Search results for: discrete state

Authors: Manlika Ratchagit

Abstract:

This paper is concerned with robust mean square stability of uncertain stochastic switched discrete time-delay systems. The system to be considered is subject to interval time-varying delays, which allows the delay to be a fast time-varying function and the lower bound is not restricted to zero. Based on the discrete Lyapunov functional, a switching rule for the robust mean square stability for the uncertain stochastic discrete time-delay system is designed via linear matrix inequalities. Finally, some examples are exploited to illustrate the effectiveness of the proposed schemes.

Keywords: robust mean square stability, discrete-time stochastic systems, hybrid systems, interval time-varying delays, Lyapunov functional, linear matrix inequalities

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Authors: Manlika Rajchakit

Abstract:

This paper is concerned with robust mean square stability of uncertain stochastic switched discrete time-delay systems. The system to be considered is subject to interval time-varying delays, which allows the delay to be a fast time-varying function and the lower bound is not restricted to zero. Based on the discrete Lyapunov functional, a switching rule for the robust mean square stability for the uncertain stochastic discrete time-delay system is designed via linear matrix inequalities. Finally, some examples are exploited to illustrate the effectiveness of the proposed schemes.

Keywords: robust mean square stability, discrete-time stochastic systems, hybrid systems, interval time-varying delays, lyapunov functional, linear matrix inequalities

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Authors: Huimin Liu, Li Xiang, Yue Liu, Jing Wang

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Background: As one of typical case for the developing countries who are stepping into the aging times globally, more and more older people in China might face the problem of which they could not maintain normal life due to the functional disability. While the government take efforts to build long-term care system and further carry out related policies for the core concept, there is still lack of strong evidence to evaluating the profile of disability states in the elderly population and its transition rate. It has been proved that disability is a dynamic condition of the person rather than irreversible so it means possible to intervene timely on them who might be in a risk of severe disability. Objective: The aim of this study was to depict the picture of the disability transferring status of the older people in China, and then find out individual characteristics that change the state of disability to provide theory basis for disability prevention and early intervention among elderly people. Methods: Data for this study came from the 2011 baseline survey and the 2013 follow-up survey of the China Health and Retirement Longitudinal Study (CHARLS). Normal ADL function, 1~2 ADLs disability,3 or above ADLs disability and death were defined from state 1 to state 4. Multi-state Markov model was applied and the four-state homogeneous model with discrete states and discrete times from two visits follow-up data was constructed to explore factors for various progressive stages. We modeled the effect of explanatory variables on the rates of transition by using a proportional intensities model with covariate, such as gender. Result: In the total sample, state 2 constituent ratio is nearly about 17.0%, while state 3 proportion is blow the former, accounting for 8.5%. Moreover, ADL disability statistics difference is not obvious between two years. About half of the state 2 in 2011 improved to become normal in 2013 even though they get elder. However, state 3 transferred into the proportion of death increased obviously, closed to the proportion back to state 2 or normal functions. From the estimated intensities, we see the older people are eleven times as likely to develop at 1~2 ADLs disability than dying. After disability onset (state 2), progression to state 3 is 30% more likely than recovery. Once in state 3, a mean of 0.76 years is spent before death or recovery. In this model, a typical person in state 2 has a probability of 0.5 of disability-free one year from now while the moderate disabled or above has a probability of 0.14 being dead. Conclusion: On the long-term care cost considerations, preventive programs for delay the disability progression of the elderly could be adopted based on the current disabled state and main factors of each stage. And in general terms, those focusing elderly individuals who are moderate or above disabled should go first.

Keywords: Markov model, elderly people, disability, transition intensity

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Authors: Elham Kazemi

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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.

Keywords: Quadratic Assignment Problem (QAP), Discrete Cuckoo Optimization Algorithm (DCOA), meta-heuristic algorithms, optimization algorithms

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Authors: Karina C. Azzolin, Luis E. Kosteski, Alisson S. Milani, Raquel C. Zydeck

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This work aims to represent Numerically tests experimentally developed in reduced scale walls with horizontal and inclined cuts by using the Lattice Discrete Element Method (LDEM) implemented On de Abaqus/explicit environment. The cuts were performed with depths of 20%, 30%, and 50% On the walls subjected to centered and eccentric loading. The parameters used to evaluate the numerical model are its strength, the failure mode, and the in-plane and out-of-plane displacements.

