, Linear Multistep Method, Self – starting, Special Second Order.

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Authors: Yasser R. Tawfic, Mohamed A. Eid

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Foundation differential settlement and supported structure tilting are an occasionally occurred engineering problem. This may be caused by overloading, changes in ground soil properties or unsupported nearby excavations. Engineering thinking points directly toward the logic solution for such problem by uplifting the settled side. This can be achieved with deep foundation elements such as micro-piles and macro-piles™, jacked piers, and helical piers, jet grouted mortar columns, compaction grout columns, cement grouting or with chemical grouting, or traditional pit underpinning with concrete and mortar. Although, some of these techniques offer economic, fast and low noise solutions, many of them are quite the contrary. For tilted structures, with the limited inclination, it may be much easier to cause a balancing settlement on the less-settlement side which shall be done carefully in a proper rate. This principal has been applied in Leaning Tower of Pisa stabilization with soil extraction from the ground surface. In this research, the authors attempt to introduce a new solution with a different point of view. So, the micro-tunneling technique is presented in here as an intended ground deformation cause. In general, micro-tunneling is expected to induce limited ground deformations. Thus, the researchers propose to apply the technique to form small size ground unsupported holes to produce the target deformations. This shall be done in four phases: 1. Application of one or more micro-tunnels, regarding the existing differential settlement value, under the raised side of the tilted structure. 2. For each individual tunnel, the lining shall be pulled out from both sides (from jacking and receiving shafts) in the slow rate. 3. If required, according to calculations and site records, an additional surface load can be applied on the raised foundation side. 4. Finally, a strengthening soil grouting shall be applied for stabilization after adjustment. A finite element based numerical model is presented to simulate the proposed construction phases for different tunneling positions and tunnels group. For each case, the surface settlements are calculated and induced plasticity points are checked. These results show the impact of the suggested procedure on the tilted structure and its feasibility. Comparing results also show the importance of the position selection and tunnels group gradual effect. Thus, a new engineering solution is presented to one of the structural and geotechnical engineering challenges.

Keywords: Differential settlement, micro-tunnel, soil-structure interaction, tilted structures.

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Authors: Kavita Burse, Manish Manoria, Vishnu P. S. Kirar

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The back propagation algorithm calculates the weight changes of artificial neural networks, and a common approach is to use a training algorithm consisting of a learning rate and a momentum factor. The major drawbacks of above learning algorithm are the problems of local minima and slow convergence speeds. The addition of an extra term, called a proportional factor reduces the convergence of the back propagation algorithm. We have applied the three term back propagation to multiplicative neural network learning. The algorithm is tested on XOR and parity problem and compared with the standard back propagation training algorithm.

Keywords: Three term back propagation, multiplicative neuralnetwork, proportional factor, local minima.

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Authors: Daniel Morariu, Lucian N. Vintan, Volker Tresp

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Text categorization is the problem of classifying text documents into a set of predefined classes. After a preprocessing step, the documents are typically represented as large sparse vectors. When training classifiers on large collections of documents, both the time and memory restrictions can be quite prohibitive. This justifies the application of feature selection methods to reduce the dimensionality of the document-representation vector. In this paper, three feature selection methods are evaluated: Random Selection, Information Gain (IG) and Support Vector Machine feature selection (called SVM_FS). We show that the best results were obtained with SVM_FS method for a relatively small dimension of the feature vector. Also we present a novel method to better correlate SVM kernel-s parameters (Polynomial or Gaussian kernel).

Keywords: Feature Selection, Learning with Kernels, SupportVector Machine, and Classification.

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Authors: Nurettin Simsek, Özlem Cakır

Abstract:

The aim of this study is to point out whether personalization of mathematical word problems could affect student achievement or not. The research was applied on two-grades students at spring semester 2008-2009. Before the treatment, students personal data were taken and given to the computer. During the treatment, paper-based personalized problems and paper-based non personalized problems were prepared by computer as the same problems and then these problems were given to students. At the end of the treatment, students- opinion was taken. As a result of this research, it was found out that there were no significant differences between learners through personalized or non-personalized materials, and also there were no significant differences between gender through personalized and non-personalized problems. However, opinion of students was highly positive through the personalized problems.

Keywords: Personalization, word problem, computer aided personalization.

