Search results for: Minimum distance jet impingement

Authors: J. Baskar Babujee, J. Senbagamalar,

Abstract:

The topological distance between a pair of vertices i and j, which is denoted by d(vi, vj), is the number of edges of the shortest path joining i and j. The Wiener index W(G) is the sum of distances between all pairs of vertices of a graph G. W(G) = i

Keywords: Graph, Degree, Distance, Pendent vertex, Wiener index, Tree.

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Authors: Wan Chian Li, Justie Su-Tzu Juan, Yi-Chun Wang, Shu-Chuan Chen

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The evolutionary tree is an important topic in bioinformation. In 2006, Chen and Lindsay proposed a new method to build the mixture tree from DNA sequences. Mixture tree is a new type evolutionary tree, and it has two additional information besides the information of ordinary evolutionary tree. One of the information is time parameter, and the other is the set of mutated sites. In 2008, Lin and Juan proposed an algorithm to compute the distance between two mixture trees. Their algorithm computes the distance with only considering the time parameter between two mixture trees. In this paper, we proposes a method to measure the similarity of two mixture trees with considering the set of mutated sites and develops two algorithm to compute the distance between two mixture trees. The time complexity of these two proposed algorithms are O(n2 × max{h(T1), h(T2)}) and O(n2), respectively

Keywords: evolutionary tree, mixture tree, mutated site, distance.

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Authors: Ali Gholami

Abstract:

In this research the Preparation of Land use map of scanner LISS III satellite data, belonging to the IRS in the Aghche region in Isfahan province, is studied carefully. For this purpose, the IRS satellite images of August 2008 and various land preparation uses in region including rangelands, irrigation farming, dry farming, gardens and urban areas were separated and identified. Therefore, the GPS and Erdas Imaging software were used and three methods of Maximum Likelihood, Mahalanobis Distance and Minimum Distance were analyzed. In each of these methods, matrix error and Kappa index were calculated and accuracy of each method, based on percentages: 53.13, 56.64 and 48.44, were obtained respectively. Considering the low accuracy of these methods in separation of land preparation use, the visual interpretation of the map was used. Finally, regional visits of 150 points were noted at random and no error was observed. It shows that the map prepared by visual interpretation is in high accuracy. Although the probable errors due to visual interpretation and geometric correction might happen but the desired accuracy of the map which is more than 85 percent is reliable.

Keywords: Land use map, Aghche Region, Erdas Imagine, satellite images

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Authors: Mohammad Asif

Abstract:

An accurate prediction of the minimum fluidization velocity is a crucial hydrodynamic aspect of the design of fluidized bed reactors. Common approaches for the prediction of the minimum fluidization velocities of binary-solid fluidized beds are first discussed here. The data of our own careful experimental investigation involving a binary-solid pair fluidized with water is presented. The effect of the relative composition of the two solid species comprising the fluidized bed on the bed void fraction at the incipient fluidization condition is reported and its influence on the minimum fluidization velocity is discussed. In this connection, the capability of packing models to predict the bed void fraction is also examined.

Keywords: Bed void fraction, Binary solid mixture, Minimumfluidization velocity, Packing models

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Authors: Tarek Mohammad

Abstract:

The amplitude response of infrared (IR) sensors depends on the reflectance properties of the target. Therefore, in order to use IR sensor for measuring distances accurately, prior knowledge of the surface must be known. This paper describes the Phong Illumination Model for determining the properties of a surface and subsequently calculating the distance to the surface. The angular position of the IR sensor is computed as normal to the surface for simplifying the calculation. Ultrasonic (US) sensor can provide the initial information on distance to obtain the parameters for this method. In addition, the experimental results obtained by using LabView are discussed. More care should be taken when placing the objects from the sensors during acquiring data since the small change in angle could show very different distance than the actual one. Since stereo camera vision systems do not perform well under some environmental conditions such as plain wall, glass surfaces, or poor lighting conditions, the IR and US sensors can be used additionally to improve the overall vision systems of mobile robots.

