Search results for: lyapunov functions

Authors: Guillermo Manuel Flores Figueroa, Juan Alejandro Vazquez Feijoo, Jose Navarro Antonio

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A continuous nonlinear system response may be obtained by an infinite sum of the so-called Volterra operators. Each operator is obtained from multidimensional convolution of nth-order between the nth-order Volterra kernel and the system input. These operators can also be obtained from the Associated Linear Equations (ALEs) that are linear models of subsystems which inputs and outputs are of the same nth-order. Each ALEs produces a particular nth-Volterra operator. As linear models a unitary impulse response can be obtained from them. This work shows the relationship between this unitary impulse responses and the corresponding order Volterra kernel.

Keywords: Volterra series, frequency response functions FRF, associated linear equations ALEs, unitary response function, Voterra kernel

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Authors: Marcin Sowa

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The paper discusses the subinterval-based numerical method for fractional derivative computations. It is now referred to by its acronym – SubIval. The basis of the method is briefly recalled. The ability of the method to be applied in time stepping solvers is discussed. The possibility of implementing a time step size adaptive solver is also mentioned. The solver is tested on a transient circuit example. In order to display the accuracy of the solver – the results have been compared with those obtained by means of a semi-analytical method called gcdAlpha. The time step size adaptive solver applying SubIval has been proven to be very accurate as the results are very close to the referential solution. The solver is currently able to solve FDE (fractional differential equations) with various derivative orders for each equation and any type of source time functions.

Keywords: numerical method, SubIval, fractional calculus, numerical solver, circuit analysis

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Authors: A. K. Sarathe, N. P. Patidar

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The Indian automobile sector is comprised of independent manufacturers and joint ventures with their foreign counterpart companies by making use of the Foreign Direct Investment policy of the Government of India. These manufacturers started capturing the hearts of Indian customers with their choice of technological and innovative product features, with quality and reliability. This transformed the automobile scene from a “sellers market to buyers market”. The potential benefits from the auto sector have been recognized by the planners, managers, and administrators of both the sectors –government and private. Generation of employment for the readily available technical workforce has been achieved not only through the manufacturers, but also through the growing ancillaries and service providers of the auto industries. The main purpose of this paper is to come up with the identification of possible working areas and associated job functions of mechanical and automobile diploma holders having employment opportunities in auto sector of India.

Keywords: automobile sector, diploma holder, employment, job description

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Authors: Talha A. Taj, Talha A. Khan, M. Imran Khalid

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Optimization techniques attract researchers to formulate a problem and determine its optimum solution. This paper presents an Enhanced Harmony Search (ENHS) algorithm for solving optimization problems. The proposed algorithm increases the convergence and is more efficient than the standard Harmony Search (HS) algorithm. The paper discusses the novel techniques in detail and also provides the strategy for tuning the decisive parameters that affects the efficiency of the ENHS algorithm. The algorithm is tested on various benchmark functions, a real world optimization problem and a constrained objective function. Also, the results of ENHS are compared to standard HS, and various other optimization algorithms. The ENHS algorithms prove to be significantly better and more efficient than other algorithms. The simulation and testing of the algorithms is performed in MATLAB.

Keywords: optimization, harmony search algorithm, MATLAB, electronic

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Authors: Diana González, Marta Gràcia, Ana Luisa Adam-Alcocer