Keywords: structural masonry, wall chases, small scale, numerical model, lattice discrete element method

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Authors: Tun Myat Aung, Ni Ni Hla

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This paper begins by describing basic properties of finite field and elliptic curve cryptography over prime field and binary field. Then we discuss the discrete logarithm problem for elliptic curves and its properties. We study the general common attacks on elliptic curve discrete logarithm problem such as the Baby Step, Giant Step method, Pollard’s rho method and Pohlig-Hellman method, and describe in detail experiments of these attacks over prime field and binary field. The paper finishes by describing expected running time of the attacks and suggesting strong elliptic curves that are not susceptible to these attacks.c

Keywords: discrete logarithm problem, general attacks, elliptic curve, prime field, binary field

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Authors: Archit Yajnik, Rustam Ali

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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.

Keywords: GHM multiwavelet, fredholm integral equations, quadrature method, function approximation

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Authors: Ashish Ashish

Abstract:

In the last few decades, the discrete chaos of difference equations has gained a massive attention of academicians and scholars due to its tremendous applications in each and every branch of science, such as cryptography, traffic control models, secure communications, weather forecasting, and engineering. In this article, a generalized logistic discrete map is established and discrete chaos is reported through period doubling bifurcation, period three orbit and Lyapunov exponent. It is interesting to see that the generalized logistic map exhibits superior chaos due to the presence of an extra degree of freedom of an ordered parameter. The period doubling bifurcation and Lyapunov exponent are demonstrated for some particular values of parameter and the discrete chaos is determined in the sense of Devaney's definition of chaos theoretically as well as numerically. Moreover, the study discusses an extended chaos based image encryption and decryption scheme in cryptography using this novel system. Surprisingly, a larger key space for coding and more sensitive dependence on initial conditions are examined for encryption and decryption of text messages, images and videos which secure the system strongly from external cyber attacks, coding attacks, statistic attacks and differential attacks.

Keywords: chaos, period-doubling, logistic map, Lyapunov exponent, image encryption

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Authors: Nidal F. Shilbayeh, Belal AbuHaija, Zainab N. Al-Qudsy

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In this paper, a hybrid blind digital watermarking system using Discrete Wavelet Transform (DWT) and Contourlet Transform (CT) has been implemented and tested. The implemented combined digital watermarking system has been tested against five common types of image attacks. The performance evaluation shows improved results in terms of imperceptibility, robustness, and high tolerance against these attacks; accordingly, the system is very effective and applicable.

Keywords: discrete wavelet transform (DWT), contourlet transform (CT), digital image watermarking, copyright protection, geometric attack

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Authors: A. S. Mousa, F. Shoman

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We present a discrete game theoretical model with homogeneous individuals who make simultaneous decisions. In this model the strategy space of all individuals is a discrete and dichotomous set which consists of two strategies. We fully characterize the coherent, split and mixed strategies that form Nash equilibria and we determine the corresponding Nash domains for all individuals. We find all strategic thresholds in which individuals can change their mind if small perturbations in the parameters of the model occurs.

Keywords: coherent strategy, split strategy, pure strategy, mixed strategy, Nash equilibrium, game theory

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Authors: Samvel H. Sargsyan

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Together with the study of electron-optical properties of nanostructures and proceeding from experiment-based data, the study of the mechanical properties of nanostructures has become quite actual. For the study of the mechanical properties of fullerene, carbon nanotubes, graphene and other nanostructures one of the crucial issues is the construction of their adequate mathematical models. Among all mathematical models of graphene or carbon nano-tubes, this so-called discrete-continuous model is specifically important. It substitutes the interactions between atoms by elastic beams or springs. The present paper demonstrates the construction of the discrete-continual beam model for carbon nanotubes or graphene, where the micropolar beam model based on the theory of moment elasticity is accepted. With the account of the energy balance principle, the elastic moment constants for the beam model, expressed by the physical and geometrical parameters of carbon nanotube or graphene, are determined. By switching from discrete-continual beam model to the continual, the models of micropolar elastic cylindrical shell and micropolar elastic plate are confirmed as continual models for carbon nanotube and graphene respectively.