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Authors: F. Alborzi, H. Vafaei, M.H. Gholami, M.M. S. Esfahani

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In this article, the design of a Supply Chain Network (SCN) consisting of several suppliers, production plants, distribution centers and retailers, is considered. Demands of retailers are considered stochastic parameters, so we generate amounts of data via simulation to extract a few demand scenarios. Then a mixed integer two-stage programming model is developed to optimize simultaneously two objectives: (1) minimization the fixed and variable cost, (2) maximization the service level. A weighting method is utilized to solve this two objective problem and a numerical example is made to show the performance of the model.

Keywords: Mixed Integer Programming, Multi-objective Optimization, Stochastic Demand, Supply Chain Design, Two Stage Programming

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Authors: B. Lorestani, M. Cheraghi, N. Yousefi

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A research project dealing with the phytoremediation of a soil polluted by some heavy metals is currently running. The case study is represented by a mining area in Hamedan province in the central west part of Iran. The potential of phytoextraction and phytostabilization of plants was evaluated considering the concentration of heavy metals in the plant tissues and also the bioconcentration factor (BCF) and the translocation factor (TF). Also the several established criteria were applied to define hyperaccumulator plants in the studied area. Results showed that none of the collected plant species were suitable for phytoextraction of Cu, Zn, Fe and Mn, but among the plants, Euphorbia macroclada was the most efficient in phytostabilization of Cu and Fe, while, Ziziphora clinopodioides, Cousinia sp. and Chenopodium botrys were the most suitable for phytostabilization of Zn and Chondrila juncea and Stipa barbata had the potential for phytostabilization of Mn. Using the most common criterion, Euphorbia macroclada and Verbascum speciosum were Fe hyperaccumulator plants. Present study showed that native plant species growing on contaminated sites may have the potential for phytoremediation.

Keywords: Bioconcentration factor, Heavy metals, Hyperaccumulator, Phytoremediation, Translocation factor

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Authors: Sera Syarmila Sameon, Rohaini Ramli

Abstract:

A trustworthy voting process in democratic is important that each vote is recorded with accuracy and impartiality. The accuracy and impartiality are tallied in high rate with biometric system. One of the sign is a fingerprint. Fingerprint recognition is still a challenging problem, because of the distortions among the different impression of the same finger. Because of the trustworthy of biometric voting technologies, it may give a great effect on numbers of voter-s participation and outcomes of the democratic process. Hence in this study, the authors are interested in designing and analyzing the Electronic Voting System and the participation of the users. The system is based on the fingerprint minutiae with the addition of person ID number. This is in order to enhance the accuracy and speed of the voting process. The new design is analyzed by conducting pilot election among a class of students for selecting their representative.

Keywords: Biometric, FAR and FRR, democratic, voting

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Authors: Youssef Khmou, Said Safi

Abstract:

In this paper, we investigated the effect of real valued transformation of the spectral matrix of the received data for Angles Of Arrival estimation problem.  Indeed, the unitary transformation of Partial Propagator (UPP) for narrowband sources is proposed and applied on Uniform Linear Array (ULA).

Monte Carlo simulations proved the performance of the UPP spectrum comparatively with Forward Backward Partial Propagator (FBPP) and Unitary Propagator (UP). The results demonstrates that when some of the sources are fully correlated and closer than the Rayleigh angular limit resolution of the broadside array, the UPP method outperforms the FBPP in both of spatial resolution and complexity.

Keywords: DOA, Uniform Linear Array, Narrowband, Propagator, Real valued transformation, Subspace, Unitary Operator.

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Authors: Vladimir Simovic, Matija Varga, Robert Svetlacic

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Scientific work analytically explores and demonstrates techniques that can animate objects and geometric characters using CSS3 language by applying proper formatting and positioning of elements. This paper presents examples of optimum application of the CSS3 descriptive language when generating general web animations (e.g., billiards and movement of geometric characters, etc.). The paper presents analytically, the optimal development and animation design with the frames within which the animated objects are. The originally developed content is based on the upgrading of existing CSS3 descriptive language animations with more complex syntax and project-oriented work. The purpose of the developed animations is to provide an overview of the interactive features of CSS3 descriptive language design for computer games and the animation of important analytical data based on the web view. It has been analytically demonstrated that CSS3 as a descriptive language allows inserting of various multimedia elements into websites for public and internal sites.

