Search results for: discrete phase modeling

Authors: P. Ruttanee, K-N. Areerak, K-L. Areerak

Abstract:

Dynamic models of power converters are normally time-varying because of their switching actions. Several approaches are applied to analyze the power converters to achieve the timeinvariant models suitable for system analysis and design via the classical control theory. The paper presents how to derive dynamic models of the power system consisting of a three-phase controlled rectifier feeding an uncontrolled buck converter by using the combination between the well known techniques called the DQ and the generalized state-space averaging methods. The intensive timedomain simulations of the exact topology model are used to support the accuracies of the reported model. The results show that the proposed model can provide good accuracies in both transient and steady-state responses.

Keywords: DQ method, Generalized state-space averaging method, Three-phase controlled rectifier, Uncontrolled buck converter, Averaging model, Modeling, Simulation.

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Authors: Wang Xing-hua, Zhong Shun-an, Zhang Zhuo

Abstract:

A kind of behavior model for discrete sampling and hold amplifier with charge transmission is analyzed. The transfer function and behavior features are based on the main AC responses of operation amplifier. The result used in pipelined and sigma-delta ADC shows the exact of model of sampling and hold amplifier, and the non-ideal factors are taken into account.

Keywords: SHA, response, behavior, transfer function.

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Authors: Sheng-Gwo Chen, Wen-Haw Chen

Abstract:

It is an important problem to compute the geodesics on a surface in many fields. To find the geodesics in practice, however, the traditional discrete algorithms or numerical approaches can only find a list of discrete points. The first author proposed in 2010 a new, elegant and accurate method, the geodesic-like method, for approximating geodesics on a regular surface. This paper will present by use of this method a computation of the Bezier geodesic-like curves on spheres.

Keywords: Geodesics, Geodesic-like curve, Spheres, Bezier.

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Authors: H. Megherbi, A. C. Megherbi, N. Megherbi, K. Benmahamed

Abstract:

This paper presents three-phase evolution search methodology to automatically design fuzzy logic controllers (FLCs) that can work in a wide range of operating conditions. These include varying load, parameter variations, and unknown external disturbances. The three-phase scheme consists of an exploration phase, an exploitation phase and a robustness phase. The first two phases search for FLC with high accuracy performances while the last phase aims at obtaining FLC providing the best compromise between the accuracy and robustness performances. Simulations were performed for direct-drive two-axis robot arm. The evolved FLC with the proposed design technique found to provide a very satisfactory performance under the wide range of operation conditions and to overcome problem associated with coupling and nonlinearities characteristics inherent to robot arms.

Keywords: Fuzzy logic control, evolutionary algorithms, robustness, exploration/exploitation phase.

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Authors: Adla Bentellis, Zizette Boufaïda

Abstract:

Nowadays, the pace of business change is such that, increasingly, new functionality has to be realized and reliably installed in a matter of days, or even hours. Consequently, more and more business processes are prone to a continuous change. The objective of the research in progress is to use the MAP model, in a conceptual modeling method for flexible and adaptive business process. This method can be used to capture the flexibility dimensions of a business process; it takes inspiration from modularity concept in the object oriented paradigm to establish a hierarchical construction of the BP modeling. Its intent is to provide a flexible modeling that allows companies to quickly adapt their business processes.

Keywords: Business Process, Business process modeling, flexibility, MAP Model.

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Authors: M. Debyeche, J.P Haton, A. Houacine

Abstract:

The paper presents a complete discrete statistical framework, based on a novel vector quantization (VQ) front-end process. This new VQ approach performs an optimal distribution of VQ codebook components on HMM states. This technique that we named the distributed vector quantization (DVQ) of hidden Markov models, succeeds in unifying acoustic micro-structure and phonetic macro-structure, when the estimation of HMM parameters is performed. The DVQ technique is implemented through two variants. The first variant uses the K-means algorithm (K-means- DVQ) to optimize the VQ, while the second variant exploits the benefits of the classification behavior of neural networks (NN-DVQ) for the same purpose. The proposed variants are compared with the HMM-based baseline system by experiments of specific Arabic consonants recognition. The results show that the distributed vector quantization technique increase the performance of the discrete HMM system.