Keywords: carbon nanotube, discrete-continual, elastic, graphene, micropolar, plate, shell

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Authors: N. Li, Y. M. Cheng

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In this paper, the classical bearing capacity problem is re-considered from discrete element analysis. In the discrete element approach, the bearing capacity problem is considered from the elastic stage to plastic stage to rupture stage (large displacement). The bearing capacity failure mechanism of a strip footing on soil is investigated, and the influence of micro-parameters on the bearing capacity of soil is also observed. It is found that the distinct element method (DEM) gives very good visualized results, and basically coincides well with that derived by the classical methods.

Keywords: bearing capacity, distinct element method, failure mechanism, large displacement

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Authors: Chao-Qing Dai

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Two families of exact travelling solutions for the discrete sine-Gordon equation are constructed based on the variable-coefficient Jacobian elliptic function method and different transformations. When the modulus of Jacobian elliptic function solutions tends to 1, soliton solutions can be obtained. Some soliton solutions degenerate into the known solutions in literatures. Moreover, dynamical properties of exact solutions are investigated. Our analysis and results may have potential values for certain applications in modern nonlinear science and textile engineering.

Keywords: exact solutions, variable-coefficient Jacobian elliptic function method, discrete sine-Gordon equation, dynamical behaviors

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Authors: Jixiang Zhang

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A dynamic of Bertrand duopoly game is analyzed, where players use different production methods and choose their prices with bounded rationality. The equilibriums of the corresponding discrete dynamical systems are investigated. The stability conditions of Nash equilibrium under a local adjustment process are studied. The stability conditions of Nash equilibrium under a local adjustment process are studied. The stability of Nash equilibrium, as some parameters of the model are varied, gives rise to complex dynamics such as cycles of higher order and chaos. On this basis, we discover that an increase of adjustment speed of bounded rational player can make Bertrand market sink into the chaotic state. Finally, the complex dynamics, bifurcations and chaos are displayed by numerical simulation.

Keywords: Bertrand duopoly model, discrete dynamical system, heterogeneous expectations, nash equilibrium

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Authors: Jixiang Zhang, Yanhua Wang

Abstract:

A dynamic of Bertrand duopoly game is analyzed, where players use different production methods and choose their prices with bounded rationality. The equilibriums of the corresponding discrete dynamical systems are investigated. The stability conditions of Nash equilibrium under a local adjustment process are studied. The stability conditions of Nash equilibrium under a local adjustment process are studied. The stability of Nash equilibrium, as some parameters of the model are varied, gives rise to complex dynamics such as cycles of higher order and chaos. On this basis, we discover that an increase of adjustment speed of bounded rational player can make Bertrand market sink into the chaotic state. Finally, the complex dynamics, bifurcations and chaos are displayed by numerical simulation.

Keywords: Bertrand duopoly model, discrete dynamical system, heterogeneous expectations, nash equilibrium

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Authors: Mohammad Amin Hamedirad, Seyed Javad Vaziri Kang Olyaei

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Project planning and control are one of the most critical issues in the management of construction projects. Traditional methods of project planning and control, such as the critical path method or Gantt chart, are not widely used for planning projects with discrete and repetitive activities, and one of the problems of project managers is planning the implementation process and optimal allocation of its resources. Massive concreting projects is also a project with discrete and repetitive activities. This study uses the concept of simulating discrete events to manage resources, which includes finding the optimal number of resources considering various limitations such as limitations of machinery, equipment, human resources and even technical, time and implementation limitations using analysis of resource consumption rate, project completion time and critical points analysis of the implementation process. For this purpose, the concept of discrete-event simulation has been used to model different stages of implementation. After reviewing the various scenarios, the optimal number of allocations for each resource is finally determined to reach the maximum utilization rate and also to reduce the project completion time or reduce its cost according to the existing constraints. The results showed that with the optimal allocation of resources, the project completion time could be reduced by 90%, and the resulting costs can be reduced by up to 49%. Thus, allocating the optimal number of project resources using this method will reduce its time and cost.