Keywords: Animation recording, web page graphics, HTML5 forms, Cascading Style Sheets 3 - CSS3, man-computer interaction, KML animation presenting format, GML, Google Earth Professional.

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Authors: Yu-Ting Tsai, Yu-da Lee, Jin H. Huang

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The nonlinear damping behavior is usually ignored in the design of a miniature moving-coil loudspeaker. But when the loudspeaker operated in air, the damping parameter varies with the voice-coil displacement corresponding due to viscous air flow. The present paper presents an identification model as inverse problem to identify the nonlinear damping parameter in the lumped parameter model for the loudspeaker. Theoretical results for the nonlinear damping are verified by using laser displacement measurement scanner. These results indicate that the damping parameter has the greatly different nonlinearity between in air and vacuum. It is believed that the results of the present work can be applied in diagnosis and sound quality improvement of a miniature loudspeaker.

Keywords: Miniature loudspeaker, non-linear damping, system identification, Lumped parameter model.

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Authors: Katrin Zaiß, Tim Schlüter, Stefan Conrad

Abstract:

Ontology Matching is a task needed in various applica-tions, for example for comparison or merging purposes. In literature,many algorithms solving the matching problem can be found, butmost of them do not consider instances at all. Mappings are deter-mined by calculating the string-similarity of labels, by recognizinglinguistic word relations (synonyms, subsumptions etc.) or by ana-lyzing the (graph) structure. Due to the facts that instances are oftenmodeled within the ontology and that the set of instances describesthe meaning of the concepts better than their meta information,instances should definitely be incorporated into the matching process.In this paper several novel instance-based matching algorithms arepresented which enhance the quality of matching results obtainedwith common concept-based methods. Different kinds of formalismsare use to classify concepts on account of their instances and finallyto compare the concepts directly.KeywordsInstances, Ontology Matching, Semantic Web

Keywords: Instances, Ontology Matching, Semantic Web

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Authors: R. Khorshidi , T. Niknam, M. Nayeripour

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Recently, distributed generation technologies have received much attention for the potential energy savings and reliability assurances that might be achieved as a result of their widespread adoption. Fueling the attention have been the possibilities of international agreements to reduce greenhouse gas emissions, electricity sector restructuring, high power reliability requirements for certain activities, and concern about easing transmission and distribution capacity bottlenecks and congestion. So it is necessary that impact of these kinds of generators on distribution feeder reconfiguration would be investigated. This paper presents an approach for distribution reconfiguration considering Distributed Generators (DGs). The objective function is summation of electrical power losses A Tabu search optimization is used to solve the optimal operation problem. The approach is tested on a real distribution feeder.

Keywords: Distributed Generator, Daily Optimal Operation, Genetic Algorithm.

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Authors: Emil G. Ioanid, Viorica Frunză, Dorina Rusu, Ana Maria Vlad, Catalin Tanase, Simona Dunca

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The application of cold Radio-Frequency (RF) plasma in the conservation of cultural heritage became important in the last decades due to the positive results obtained in decontamination treatments. This paper presents an equipment especially designed for cold RF plasma application on paper documents, developed within a research project. The equipment consists in two modules: the first one is designed for decontamination and cleaning treatments of any type of paper supports, while the second one can be used for coating friable papers with adequate polymers, for protection purposes. All these operations are carried out in cold radio-frequency plasma, working in gaseous nitrogen, at low pressure. In order to optimize the equipment parameters ancient paper samples infested with microorganisms have been treated in nitrogen plasma and the decontamination effects, as well as changes in surface properties (color, pH) were assessed. The microbiological analysis revealed complete decontamination at 6 minutes treatment duration; only minor modifications of the surface pH were found and the colorimetric analysis showed a slight yellowing of the support.

Keywords: Cultural heritage, nitrogen plasma, paper support.