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Incorporating digital tools into educational practices represents a valuable approach for enriching the quality of teachers' educational practices in oral competence and fostering improvements in student learning outcomes. This study aims to promote a collaborative and culturally sensitive approach to professional development between teachers and a speech therapist to enhance their self-awareness and reflection on high-quality educational practices that integrate school components to strengthen children’s oral communication and pragmatic skills. The study involved five bilingual teachers fluent in both English and Spanish, with three specializing in special education and two in general education. It focused on Spanish-English bilingual students, aged 3-6, who were experiencing speech delays or disorders in a New York City public school, with the collaboration of a speech therapist. Using EVALOE-DSS (Assessment Scale of Oral Language Teaching in the School Context - Decision Support System), teachers conducted self-assessments of their teaching practices, reflect and make-decisions throughout six classes from March to June, focusing on students' communicative competence across various activities. Concurrently, the speech therapist observed and evaluated six classes per teacher using EVALOE-DSS during the same period. Additionally, professional development meetings were held monthly between the speech therapist and teachers, centering on discussing classroom interactions, instructional strategies, and the progress of both teachers and students in their classes. Findings highlight the digital tool EVALOE-DSS's value in analyzing communication patterns and trends among bilingual children in inclusive settings. It helps in identifying improvement areas through teacher and speech therapist collaboration. After self-reflection meetings, teachers demonstrated increased awareness of student needs in oral language and pragmatic skills. They also exhibited enhanced utilization of strategies outlined in EVALOE-DSS, such as actively guiding and orienting students during oral language activities, promoting student-initiated communicative interactions, teaching students how to seek and provide information, and managing turn-taking to ensure inclusive participation. Teachers participating in the professional development program have shown positive progress in assessing their classes across all dimensions of the training tool, including instructional design, teacher conversation management, pupil conversation management, communicative functions, teacher strategies, and pupil communication functions. This includes aspects related to both teacher actions and child actions, particularly in child language development. This progress underscores the effectiveness of individual reflection (conducted weekly or biweekly using EVALOE-DSS) as well as collaborative reflection among teachers and the speech therapist during meetings. The EVALOE-SSD has proven effective in supporting teachers' self-reflection, decision-making, and classroom changes, leading to improved development of students' oral language and pragmatic skills. It has facilitated culturally sensitive evaluations of communication among bilingual children, cultivating collaboration between teachers and speech therapist to identify areas of growth. Participants in the professional development program demonstrated substantial progress across all dimensions assessed by EVALOE-DSS. This included improved management of pupil communication functions, implementation of effective teaching strategies, and better classroom dynamics. Regular reflection sessions using EVALOE-SSD supported continuous improvement in instructional practices, highlighting its role in fostering reflective teaching and enriching student learning experiences. Overall, EVALOE-DSS has proven invaluable for enhancing teaching effectiveness and promoting meaningful student interactions in diverse educational settings.

Keywords: bilingual students, collaboration, culturally sensitive, oral communication skills, self-reflection

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Authors: Aouiche Abdelaziz, Soudani Mouhamed Salah, Aouiche El Moundhe

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The present paper addresses the utilization of Artificial Neural Networks (ANNs) and Fuzzy Inference Systems (FISs) for the identification and control of dynamical systems with some degree of uncertainty. Because ANNs and FISs have an inherent ability to approximate functions and to adapt to changes in input and parameters, they can be used to control systems too complex for linear controllers. In this work, we show how ANNs and FISs can be put in order to form nets that can learn from external data. In sequence, it is presented structures of inputs that can be used along with ANNs and FISs to model non-linear systems. Four systems were used to test the identification and control of the structures proposed. The results show the ANNs and FISs (Back Propagation Algorithm) used were efficient in modeling and controlling the non-linear plants.

Keywords: non-linear systems, fuzzy set Models, neural network, control law

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Authors: Krongthong Khairiree

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The purpose of this research study is to explore mathematics project-based learning approach and the use of technology in the context of school mathematics in Thailand. Data of the study were collected from 6 sample secondary schools and the students were 6-14 years old. Research findings show that through mathematics project-based learning approach and the use of GSP, students were able to make mathematics learning fun and challenging. Based on the students’ interviews they revealed that, with GSP, they were able to visualize and create graphical representations, which will enable them to develop their mathematical thinking skills, concepts and understanding. The students had fun in creating variety of graphs of functions which they can not do by drawing on graph paper. In addition, there are evidences to show the students’ abilities in connecting mathematics to real life outside the classroom and commercial products, such as weaving, patterning of broomstick, and ceramics design.

Keywords: mathematics, project-based learning, Geometer’s Sketchpad (GSP), commercial products

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Authors: Aliyu Uthman Abdulkadir

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Insurgent reports in Nigeria have become a recurring focus in the media due to the significance of language choices. This paper investigates these reports with the aim of identifying various pragmatic practices and exploring the role of the media in shaping public perception of insurgency. Three Nigerian newspapers The Punch, This Day, and The Guardian were selected for analysis between December 2022 and January 2023. Five media reports were examined to uncover the pragmatic functions embedded in the discourse. The study reveals that the media employ implicit acts such as exposing, sensitizing, informing, castigating, reprimanding, and shaming to depict insurgent activities in the country. The analysis also highlights how the use of presupposed ideologies enhances the delivery and acceptance of information related to insurgent actions. The study concludes that the media's portrayal of insurgency is often biased, as reflected in the data analysis.