Keywords: Hidden Markov Model, Vector Quantization, Neural Network, Speech Recognition, Arabic Language

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Authors: Taiki Baba, Tomoaki Hashimoto

Abstract:

The random dither quantization method enables us to achieve much better performance than the simple uniform quantization method for the design of quantized control systems. Motivated by this fact, the stochastic model predictive control method in which a performance index is minimized subject to probabilistic constraints imposed on the state variables of systems has been proposed for linear feedback control systems with random dither quantization. In other words, a method for solving optimal control problems subject to probabilistic state constraints for linear discrete-time control systems with random dither quantization has been already established. To our best knowledge, however, the feasibility of such a kind of optimal control problems has not yet been studied. Our objective in this paper is to investigate the feasibility of stochastic model predictive control problems for linear discrete-time control systems with random dither quantization. To this end, we provide the results of numerical simulations that verify the feasibility of stochastic model predictive control problems for linear discrete-time control systems with random dither quantization.

Keywords: Model predictive control, stochastic systems, probabilistic constraints, random dither quantization.

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Authors: Souhir Ben Amor, Heni Boubaker, Lotfi Belkacem

Abstract:

This aims of this paper is to forecast the electricity spot prices. First, we focus on modeling the conditional mean of the series so we adopt a generalized fractional -factor Gegenbauer process (k-factor GARMA). Secondly, the residual from the -factor GARMA model has used as a proxy for the conditional variance; these residuals were predicted using two different approaches. In the first approach, a local linear wavelet neural network model (LLWNN) has developed to predict the conditional variance using the Back Propagation learning algorithms. In the second approach, the Gegenbauer generalized autoregressive conditional heteroscedasticity process (G-GARCH) has adopted, and the parameters of the k-factor GARMA-G-GARCH model has estimated using the wavelet methodology based on the discrete wavelet packet transform (DWPT) approach. The empirical results have shown that the k-factor GARMA-G-GARCH model outperform the hybrid k-factor GARMA-LLWNN model, and find it is more appropriate for forecasts.

Keywords: k-factor, GARMA, LLWNN, G-GARCH, electricity price, forecasting.

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Authors: P. Joyce Beryl Princess, Y. Harold Robinson

Abstract:

Image Processing is a structure of Signal Processing for which the input is the image and the output is also an image or parameter of the image. Image Resolution has been frequently referred as an important aspect of an image. In Image Resolution Enhancement, images are being processed in order to obtain more enhanced resolution. To generate highly resoluted image for a low resoluted input image with high PSNR value. Stationary Wavelet Transform is used for Edge Detection and minimize the loss occurs during Downsampling. Inverse Discrete Wavelet Transform is to get highly resoluted image. Highly resoluted output is generated from the Low resolution input with high quality. Noisy input will generate output with low PSNR value. So Noisy resolution enhancement technique has been used for adaptive sub-band thresholding is used. Downsampling in each of the DWT subbands causes information loss in the respective subbands. SWT is employed to minimize this loss. Inverse Discrete wavelet transform (IDWT) is to convert the object which is downsampled using DWT into a highly resoluted object. Used Image denoising and resolution enhancement techniques will generate image with high PSNR value. Our Proposed method will improve Image Resolution and reached the optimized threshold.

Keywords: Image Processing, Inverse Discrete wavelet transform, PSNR.

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Authors: Kejun Zhuang, Zhaohui Wen

Abstract:

By incorporating a prey refuge, this paper proposes new discrete Leslie–Gower predator–prey systems with and without Allee effect. The existence of fixed points are established and the stability of fixed points are discussed by analyzing the modulus of characteristic roots.

Keywords: Leslie-Gower, predator–prey model, prey refuge, allee effect.

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Authors: Jan Newmarch

Abstract:

With the increasing spread of computers and the internet among culturally, linguistically and geographically diverse communities, issues of internationalization and localization and becoming increasingly important. For some of the issues such as different scales for length and temperature, there is a well-developed measurement theory. For others such as date formats no such theory will be possible. This paper fills a gap by developing a measurement theory for a class of scales previously overlooked, based on discrete and interval-valued scales such as spanner and shoe sizes. The paper gives a theoretical foundation for a class of data representation problems.

Keywords: Data representation, internationalisation, localisation, measurement theory.