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Authors: J. P. Méreaux, E. Lamy, J. C. Lopez

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In the face of new regulations related to waste management, it has become essential to understand the organizational process that accompanies this change. Through an experiment on the sorting of food waste in the community of Grand Reims, this research explores the acceptability, the behavior and the tools needed to manage the change. Our position within a private company, SUEZ, a key player in the waste management sector, has allowed us to set up a driven team with concerned public organizations. The research was conducted through a theoretical study combined with semi-structured interviews. This qualitative method allowed us to conduct exchanges with users to assess the motivations and obstacles linked to the sorting of bio-waste. The results revealed the action levers necessary for the project's sustainability. Making the sorting gestures accessible and simplified makes it possible to target all populations. Playful communication adapted to each type of persona allows the user and stakeholders to be placed at the heart of the strategy. These recommendations are spotlighted thanks to the combination of theoretical and operational contributions, with the aim of facilitating the new public management and inducing the notion of performance while providing an example of added value.

Keywords: Bio-waste, Corporate Social Responsibility, CSR, stakeholders, public policies.

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Authors: S. Soltan, A. Zarnani, R. AliMohammadzadeh, M. Rahgozar

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XML has become a popular standard for information exchange via web. Each XML document can be presented as a rooted, ordered, labeled tree. The Node label shows the exact position of a node in the original document. Region and Dewey encoding are two famous methods of labeling trees. In this paper, we propose a new insert friendly labeling method named IFDewey based on recently proposed scheme, called Extended Dewey. In Extended Dewey many labels must be modified when a new node is inserted into the XML tree. Our method eliminates this problem by reserving even numbers for future insertion. Numbers generated by Extended Dewey may be even or odd. IFDewey modifies Extended Dewey so that only odd numbers are generated and even numbers can then be used for a much easier insertion of nodes.

Keywords: XML, tree labeling, query processing.

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Authors: Yanmin Zhao, Siu Ming Yiu

Abstract:

To claim the ownership for an executable program is a non-trivial task. An emerging direction is to add a watermark to the program such that the watermarked program preserves the original program’s functionality and removing the watermark would heavily destroy the functionality of the watermarked program. In this paper, the first watermarking signature scheme with the watermark and the constraint function hidden in the symmetric key setting is constructed. The scheme uses well-known techniques of lattice trapdoors and a lattice evaluation. The watermarking signature scheme is unforgeable under the Short Integer Solution (SIS) assumption and satisfies other security requirements such as the unremovability security property.

Keywords: Short integer solution problem, signatures, the symmetric-key setting, watermarking schemes.

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Authors: Viktor Achter, Sebastian Breuers, Marc Seifert, Ulrich Lang, Joachim Götze, Bernd Reuther, Paul Müller

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Within the last years, several technologies have been developed to help building e-learning portals. Most of them follow approaches that deliver a vast amount of functionalities, suitable for class-like learning. The SuGI project, as part of the D-Grid (funded by the BMBF), targets on delivering a highly scalable and sustainable learning solution to provide materials (e.g. learning modules, training systems, webcasts, tutorials, etc.) containing knowledge about Grid computing to the D-Grid community. In this article, the process of the development of an e-learning portal focused on the requirements of this special user group is described. Furthermore, it deals with the conceptual and technical design of an e-learning portal, addressing the special needs of heterogeneous target groups. The main focus lies on the quality management of the software development process, Web templates for uploading new contents, the rich search and filter functionalities which will be described from a conceptual as well as a technical point of view. Specifically, it points out best practices as well as concepts to provide a sustainable solution to a relatively unknown and highly heterogeneous community.

Keywords: D-Grid, e-learning, e-science, Grid computing, SuGI.

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Authors: R. Dariani, S. Schmidt, R. Kasper

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Based on a non-linear single track model which describes the dynamics of vehicle, an optimal path planning strategy is developed. Real time optimization is used to generate reference control values to allow leading the vehicle alongside a calculated lane which is optimal for different objectives such as energy consumption, run time, safety or comfort characteristics. Strict mathematic formulation of the autonomous driving allows taking decision on undefined situation such as lane change or obstacle avoidance. Based on position of the vehicle, lane situation and obstacle position, the optimization problem is reformulated in real-time to avoid the obstacle and any car crash.

Keywords: Autonomous driving, Obstacle avoidance, Optimal control, Path planning.

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Authors: M. H. Abedi, A. Jalilvand

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The economic development benefits of wind energy may be the most tangible basis for the local and state officials’ interests. In addition to the direct salaries associated with building and operating wind projects, the wind energy industry provides indirect jobs and benefits. The optimal planning of a wind farm is one most important topic in renewable energy technology. Many methods have been implemented to optimize the cost and output benefit of wind farms, but the contribution of this paper is mentioning different types of service providers and also time of installation of wind turbines during planning horizon years. Genetic algorithm (GA) is used to optimize the problem. It is observed that an appropriate layout of wind farm can cause to minimize the different types of cost.