Keywords: Distance Measurement, Infrared sensor, Surface properties, Ultrasonic sensor.

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Authors: Eleni Giouli

Abstract:

Distance learning provides a flexible approach to education, enabling busy learners to complete their coursework at their own pace, on their own schedule, and from a convenient location. This flexibility combined with a series of other issues; make the benefits of lifelong distance learning numerous. The purpose of the paper is to investigate whether distance education can contribute to the improvement of adult skills in Greece, highlighting in this way the necessity of the lifelong distance learning. To investigate this goal, a questionnaire is constructed and analyzed based on responses from 3,016 attendees of lifelong distance learning programs in the e-learning of the National and Kapodistrian University of Athens in Greece. In order to do so, a series of relationships is examined including the effects of a) the gender, b) the previous educational level, c) the current employment status, and d) the method used in the distance learning program, on the development of new general, technical, administrative, social, cultural, entrepreneurial and green skills. The basic conclusions that emerge after using a binary logistic framework are that the following factors are critical in order to develop new skills: the gender, the education level and the educational method used in the lifelong distance learning program. The skills more significantly affected by those factors are the acquiring new skills in general, as well as acquiring general, language and cultural, entrepreneurial and green skills, while for technical and social skills only gender and educational method play a crucial role. Moreover, routine skills and social skills are not affected by the four factors included in the analysis.

Keywords: Adult skills, distance learning, education, lifelong learning.

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Authors: Jamel Miri, Bechir Nsiri, Ridha Bouallegue

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Ultra WidBand-IR physical layer technology has seen a great development during the last decade which makes it a promising candidate for short range wireless communications, as they bring considerable benefits in terms of connectivity and mobility. However, like all wireless communication they suffer from vulnerabilities in terms of security because of the open nature of the radio channel. To face these attacks, distance bounding protocols are the most popular counter measures. In this paper, we presented a protocol based on distance bounding to thread the most popular attacks: Distance Fraud, Mafia Fraud and Terrorist fraud. In our work, we study the way to adapt the best secure distance bounding protocols to mapping code of ultra-wideband (TH-UWB) radios. Indeed, to ameliorate the performances of the protocol in terms of security communication in TH-UWB, we combine the modified protocol to ultra-wideband impulse radio technology (IR-UWB). The security and the different merits of the protocols are analyzed.

Keywords: Distance bounding, mapping code ultra-wideband, Terrorist Fraud.

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Authors: Yu-Chen Kuo, Shih-Chieh Lin

Abstract:

Sensor relocation is to repair coverage holes caused by node failures. One way to repair coverage holes is to find redundant nodes to replace faulty nodes. Most researches took a long time to find redundant nodes since they randomly scattered redundant nodes around the sensing field. To record the precise position of sensor nodes, most researches assumed that GPS was installed in sensor nodes. However, high costs and power-consumptions of GPS are heavy burdens for sensor nodes. Thus, we propose a fast sensor relocation algorithm to arrange redundant nodes to form redundant walls without GPS. Redundant walls are constructed in the position where the average distance to each sensor node is the shortest. Redundant walls can guide sensor nodes to find redundant nodes in the minimum time. Simulation results show that our algorithm can find the proper redundant node in the minimum time and reduce the relocation time with low message complexity.

Keywords: Coverage, distributed algorithm, sensor relocation, wireless sensor networks.