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Authors: Francis William S, Imtiaz Ahmed Choudhury, Ananda Kumar Eriki, A. John Rajan

Abstract:

Diesel Engines emit complex mixtures of inorganic and organic compounds in the form of both solid and vapour phase particles. Most of the particulates released are ultrafine nanoparticles which are detrimental to human health and can easily enter the body by respiration. The emissions standards on particulate matter release from diesel engines are constantly upgraded within the European Union and with future regulations based on the particles numbers released instead of merely mass, the need for effective aftertreatment devices will increase. Standard particulate filters in the form of wall flow filters can have problems with high soot accumulation, producing a large exhaust backpressure. A potential solution would be to combine the standard filter with a flow through filter to reduce the load on the wall flow filter. In this paper soot particle trapping has been simulated in different continuous flow filters of monolithic structure including the use of promoters, at laminar flow conditions. An Euler Lagrange model, the discrete phase model in Ansys used with user defined functions for forces acting on particles. A method to quickly screen trapping of 5 nm and 10 nm particles in different catalysts designs with tracers was also developed. Simulations of square duct monoliths with promoters show that the strength of the vortices produced are not enough to give a high amount of particle deposition on the catalyst walls. The smallest particles in the simulations, 5 and 10 nm particles were trapped to a higher extent, than larger particles up to 1000 nm, in all studied geometries with the predominant deposition mechanism being Brownian diffusion. The comparison of the different filters designed with a wall flow filter does show that the options for altering a design of a flow through filter, without imposing a too large pressure drop penalty are good.

Keywords: Diesel Engine trap, thermophoresis, Exhaust pipe, PM-Simulation modeling.

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Authors: Seong-Su Jeon, Seong-Jin Kim, Goon-Cherl Park

Abstract:

In this study, direct numerical simulation for the bubble condensation in the subcooled boiling flow was performed. The main goal was to develop the CFD modeling for the bubble condensation and to evaluate the accuracy of the VOF model with the developed CFD modeling. CFD modeling for the bubble condensation was developed by modeling the source terms in the governing equations of VOF model using UDF. In the modeling, the amount of condensation was determined using the interfacial heat transfer coefficient obtained from the bubble velocity, liquid temperature and bubble diameter every time step. To evaluate the VOF model using the CFD modeling for the bubble condensation, CFD simulation results were compared with SNU experimental results such as bubble volume and shape, interfacial area, bubble diameter and bubble velocity. Simulation results predicted well the behavior of the actual condensing bubble. Therefore, it can be concluded that the VOF model using the CFD modeling for the bubble condensation will be a useful computational fluid dynamics tool for analyzing the behavior of the condensing bubble in a wide range of the subcooled boiling flow.

Keywords: Bubble condensation, CFD modeling, Subcooled boiling flow, VOF model.

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Authors: Adam Czornik, Aleksander Nawrat

Abstract:

This paper studies the problem of exponential stability of perturbed discrete linear systems with periodic coefficients. Assuming that the unperturbed system is exponentially stable we obtain conditions on the perturbations under which the perturbed system is exponentially stable.

Keywords: Exponential stability, time-varying linear systems, periodic systems.

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Authors: Antoaneta Stefanova, Rositsa Gencheva, Pavlin Groudev

Abstract:

This paper presents the modeling approach in SBO sequence for VVER 1000 reactors and describes the reactor core behavior at late in-vessel phase in case of late reflooding by HPIS and gives preliminary results for the ASTECv2 validation. The work is focused on investigation of plant behavior during total loss of power and the operator actions. The main goal of these analyses is to assess the phenomena arising during the Station blackout (SBO) followed by primary side high pressure injection system (HPIS) reflooding of already damaged reactor core at very late “in-vessel” phase. The purpose of the analyses is to define how the later HPIS switching on can delay the time of vessel failure or possibly avoid vessel failure. The times for HPP injection were chosen based on previously performed investigations.

Keywords: VVER, operator action validation, reflooding of overheated reactor core, ASTEC computer code.

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Authors: R. S. Hassani, S. Chikh, L. Tadrist, S. Radev

Abstract:

A bubbly flow in a vertical miniature tube is analyzed theoretically. The liquid and gas phase are co-current flowing upward. The gas phase is injected via a nozzle whose inner diameter is 0.11mm and it is placed on the axis of the tube. A force balance is applied on the bubble at its detachment. The set of governing equations are solved by use of Mathematica software. The bubble diameter and the bubble generation frequency are determined for various inlet phase velocities represented by the inlet mass quality. The results show different behavior of bubble growth and detachment depending on the tube size.