Keywords: Renewable energy, wind farm, optimization, planning.

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Authors: F. Behrouzi, A. Maimun, M. Nakisa

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The growth of population, rising fossil fuel prices (limited and decreasing day by day), pollution problem due to use of fossil fuels and increasing electrical demand are important factors that encourage the use of green and renewable energy technologies. Among the different renewable energy technologies, hydro power generation (large and small scale) is the prime choice in terms of contribution to the world's electricity generation by using water current turbines. Currently, researchers mainly focused on design and development of different kind of turbines to capture hydropower to generate electricity as clean and reliable energy. This paper is a review of the status of research on water current turbines carried out to generate electricity from hydrokinetic energy especially in places where there is no electricity, but there is access to flowing water.

Keywords: Turbines, Renewable Energy, Hydropower.

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Authors: Sandeep Singh Gill, Rajeevan Chandel, Ashwani Chandel

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This paper presents a comparative study of Ant Colony and Genetic Algorithms for VLSI circuit bi-partitioning. Ant colony optimization is an optimization method based on behaviour of social insects [27] whereas Genetic algorithm is an evolutionary optimization technique based on Darwinian Theory of natural evolution and its concept of survival of the fittest [19]. Both the methods are stochastic in nature and have been successfully applied to solve many Non Polynomial hard problems. Results obtained show that Genetic algorithms out perform Ant Colony optimization technique when tested on the VLSI circuit bi-partitioning problem.

Keywords: Partitioning, genetic algorithm, ant colony optimization, non-polynomial hard, netlist, mutation.

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Authors: Changchun Shen, Shouming Zhong

Abstract:

This paper investigates the problem of absolute stability and robust stability of a class of Lur-e systems with neutral type and time-varying delays. By using Lyapunov direct method and linear matrix inequality technique, new delay-dependent stability criteria are obtained and formulated in terms of linear matrix inequalities (LMIs) which are easy to check the stability of the considered systems. To obtain less conservative stability conditions, an operator is defined to construct the Lyapunov functional. Also, the free weighting matrices approach combining a matrix inequality technique is used to reduce the entailed conservativeness. Numerical examples are given to indicate significant improvements over some existing results.

Keywords: Lur'e system, linear matrix inequalities, Lyapunov, stability.

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Authors: Amir Vahid Dastjerdi, Kamalrulnizam Abu Bakar

Abstract:

The first generation of Mobile Agents based Intrusion Detection System just had two components namely data collection and single centralized analyzer. The disadvantage of this type of intrusion detection is if connection to the analyzer fails, the entire system will become useless. In this work, we propose novel hybrid model for Mobile Agent based Distributed Intrusion Detection System to overcome the current problem. The proposed model has new features such as robustness, capability of detecting intrusion against the IDS itself and capability of updating itself to detect new pattern of intrusions. In addition, our proposed model is also capable of tackling some of the weaknesses of centralized Intrusion Detection System models.

Keywords: Distributed Intrusion Detection System, Mobile Agents, Network Security.

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Authors: A. Escoffier, A. Albrecht, F. Consigny

Abstract:

In order to host the Football Euro in 2016, many French cities have launched architectural competitions in recent years to improve the quality of their stadiums. The winning project in Nice was designed by Wilmotte architects together with Elioth structural engineers. It has a capacity of 35,000 seats.Its roof structure consists of a complex 3D shape timber and steel lattice and is covered by 25,000m² of ETFE, 10,500m² of PES-PVC fabric and 8,500m² of photovoltaic panels.

This paper focuses on the ETFE part of the cover. The stadium is one of the first constructions to use flat single layer ETFE on such a big area. Due to its relatively recent appearance in France, ETFE structures are not yet covered by any regulations and the existing codes for fabric structures cannot be strictly applied. Rather, they are considered as cladding systems and therefore have to be approved by an “Appréciation Technique d’Expérimentation” (ATEx), during which experimental tests have to be performed. We explain the method that we developed to justify the ETFE, which eventually led to bi-axial tests to clarify the allowable stress in the film.