Keywords: insurgency, pragmatic acts, bias, framing, ideoligies

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Authors: S. Ansari Sadrabadi, G. H. Rahimi

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In this paper, thick walled Cylindrical tanks or tubes made of functionally graded material under internal pressure and temperature gradient are studied. Material parameters have been considered as power functions. They play important role in the elastoplastic behavior of these materials. To clarify their role, different materials with different parameters have been used under temperature gradient. Finally, their effect and loading effect have been determined in first yield point. Also, the important role of temperature gradient was also shown. At the end the study has been results obtained from changes in the elastic modulus and yield stress. Also special attention is also given to the effects of this internal pressure and temperature gradient in the creation of tensile and compressive stresses.

Keywords: FGM, cylindrical pressure tubes, small deformation theory, yield onset, thermal loading

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Authors: Yu Qin

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In order to tackle the ever-growing problem of the lack of parking space, this paper presents the design and implementation of a smart unmanned parking system that is based on RFID (radio frequency identification) technology and Wireless communication technology. This system uses RFID technology to achieve the identification function (transmitted by 2.4 G wireless module) and is equipped with an STM32L053 micro controller as the main control chip of the smart vehicle. This chip can accomplish automatic parking (in/out), charging and other functions. On this basis, it can also help users easily query the information that is stored in the database through the Internet. Experimental tests have shown that the system has the features of low power consumption and stable operation, among others. It can effectively improve the level of automation control of the parking lot management system and has enormous application prospects.

Keywords: RFID, embedded system, unmanned, parking management

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Authors: M. Guemmadi, A. Ouibrahim

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This numerical investigation aims to evaluate how a viscoelastic lubricant described by a generalized Maxwell model, affects the pressure distribution, the load capacity and thermal effect in a journal bearing lubrication. We use for the purpose the CFD package software completed by adapted user define functions (UDFs) to solve the coupled equations of momentum, of energy and of the viscoelastic model (generalized Maxwell model). Two parameters, viscosity and relaxation time are involved to show how viscoelasticity substantially affect the pressure distribution, the load capacity and the thermal transfer by comparison to Newtonian lubricant. These results were also compared with the available published results.

Keywords: journal bearing, lubrication, Maxwell model, viscoelastic fluids, computational modelling, load capacity

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Authors: Hassan Abubakar

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This paper will serve as a guide to good design and implementation of wireless networking for campus institutions in Nigeria. It can be implemented throughout the primary, secondary and tertiary institutions. This paper describe the some technical functions, standard configurations and layouts of the 802.11 wireless LAN(Local Area Network) that can be implemented across the campus network. The paper also touches upon the wireless infrastructure standards involved with enhanced services, such as voice over wireless and wireless guest hotspot. The paper also touch the benefits derived from implementing campus wireless network and share some lights on how to arrive at the success in increasing the performance of wireless and using the campus wireless to share resources like software applications, printer and documents.

Keywords: networking, standards, wireless local area network (WLAN), radio frequency (RF), campus

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Authors: N. W. Paul, M. Banerjee

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Discussing the nexus between global health policy and local practices, this article addresses the recent Ebola outbreak as a role model for narrative co-constructions of epidemic risk. We will demonstrate in how far a theory-driven and methodologically rooted analysis of narrativity can help to improve mechanisms of prevention and intervention whenever epidemic risk needs to be addressed locally in order to contribute to global health. Analyzing the narrative transformation of Ebola, we will also address issues of transcultural problem-solving and of normative questions at stake. In this regard, we seek to contribute to a better understanding of a key question of global health and justice as well as to the underlying ethical questions. By highlighting and analyzing the functions of narratives, this paper provides a translational approach to refine our practices by which we address epidemic risk, be it on the national, the transnational or the global scale.

Keywords: ebola, epidemic risk, medical ethics, medical humanities

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Authors: Wahiba Mohguen, Raïs El’hadi Bekka

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This paper presents a denoising method called EMD-Custom that was based on Empirical Mode Decomposition (EMD) and the modified Customized Thresholding Function (Custom) algorithms. EMD was applied to decompose adaptively a noisy signal into intrinsic mode functions (IMFs). Then, all the noisy IMFs got threshold by applying the presented thresholding function to suppress noise and to improve the signal to noise ratio (SNR). The method was tested on simulated data and real ECG signal, and the results were compared to the EMD-Based signal denoising methods using the soft and hard thresholding. The results showed the superior performance of the proposed EMD-Custom denoising over the traditional approach. The performances were evaluated in terms of SNR in dB, and Mean Square Error (MSE).