Keywords: Two phase flow, bubble growth, minichannel, generation frequency.

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Authors: Daehwan Kim, Jaemin Kim, Seongwon Cho, Yongsuk Jang, Sun-Tae Chung, Boo-Gyoun Kim

Abstract:

AAM has been successfully applied to face alignment, but its performance is very sensitive to initial values. In case the initial values are a little far distant from the global optimum values, there exists a pretty good possibility that AAM-based face alignment may converge to a local minimum. In this paper, we propose a progressive AAM-based face alignment algorithm which first finds the feature parameter vector fitting the inner facial feature points of the face and later localize the feature points of the whole face using the first information. The proposed progressive AAM-based face alignment algorithm utilizes the fact that the feature points of the inner part of the face are less variant and less affected by the background surrounding the face than those of the outer part (like the chin contour). The proposed algorithm consists of two stages: modeling and relation derivation stage and fitting stage. Modeling and relation derivation stage first needs to construct two AAM models: the inner face AAM model and the whole face AAM model and then derive relation matrix between the inner face AAM parameter vector and the whole face AAM model parameter vector. In the fitting stage, the proposed algorithm aligns face progressively through two phases. In the first phase, the proposed algorithm will find the feature parameter vector fitting the inner facial AAM model into a new input face image, and then in the second phase it localizes the whole facial feature points of the new input face image based on the whole face AAM model using the initial parameter vector estimated from using the inner feature parameter vector obtained in the first phase and the relation matrix obtained in the first stage. Through experiments, it is verified that the proposed progressive AAM-based face alignment algorithm is more robust with respect to pose, illumination, and face background than the conventional basic AAM-based face alignment algorithm.

Keywords: Face Alignment, AAM, facial feature detection, model matching.

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Authors: Abderrahmane Amrouche, Jean Michel Rouvaen

Abstract:

In this paper we present an efficient system for independent speaker speech recognition based on neural network approach. The proposed architecture comprises two phases: a preprocessing phase which consists in segmental normalization and features extraction and a classification phase which uses neural networks based on nonparametric density estimation namely the general regression neural network (GRNN). The relative performances of the proposed model are compared to the similar recognition systems based on the Multilayer Perceptron (MLP), the Recurrent Neural Network (RNN) and the well known Discrete Hidden Markov Model (HMM-VQ) that we have achieved also. Experimental results obtained with Arabic digits have shown that the use of nonparametric density estimation with an appropriate smoothing factor (spread) improves the generalization power of the neural network. The word error rate (WER) is reduced significantly over the baseline HMM method. GRNN computation is a successful alternative to the other neural network and DHMM.

Keywords: Speech Recognition, General Regression NeuralNetwork, Hidden Markov Model, Recurrent Neural Network, ArabicDigits.

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Authors: Samba Raju, Chiluveru, Manoj Tripathy

Abstract:

Intelligibility is an essential characteristic of a speech signal, which is used to help in the understanding of information in speech signal. Background noise in the environment can deteriorate the intelligibility of a recorded speech. In this paper, we presented a simple variance subtracted - variable level discrete wavelet transform, which improve the intelligibility of speech. The proposed algorithm does not require an explicit estimation of noise, i.e., prior knowledge of the noise; hence, it is easy to implement, and it reduces the computational burden. The proposed algorithm decides a separate decomposition level for each frame based on signal dominant and dominant noise criteria. The performance of the proposed algorithm is evaluated with speech intelligibility measure (STOI), and results obtained are compared with Universal Discrete Wavelet Transform (DWT) thresholding and Minimum Mean Square Error (MMSE) methods. The experimental results revealed that the proposed scheme outperformed competing methods

Keywords: Discrete Wavelet Transform, speech intelligibility, STOI, standard deviation.

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Authors: Song Meng, Dianfu Ma, Yongwang Zhao, Jianxin Li

Abstract:

Cooperative visual modeling is more and more necessary in our complicated world. A collaborative environment which supports interactive operation and communication is required to increase work efficiency. We present a collaborative visual modeling framework which collaborative platform could be built on. On this platform, cooperation and communication is available for designers from different regions. This framework, which is different from other collaborative frameworks, contains a uniform message format, a message handling mechanism and other functions such as message pretreatment and Role-Communication-Token Access Control (RCTAC). We also show our implementation of this framework called Orchestra Designer, which support BPLE workflow modeling cooperatively online.