Keywords: simulation, massive concreting, discrete event simulation, resource management

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Authors: Michalis Linardakis, Vasilis Grammatikopoulos, Athanasios Gregoriadis, Kalliopi Trouli

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Discrete choice models belong to the family of Conjoint analysis that are applied on the preferences of the respondents towards a set of scenarios that describe alternative choices. The scenarios have been pre-designed to cover all the attributes of the alternatives that may affect the choices. In this study, we examine how preschool educators integrate physical activities into their everyday teaching practices through the use of discrete choice models. One of the advantages of discrete choice models compared to other more traditional data collection methods (e.g. questionnaires and interviews that use ratings) is that the respondent is called to select among competitive and realistic alternatives, rather than objectively rate each attribute that the alternatives may have. We present the effort to construct and choose representative attributes that would cover all possible choices of the respondents, and the scenarios that have arisen. For the purposes of the study, we used a sample of 50 preschool educators in Greece that responded to 4 scenarios (from the total of 16 scenarios that the orthogonal design resulted), with each scenario having three alternative teaching practices. Seven attributes of the alternatives were used in the scenarios. For the analysis of the data, we used multinomial logit model with random effects, multinomial probit model and generalized mixed logit model. The conclusions drawn from the estimated parameters of the models are discussed.

Keywords: conjoint analysis, discrete choice models, educational data, multivariate statistical analysis

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Authors: Iqbal Felani

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A Coal-Fired Power Plant has some integrated facilities to handle coal from three separated coal yards to eight units power plant’s bunker. But nowadays the facilities are not reliable enough for supporting the system. Management planned to invest some facilities to increase the reliability. They also had a plan to make single spesification of coal used all of the units, called Single Quality Coal (SQC). This simulation would compare before and after improvement with two scenarios i.e First In First Out (FIFO) and Last In First Out (LIFO). Some parameters like stay time, reorder point and safety stock is determined by the simulation. Discrete event simulation based software, Flexsim 5.0, is used to help the simulation. Based on the simulation, Single Quality Coal with FIFO scenario has the shortest staytime with 8.38 days.

Keywords: Coal Yard Management, Discrete event simulation First In First Out, Last In First Out.

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Authors: Mohamed Hassan Abdullahi

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Decomposition is used as a synthesis tool in several physical systems. It can also be used for tearing and restructuring, which is large-scale system analysis. On the other hand, the commutativity of series-connected systems has fascinated the interest of researchers, and its advantages have been emphasized in the literature. The presentation looks into the necessary conditions for decomposing any third-order discrete-time linear time-varying system into a commutative pair of first- and second-order systems. Additional requirements are derived in the case of nonzero initial conditions. MATLAB simulations are used to verify the findings. The work is unique and is being published for the first time. It is critical from the standpoints of synthesis and/or design. Because many design techniques in engineering systems rely on tearing and reconstruction, this is the process of putting together simple components to create a finished product. Furthermore, it is demonstrated that regarding sensitivity to initial conditions, some combinations may be better than others. The results of this work can be extended for the decomposition of fourth-order discrete-time linear time-varying systems into lower-order commutative pairs, as two second-order commutative subsystems or one first-order and one third-order commutative subsystems.