Keywords: customized thresholding, ECG signal, EMD, hard thresholding, soft-thresholding

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Authors: Sandeep Singh

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In this article, two classes of iterative schemes are proposed for approximating solutions of nonlinear systems of equations whose orders of convergence are six and eight respectively. Sixth order scheme requires the evaluation of two vector-functions, two first Fr'echet derivatives and three matrices inversion per iteration. This three-step sixth-order method is further extended to eighth-order method which requires one more step and the evaluation of one extra vector-function. Moreover, computational efficiency is compared with some other recently published methods in which we found, our methods are more efficient than existing numerical methods for higher and medium size nonlinear system of equations. Numerical tests are performed to validate the proposed schemes.

Keywords: Nonlinear systems, Computational complexity, order of convergence, Jarratt-type scheme

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Authors: Agata Matuszewska, Małgorzata Syczewska

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The movement of the upper limbs contributes significantly to balance control while walking in humans. However, the impact of different arm swing modes on gait stability is yet to be determined. This work intends to establish numerical parameters for assessing the arm swing. Nineteen people, comprising fifteen young, healthy individuals, two middle-aged individuals, and two individuals with dysfunctions, were analyzed using the movement analysis system. Proposed parameters such as ASᵢₐ (reflecting the arm swing amplitude) and Pearson’s correlation coefficient between the right and left upper limbs can be used to classify the type of movement task each participant performs. The results indicate that the ASᵢₐ parameter could potentially detect any abnormalities in upper limb functions, which may be due to musculoskeletal disorders or other malfunctions.

Keywords: arm swing, human balance, interlimb coordination, motion analysis system

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Authors: A. Boumedine, K. Benfriha, S. Lecheb

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Manufacturing methods for products and structures made of composite materials reduce the number of parts and integrate technical functions, this advantage of composite materials leads to a lot of innovation but also to a reduction of costs and a gain in quality. A material has attributes: its density, it’s resistance, it’s cost, it’s resistance to corrosion. For the design of a product, a certain profile of these attributes is required: low density, resistance removed, low cost. The problem is then to identify this attribute profile and to compare it with those of the materials, in order to find the one that comes closest. The aim of this work is to demonstrate the feasibility of characterizing a mini turbine made of 3D printed fiber-filled composite material by the process of additive manufacturing, then compare the performance of the alloy turbine with the composite turbine according to the results of the simulation by Abaqus software.

Keywords: additive manufacturing, composite materials, design, 3D printer, turbine

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Authors: Ameen Jameel Alawneh

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A Mobile ad-hoc network (MANET) is a collection of wireless mobile hosts that dynamically form a temporary network without the aid of a system administrator. It has neither fixed infrastructure nor wireless ad hoc sessions. It inherently reaches several nodes with a single transmission, and each node functions as both a host and a router. The network maybe represented as a set of clusters each managed by clusterhead. The cluster size is not fixed and it depends on the movement of nodes. We proposed a clusterhead size algorithm (CHSize). This clustering algorithm can be used by several routing algorithms for ad hoc networks. An elected clusterhead is assigned for communication with all other clusters. Analysis and simulation of the algorithm has been implemented using GloMoSim networks simulator, MATLAB and MAPL11 proved that the proposed algorithm achieves the goals.

Keywords: simulation, MANET, Ad-hoc, cluster head size, linked cluster algorithm, loss and dropped packets

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Authors: Mohd Yusri Ibrahim

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Although it is known that interpersonal skills are essential for personal development, the debate however continues as to how to measure those skills, especially in youths. This study was conducted to develop a measurement model of interpersonal skills by suggesting three construct namely personal, skills and relationship; six function namely self, perception, listening, conversation, emotion and conflict management; and 30 behaviours as indicators. This cross-sectional survey by questionnaires was applied in east side of peninsula of Malaysia for 150 respondents, and analyzed by structural equation modelling (SEM) by AMOS. The suggested constructs, functions and indicators were consider accepted as measurement elements by observing on regression weight for standard loading, average variance extracted (AVE) for convergent validity, square root of AVE for discriminant validity, composite reliability (CR), and at least three fit indexes for model fitness. Finally, a measurement model of interpersonal skill for youth was successfully developed.