Keywords: colllaborative framework; visual modeling; message handling mechanism

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Authors: Grigorios D. Dimitroulakos, N. D. Zervas, N. Sklavos, Costas E. Goutis

Abstract:

In this paper, the implementation of low power, high throughput convolutional filters for the one dimensional Discrete Wavelet Transform and its inverse are presented. The analysis filters have already been used for the implementation of a high performance DWT encoder [15] with minimum memory requirements for the JPEG 2000 standard. This paper presents the design techniques and the implementation of the convolutional filters included in the JPEG2000 standard for the forward and inverse DWT for achieving low-power operation, high performance and reduced memory accesses. Moreover, they have the ability of performing progressive computations so as to minimize the buffering between the decomposition and reconstruction phases. The experimental results illustrate the filters- low power high throughput characteristics as well as their memory efficient operation.

Keywords: Discrete Wavelet Transform; JPEG2000 standard; VLSI design; Low Power-Throughput-optimized filters

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Authors: Zoltan Theisz, Gergely Mezei

Abstract:

Modern industrial automation relies on service oriented concepts of Internet of Things (IoT) device modeling in order to provide a flexible and extendable environment for service meta-repository. However, state-of-the-art meta-modeling techniques prefer design-time modeling, which results in a heavy usage of class sometimes unnecessary static subtyping. Although this approach benefits from clear-cut object-oriented design principles, it also seals the model repository for further dynamic extensions. In this paper, a dynamic multi-level modeling approach is introduced that enables dynamic subtyping through a more relaxed partial instantiation mechanism. The approach is demonstrated on a simple sensor network example.

Keywords: Meta-modeling, dynamic subtyping, DMLA, industrial automation, arrowhead.

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Authors: M. Yousefi, N. Nasirzadeh

Abstract:

In this paper, behavior of an oscillator under injection of another signal has been investigated. Also, variation of output signal amplitude versus injected signal phase variation, the effect of varying the amplitude of injected signal and quality factor of the oscillator has been investigated. The results show that the locking time depends on phase and the best locking time happens at 180-degrees phase. Also, the effect of injected lock has been discussed. Simulations show that the locking time decreases with signal injection to bulk. Locking time has been investigated versus various phase differences. The effect of phase and amplitude changes on locking time of a typical LC oscillator in 180 nm technology has been investigated.

Keywords: Injection-lock oscillator, oscillator, analysis, phase modulation.

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Authors: Hidetoshi Nonaka

Abstract:

This paper presents an interactive modeling system of uniform polyhedra using the isomorphic graphs. Especially, Kepler-Poinsot solids are formed by modifications of dodecahedron and icosahedron.

Keywords: Kepler-Poinsot solid, Shape modeling, Polyhedralgraph, Graph drawing.

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Authors: Kamaldeep Joshi, Rajkumar Yadav, Ashok Kumar Yadav

Abstract:

Digital Watermarking is a procedure to prevent the unauthorized access and modification of personal data. It assures that the communication between two parties remains secure and their communication should be undetected. This paper investigates the consequence of the watermark strength of the grayscale image using a Discrete Wavelet Transformation (DWT) additive technique. In this method, the gray scale host image is divided into four sub bands: LL (Low-Low), HL (High-Low), LH (Low-High), HH (High-High) and the watermark is inserted in an LL sub band using DWT technique. As the image is divided into four sub bands, a watermark of equal size of the LL sub band has been inserted and the results are discussed. LL represents the average component of the host image which contains the maximum information of the image. Two kinds of experiments are performed. In the first, the same watermark is embedded in different images and in the later on the strength of the watermark varies by a factor of s i.e. (s=10, 20, 30, 40, 50) and it is inserted in the same image.

Keywords: Watermarking, discrete wavelet transform, scaling factor, steganography.