Keywords: commutativity, decomposition, discrete time-varying systems, systems

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Authors: Anders Schou Simonsen, Thomas Condra, Kim Sørensen

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Various processes are modelled using a discrete phase, where particles are seeded from a source. Such particles can represent liquid water droplets, which are affecting the continuous phase by exchanging thermal energy, momentum, species etc. Discrete phases are typically modelled using parcel, which represents a collection of particles, which share properties such as temperature, velocity etc. When coupling the phases, the exchange rates are integrated over the cell, in which the parcel is located. This can cause spikes and fluctuating exchange rates. This paper presents an alternative method of coupling a discrete and a continuous plug flow phase. This is done using triangular parcels, which span between nodes following the dynamics of single droplets. Thus, the triangular parcels are propagated using the corner nodes. At each time step, the exchange rates are spatially integrated over the surface of the triangular parcels, which yields a smooth continuous exchange rate to the continuous phase. The results shows that the method is more stable, converges slightly faster and yields smooth exchange rates compared with the steam tube approach. However, the computational requirements are about five times greater, so the applicability of the alternative method should be limited to processes, where the exchange rates are important. The overall balances of the exchanged properties did not change significantly using the new approach.

Keywords: CFD, coupling, discrete phase, parcel

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Authors: Ozden Saygili, Eser Cakti

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Mosques and minarets can be severely damaged as a result of earthquakes. Non-linear behavior of minarets of Mihrimah Sultan and Süleymaniye Mosques and the minaret of St. Sophia are analyzed to investigate seismic response, damage and failure mechanisms of minarets during earthquake. Selected minarets have different height and diameter. Discrete elements method was used to create the numerical minaret models. Analyses were performed using sine waves. Two parameters were used for evaluating the results: the maximum relative dislocation of adjacent drums and the maximum displacement at the top of the minaret. Both parameters were normalized by the drum diameter. The effects of minaret geometry on seismic behavior were evaluated by comparing the results of analyses.

Keywords: discrete element method, earthquake safety, nonlinear analysis, masonry structures

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Authors: H. El-Kamchouchi, Heba Gaber, Fatma Ahmed, Dalia H. El-Kamchouchi

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A digital signature is an electronic signature form used by an original signer to sign a specific document. When the original signer is not in his office or when he/she travels outside, he/she delegates his signing capability to a proxy signer and then the proxy signer generates a signing message on behalf of the original signer. The two parties must be able to authenticate one another and agree on a secret encryption key, in order to communicate securely over an unreliable public network. Authenticated key agreement protocols have an important role in building a secure communications network between the two parties. In this paper, we present a secure proxy signature scheme over an efficient and secure authenticated key agreement protocol based on factoring and discrete logarithm problem.

Keywords: discrete logarithm, factoring, proxy signature, key agreement

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Authors: Liang-Ta Cheng, Ching-Yu Yang

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Based on fast discrete cosine transform (FDCT), the authors present a high capacity and high perceived quality method for electrocardiogram (ECG) signal. By using a simple adjusting policy to the 1-dimentional (1-D) DCT coefficients, a large volume of secret message can be effectively embedded in an ECG host signal and be successfully extracted at the intended receiver. Simulations confirmed that the resulting perceived quality is good, while the hiding capability of the proposed method significantly outperforms that of existing techniques. In addition, our proposed method has a certain degree of robustness. Since the computational complexity is low, it is feasible for our method being employed in real-time applications.

Keywords: data hiding, ECG steganography, fast discrete cosine transform, 1-D DCT bundle, real-time applications

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Authors: Tomas Menard

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The design of control-law for engineering systems has been investigated for many decades. While many results are concerned with continuous systems with continuous output, nowadays, many controlled systems have to transmit their output measurements through network, hence making it discrete-time. But it is well known that the sampling of a system whose control-law is based on the continuous output may render the system unstable, especially when this sampling period is long compared to the system dynamics. The control design then has to be adapted in order to cope with this issue. In this paper, we consider systems which can be modeled as double integrator with uncertainty on the input since many mechanical systems can be put under such form. We present a control scheme based on an observer using only discrete time measurement and which provides continuous time estimation of the state, combined with a continuous control law, which stabilized a system with second-order dynamics even in the presence of uncertainty. It is further shown that arbitrarily long sampling periods can be dealt with properly setting the control scheme parameters.