Keywords: interpersonal communication, interpersonal skill, youth, communication skill

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Authors: Leonid Borisov, Yuri Orlov

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We obtain explicit formulas of finitely multiple approximations of the equilibrium density matrix for the case of the harmonic oscillator using Chernoff's theorem and the notion of semigroup which is Chernoff equivalent to average semigroup. Also we found explicit formulas for the corresponding approximate Wigner functions and average values of the observable. We consider a superposition of τ -quantizations representing a wide class of linear quantizations. We show that the convergence of the approximations of the average values of the observable is not uniform with respect to the Gibbs parameter. This does not allow to represent approximate expression as the sum of the exact limits and small deviations evenly throughout the temperature range with a given order of approximation.

Keywords: Chernoff theorem, Feynman formulas, finitely multiple approximation, harmonic oscillator, Wigner function

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Authors: Changho Han, Jaemin Lee, Li Hua, Seokkwon Jeong

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Design of membership function ranges in fuzzy logic control (FLC) is presented for robust control of a variable speed refrigeration system (VSRS). The criterion values of the membership function ranges can be carried out from the static experimental data, and two different values are offered to compare control performance. Some simulations and real experiments for the VSRS were conducted to verify the validity of the designed membership functions. The experimental results showed good agreement with the simulation results, and the error change rate and its sampling time strongly affected the control performance at transient state of the VSRS.

Keywords: variable speed refrigeration system, fuzzy logic control, membership function range, control performance

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Authors: Raphael Zanella, Todd A. Oliver, Karl W. Schulz

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This work deals with the finite element approximation of axisymmetric compressible flows with swirl velocity. We are interested in problems where the flow, while weakly dependent on the azimuthal coordinate, may have a strong azimuthal velocity component. We describe the approximation of the compressible Navier-Stokes equations with H1-conformal spaces of axisymmetric functions. The weak formulation is implemented in a C++ solver with explicit time marching. The code is first verified with a convergence test on a manufactured solution. The verification is completed by comparing the numerical and analytical solutions in a Poiseuille flow case and a Taylor-Couette flow case. The code is finally applied to the problem of a swirling subsonic air flow in a plasma torch geometry.

Keywords: axisymmetric problem, compressible Navier-Stokes equations, continuous finite elements, swirling flow

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Authors: Abdul Azeez Badmus

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The extent to which women are allowed to express their liberty and freedom are varying in human society. Islam’s basic view of women postulates a complimentary function as the creator has created every living thing in pairs based on the foundation of complimentary functions, so the human species is not exceptional. It is also ubiquitous contention whether women should participate in politics or not; the limit to which women should participate in politics is another problem. This paper attempts to suggest a possible solution to the questions mentioned above, to review the level of participation of women in politics since the beginning of Islam and the Quranic injunctions that seem to have allowed or disallowed that. The paper adopts a historical and analytical approach, with special reference to the Quranic, Sunna, juristic opinions, and historical events. The paper advocates for proper reference to the authentic Islamic sources in determining the right and obligations of women in society.

Keywords: politics, right, challenges, Qur’ān, perspective

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Authors: Ya-Mei Chang

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This research is intended to apply spatio-temporal point process methods to the dengue fever data in Tainan. The spatio-temporal intensity function of the dataset is assumed to be separable. The kernel estimation is a widely used approach to estimate intensity functions. The intensity function is very helpful to study the relation of the spatio-temporal point process and some covariates. The covariate effects might be nonlinear. An nonparametric smoothing estimator is used to detect the nonlinearity of the covariate effects. A fitted parametric model could describe the influence of the covariates to the dengue fever. The correlation between the data points is detected by the K-function. The result of this research could provide useful information to help the government or the stakeholders making decisions.

Keywords: dengue fever, spatial point process, kernel estimation, covariate effect

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Authors: Reza Mollapourasl, Majid Haghi

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In this study, we develop local meshfree methods known as radial basis function-generated finite difference (RBF-FD) method and Hermite finite difference (RBF-HFD) method to design stencil weights and spatial discretization for Helmholtz equation. The convergence and stability of schemes are investigated numerically in three dimensions with irregular shaped domain. These localized meshless methods incorporate the advantages of the RBF method, finite difference and Hermite finite difference methods to handle the ill-conditioning issue that often destroys the convergence rate of global RBF methods. Moreover, numerical illustrations show that the proposed localized RBF type methods are efficient and applicable for problems with complex geometries. The convergence and accuracy of both schemes are compared by solving a test problem.