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Authors: Shu-Hui Jan, Hui-Ping Tserng, Shih-Ping Ho

Abstract:

Construction project control attempts to obtain real-time as-built schedule information and to eliminate project delays by effectively enhancing dynamic schedule control and management. Suitable platforms for enhancing an as-built schedule visually during the construction phase are necessary and important for general contractors. As the application of building information modeling (BIM) becomes more common, schedule management integrated with the BIM approach becomes essential to enhance visual construction management implementation for the general contractor during the construction phase. To enhance visualization of the updated as-built schedule for the general contractor, this study presents a novel system called the Construction BIM-assisted Schedule Management (ConBIM-SM) system for general contractors in Taiwan. The primary purpose of this study is to develop a web ConBIM-SM system for the general contractor to enhance visual as-built schedule information sharing and efficiency in tracking construction as-built schedule. Finally, the ConBIM-SM system is applied to a case study of a commerce building project in Taiwan to verify its efficacy and demonstrate its effectiveness during the construction phase. The advantages of the ConBIM-SM system lie in improved project control and management efficiency for general contractors, and in providing BIM-assisted as-built schedule tracking and management, to access the most current as-built schedule information through a web browser. The case study results show that the ConBIM-SM system is an effective visual as-built schedule management platform integrated with the BIM approach for general contractors in a construction project.

Keywords: BIM, Building information modeling, construction schedule management, as-built schedule management, BIM schedule updating mechanism.

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Authors: A. Retbi, M. Khalidi Idrissi, S. Bennani

Abstract:

The success of an e-learning system is highly dependent on the quality of its educational content and how effective, complete, and simple the design tool can be for teachers. Educational modeling languages (EMLs) are proposed as design languages intended to teachers for modeling diverse teaching-learning experiences, independently of the pedagogical approach and in different contexts. However, most existing EMLs are criticized for being too abstract and too complex to be understood and manipulated by teachers. In this paper, we present a visual EML that simplifies the process of designing learning scenarios for teachers with no programming background. Based on the conceptual framework of the activity theory, our resulting visual EML focuses on using Domainspecific modeling techniques to provide a pedagogical level of abstraction in the design process.

Keywords: Educational modeling language, Domain Specific Modeling, authoring systems, learning scenario.

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Authors: R. Sabre, W. Horrigue, J. C. Simon

Abstract:

This paper is interested in two difficulties encountered in practice when observing a continuous time process. The first is that we cannot observe a process over a time interval; we only take discrete observations. The second is the process frequently observed with a constant additive error. It is important to give an estimator of the spectral density of such a process taking into account the additive observation error and the choice of the discrete observation times. In this work, we propose an estimator based on the spectral smoothing of the periodogram by the polynomial Jackson kernel reducing the additive error. In order to solve the aliasing phenomenon, this estimator is constructed from observations taken at well-chosen times so as to reduce the estimator to the field where the spectral density is not zero. We show that the proposed estimator is asymptotically unbiased and consistent. Thus we obtain an estimate solving the two difficulties concerning the choice of the instants of observations of a continuous time process and the observations affected by a constant error.

Keywords: Spectral density, stable processes, aliasing, periodogram.

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Authors: Hee Suk Seo, Tae Kyung Kim

Abstract:

As network components grow larger and more diverse, and as securing them on a host-by-host basis grow more difficult, more sites are turning to a network security model. We concentrate on controlling network access to various hosts and the services they offer, rather than on securing them one by one with a network security model. We present how the policy rules from vulnerabilities stored in SVDB (Simulation based Vulnerability Data Base) are inducted, and how to be used in PBN. In the network security environment, each simulation model is hierarchically designed by DEVS (Discrete EVent system Specification) formalism.

Keywords: SVDB, PBN, DEVS, Network security.

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Authors: A. Jawahar

Abstract:

In this paper, the dependence of soliton pulses with respect to phase in a 10Gbps, single channel, dispersion uncompensated telecommunication system was studied. The characteristic feature of periodic soliton interaction was noted at the Interaction point (I=6202.5Km) in one collision length of L=12405.1 Km. The interaction point is located for 10Gbps system with an initial relative spacing (qo) of soliton as 5.28 using Perturbation theory. It is shown that, when two in-phase solitons are launched, they interact at the point I=6202.5 Km, but the interaction could be restricted with introduction of different phase initially. When the phase of the input solitons increases, the deviation of soliton pulses at the ‘I’ also increases. We have successfully demonstrated this effect in a telecommunication set-up in terms of Quality factor (Q), where the Q=0 for in-phase soliton. The Q was noted to be 125.9, 38.63, 47.53, 59.60, 161.37, and 78.04 for different phases such as 10o, 20o, 30o, 45o, 60o and 90o degrees respectively at Interaction point (I).

Keywords: Soliton interaction, Initial relative spacing, phase, Perturbation theory and telecommunication system.

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