Keywords: dynamical system, control law design, sampled output, observer design

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Authors: Mwafak Shakoor

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The purpose of this study was to reduce patient waiting times, improve system throughput and improve resources utilization in radiology department. A discrete event simulation model was developed using Arena simulation software to investigate different alternatives to improve the overall system delivery based on adding resource scenarios due to the linkage between patient waiting times and resource availability. The study revealed that there is no addition investment need to procure additional scanner but hospital management deploy managerial tactics to enhance machine utilization and reduce the long waiting time in the department.

Keywords: discrete event simulation, radiology department, arena, waiting time, healthcare modeling, computed tomography

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Authors: F. J. Ma, A. K. H. Kwan

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Crack caused by shrinkage movement of concrete is a serious problem especially when restraint is provided. It may cause severe serviceability and durability problems. The existing prediction methods for crack width of concrete due to shrinkage movement are mainly numerical methods under simplified circumstances, which do not agree with each other. To get a more unified prediction method applicable to more sophisticated circumstances, finite element crack width analysis for shrinkage effect should be developed. However, no existing finite element analysis can be carried out to predict the crack width of concrete due to shrinkage movement because of unsolved reasons of conventional finite element analysis. In this paper, crack width analysis implemented by finite element analysis is presented with pseudo-discrete crack model, which combines traditional smeared crack model and newly proposed crack queuing algorithm. The proposed pseudo-discrete crack model is capable of simulating separate and single crack without adopting discrete crack element. And the improved finite element analysis can successfully simulate the stress redistribution when concrete is cracked, which is crucial for predicting crack width, crack spacing and crack number.

Keywords: crack queuing algorithm, crack width analysis, finite element analysis, shrinkage effect

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Authors: Shin Je Lee, Go Bong Choi, Jeong Cheol Seo, Jong Min Lee, Gibaek Lee

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Water pipe network is installed underground and once equipped; it is difficult to recognize the state of pipes when the leak or burst happens. Accordingly, post management is often delayed after the fault occurs. Therefore, the systematic fault management system of water pipe network is required to prevent the accident and minimize the loss. In this work, we develop online fault detection system of water pipe network using data of pipes such as flow rate or pressure. The transient model describing water flow in pipelines is presented and simulated using Matlab. The fault situations such as the leak or burst can be also simulated and flow rate or pressure data when the fault happens are collected. Faults are detected using statistical methods of fast Fourier transform and discrete wavelet transform, and they are compared to find which method shows the better fault detection performance.

Keywords: fault detection, water pipeline model, fast Fourier transform, discrete wavelet transform

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Authors: Tomoaki Hashimoto

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Model predictive control is a kind of optimal feedback control in which control performance over a finite future is optimized with a performance index that has a moving initial time and a moving terminal time. This paper examines the stability of model predictive control for linear discrete-time systems with additive stochastic disturbances. A sufficient condition for the stability of the closed-loop system with model predictive control is derived by means of a linear matrix inequality. The objective of this paper is to show the results of computational simulations in order to verify the validity of the obtained stability condition.

Keywords: computational simulations, optimal control, predictive control, stochastic systems, discrete-time systems

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Authors: Rogelio Luck, Yucheng Liu

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This paper proposes the continuous-time singular value decomposition (SVD) for the impulse response function, a special kind of Green’s functions e⁻⁽ᵗ⁻ ᵀ⁾, in order to find a set of singular functions and singular values so that the convolutions of such function with the set of singular functions on a specified domain are the solutions to the inhomogeneous differential equations for those singular functions. A numerical example was illustrated to verify the proposed method. Besides the continuous-time SVD, a discrete-time SVD is also presented for the impulse response function, which is modeled using a Toeplitz matrix in the discrete system. The proposed method has broad applications in signal processing, dynamic system analysis, acoustic analysis, thermal analysis, as well as macroeconomic modeling.

Keywords: singular value decomposition, impulse response function, Green’s function , Toeplitz matrix , Hankel matrix

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Authors: Tarek Aboueldahab, Hanan Farag

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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.

Keywords: cost minimization, multi-vehicle routing problem, passive congregation, discrete swarm, passive congregation

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