Keywords: radial basis functions, Hermite finite difference, Helmholtz equation, stability

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Authors: Burak Erkayman

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The manufacturing and service organizations are in the need of using ERP systems to integrate many functions from purchasing to storage, production planning to calculation of costs. Using ERP systems by the integration in the level of information provides companies remarkable advantages in terms of profitability, productivity and efficiency in processes. Cloud computing is one of the most significant changes in information and communication technology. The developments in Cloud Computing attract business world to take advantage of this field. Cloud Computing means much more storage area, more cost saving and faster data transfer rate. In addition to these, it presents new business models, new field of study and practicable solutions for anyone’s use. These developments make inevitable the implementation of ERP systems to cloud environment. In this study, the performance of ERP systems in cloud environment is analyzed through various performance criteria and a comparison between traditional and cloud-ERP systems is presented. At the end of study the transformation and the future of ERP systems is discussed.

Keywords: cloud-ERP, ERP system performance, information system transformation

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Authors: P. Zamani, A. Taleshi Anbouhi, M. R. Ashory, S. Mohajerzadeh, M. M. Khatibi

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Modal analysis is a developing science in the experimental evaluation of dynamic properties of the structures. Mechanical devices such as accelerometers are one of the sources of lack of quality in measuring modal testing parameters. In this paper, eliminating the accelerometer’s mass effect of the frequency response of the structure is studied. So, a strategy is used for eliminating the mass effect by using sensitivity analysis. In this method, the amount of mass change and the place to measure the structure’s response with least error in frequency correction is chosen. Experimental modal testing is carried out on a steel plate and the effect of accelerometer’s mass is omitted using this strategy. Finally, a good agreement is achieved between numerical and experimental results.

Keywords: accelerometer mass, frequency response function, modal analysis, sensitivity analysis

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Authors: Kailash C. Madan

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We study the steady state behaviour of a batch arrival single server queue in which the first service with general service times is compulsory and the second service with general service times is optional. We term such a two phase service as generalized Coxian-2 service. Just after completion of a service the server must take a vacation of random length of time with general vacation times. We obtain steady state probability generating functions for the queue size as well as the steady state mean queue size at a random epoch of time in explicit and closed forms. Some particular cases of interest including some known results have been derived.

Keywords: batch arrivals, compound Poisson process, generalized Coxian-2 service, steady state

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Authors: F. Karim

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In this paper, triangular lattice index-guiding photonic crystal fibers (PCFs) are synthesized to compensate the chromatic dispersion of a single mode fiber (SMF-28) for an 80 km optical link operating at 1.55 µm, by using the directed tabu search algorithm. Hole-to-hole distance, circular air-hole diameter, solid-core diameter, ring number and PCF length parameters are optimized for this purpose. Three Synthesized PCFs with different physical parameters are compared in terms of their objective functions values, residual dispersions and compensation ratios.

Keywords: triangular lattice index-guiding photonic crystal fiber, dispersion compensation, directed tabu search, synthesis

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Authors: Mohamed S. Sasi, Adel M. Mlitan, Abdulfattah M. Alkherraz

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This research project aims to investigate difference in relative rates concerning phosphoryl transfer relevant to biological catalysis of DNA and RNA in the pH-independent reactions. Activated Models of DNA and RNA for alkyl-aryl phosphate diesters (with 4-nitrophenyl as a good leaving group) have successfully been prepared to gather kinetic parameters. Eyring plots for the pH–independent hydrolysis of 1 and 2 were established at different temperatures in the range 100–160 °C. These measurements have been used to provide a better estimate for the difference in relative rates between the reactivity of DNA and RNA cleavage. Eyring plot gave an extrapolated rate of kH2O = 1 × 10-10 s -1 for 1 (RNA model) and 2 (DNA model) at 25°C. Comparing the reactivity of RNA model and DNA model shows that the difference in relative rates in the pH-independent reactions is surprisingly very similar at 25°. This allows us to obtain chemical insights into how biological catalysts such as enzymes may have evolved to perform their current functions.

Keywords: DNA and RNA models, relative rates, reactivity, phosphoryl transfe

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