Keywords: Biaxial test, creep, ETFE, single layer, stadium roof.

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Authors: Eli Hustad, Aurilla A. Bechina

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The purpose of this research is to increase our knowledge as regards how Small-and-Medium-Sized Enterprises (SMEs) tackle ERP implementation projects to achieve successful adoption and use of these systems within the organization. SMEs have scare resources to handle these kinds of projects which have proved to be risky and costly. There are several studies focusing on ERP implementation in larger companies, however, few studies report on challenges experienced by SMEs. Our research seeks to bridge this gap. Through a multiple case study of four companies, we identified challenges and critical elements within the different phases (pre-implementation, implementation and post-implementation) of the ERP life cycle. To interpret our findings, we utilize a well-know ERP life cycle model and critical success factors developed for larger companies which are reported in former research literature. We discuss if these models are relevant for SMEs and suggest additional critical elements identified in this study to make a framework more adapted to the SME context.

Keywords: ERP implementation challenges, ERP implementation framework, ERP life cycle model, Small-and- Medium-Sized Enterprises, ERP critical success factors

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Authors: Zuzana Crhová, Marie Paseková

Abstract:

This contribution aims to compare legislation adjusting the course of insolvency proceedings in France, Germany and Slovakia. On the basis of an investigation of the legislative adjustment of this problem, an attempt is made to ascertain in the given countries the extent to which the outcome of the entire proceedings is influenced by legislation and to determine the fundamental moments that influence costs, recovery rate and the duration of proceedings. A comparative analysis was utilized in order to achieve the set goal. The results of the survey could be used to improve legislation so as to lead in the best and most expedient way to a departure from the market of those subjects that are for economic reasons unable to continue with their activities whilst burdening the entire process with the lowest possible costs, which would lead to a high level of satisfaction for creditors. 

Keywords: Costs, Insolvency Proceeding, Recovery Rate, Time.

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Authors: Khairil Iskandar Othman, Zarina Bibi Ibrahim, Mohamed Suleiman

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A parallel block method based on Backward Differentiation Formulas (BDF) is developed for the parallel solution of stiff Ordinary Differential Equations (ODEs). Most common methods for solving stiff systems of ODEs are based on implicit formulae and solved using Newton iteration which requires repeated solution of systems of linear equations with coefficient matrix, I - hβJ . Here, J is the Jacobian matrix of the problem. In this paper, the matrix operations is paralleled in order to reduce the cost of the iterations. Numerical results are given to compare the speedup and efficiency of parallel algorithm and that of sequential algorithm.

Keywords: Backward Differentiation Formula, block, ordinarydifferential equations.

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Authors: R. Kongnuy, E. Naowanich

Abstract:

In this paper we develop and analyze the model for the spread of Leptospirosis by age group in Thailand, between 1997 and 2010 by using mathematical modeling and computer simulation. Leptospirosis is caused by pathogenic spirochetes of the genus Leptospira. It is a zoonotic disease of global importance and an emerging health problem in Thailand. In Thailand, leptospirosis is a reportable disease, the top three age groups are 23.31% in 35-44 years olds group, 22.76% in 25-34 year olds group, 17.60% in 45-54 year olds group from reported leptospirosis between 1997 and 2010, with a peak in 35-44 year olds group. Our paper, the Leptosipirosis transmission by age group in Thailand is studied on the mathematical model. Some analytical and simulation results are presented.

Keywords: Age Group, Equilibrium State, Leptospirosis, Mathematical Modeling.

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Authors: Mohammad Taghi Darvishi, Samad Kheybari

Abstract:

In this article, we are dealing with a model consisting of a classical Van der Pol oscillator coupled gyroscopically to a linear oscillator. The major problem is analyzed. The regular dynamics of the system is considered using analytical methods. In this case, we provide an approximate solution for this system using parameter-expansion method. Also, we find approximate values for frequencies of the system. In parameter-expansion method the solution and unknown frequency of oscillation are expanded in a series by a bookkeeping parameter. By imposing the non-secularity condition at each order in the expansion the method provides different approximations to both the solution and the frequency of oscillation. One iteration step provides an approximate solution which is valid for the whole solution domain.

Keywords: Parameter-expansion method, classical Van der Pol oscillator.

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