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Authors: Mohamed Mahmoud Hasan, Hoda M. O. Mokhtar

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With the gigantic increment in portable applications use and the spread of positioning and location-aware technologies that we are seeing today, new procedures and methodologies for location-based strategies are required. Location recommendation is one of the highly demanded location-aware applications uniquely with the wide accessibility of social network applications that are location-aware including Facebook check-ins, Foursquare, and others. In this paper, we aim to present a new methodology for location recommendation. The proposed approach coordinates customary spatial traits alongside other essential components including shortest distance, and user interests. We also present another idea namely, "activity trajectory" that represents trajectory that fulfills the set of activities that the user is intrigued to do. The approach dispatched acquaints the related distance value to select trajectory(ies) with minimum cost value (distance) and spatial-area to prune unneeded directions. The proposed calculation utilizes the idea of movement direction to prescribe most comparable N-trajectory(ies) that matches the client's required action design with least voyaging separation. To upgrade the execution of the proposed approach, parallel handling is applied through the employment of a MapReduce based approach. Experiments taking into account genuine information sets were built up and tested for assessing the proposed approach. The exhibited tests indicate how the proposed approach beets different strategies giving better precision and run time.

Keywords: Location-based recommendation, map-reduce, recommendation system, trajectory search.

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Authors: Elsayeda M. Elgaml, Dina M. Ibrahim, Elsayed A. Sallam

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Association rule mining is one of the most important fields of data mining and knowledge discovery. In this paper, we propose an efficient multiple support frequent pattern growth algorithm which we called “MSFP-growth” that enhancing the FPgrowth algorithm by making infrequent child node pruning step with multiple minimum support using maximum constrains. The algorithm is implemented, and it is compared with other common algorithms: Apriori-multiple minimum supports using maximum constraints and FP-growth. The experimental results show that the rule mining from the proposed algorithm are interesting and our algorithm achieved better performance than other algorithms without scarifying the accuracy. 

Keywords: Association Rules, FP-growth, Multiple minimum supports, Weka Tool

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Authors: Taylor J. Canfield, Kathe A. Gabel

Abstract:

The aim of this studywas toinvestigate the effect ofrunning classification (sprint, middle, and long distance)and two distances on blood lactate (BLa), heart rate (HR), and rating of perceived exertion (RPE) Borg scale ratings in collegiate athletes. On different days, runners (n = 15) ran 400m and 1600m at a five min mile pace, followed by a two min 6mph jog, and a two min 3mph walk as part of the cool down. BLa, HR, and RPE were taken at baseline, post-run, plus 2 and 4 min recovery times. The middle and long distance runners exhibited lower BLa concentrations than sprint runners after two min of recovery post 400 m runs, immediately after, and two and four min recovery periods post 1600 m runs. When compared to sprint runners, distance runners may have exhibited the ability to clear BLa more quickly, particularly after running 1600 m.

Keywords: Blood lactate, HR, RPE, running.

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Authors: S.G. Srivani, Chandrasekhar Reddy Atla, K.P.Vittal

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This paper presents the development of adaptive distance relay for protection of parallel transmission line with mutual coupling. The proposed adaptive relay, automatically adjusts its operation based on the acquisition of the data from distance relay of adjacent line and status of adjacent line from line circuit breaker IED (Intelligent Electronic Device). The zero sequence current of the adjacent parallel transmission line is used to compute zero sequence current ratio and the mutual coupling effect is fully compensated. The relay adapts to changing circumstances, like failure in communication from other relays and non - availability of adjacent transmission line. The performance of the proposed adaptive relay is tested using steady state and dynamic test procedures. The fault transients are obtained by simulating a realistic parallel transmission line system with mutual coupling effect in PSCAD. The evaluation test results show the efficacy of adaptive distance relay over the conventional distance relay.

Keywords: Adaptive relaying, distance measurement, mutualcoupling, quadrilateral trip characteristic, zones of protection.

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Authors: A. Pashayev, C. Ardil, R. Sadiqov

Abstract:

Various methods of geofield parameters restoration (by algebraic polynoms; filters; rational fractions; interpolation splines; geostatistical methods – kriging; search methods of nearest points – inverse distance, minimum curvature, local – polynomial interpolation; neural networks) have been analyzed and some possible mistakes arising during geofield surface modeling have been presented.

Keywords: interpolation methods, geofield parameters, neural networks.

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Authors: Alan Luo, Hunter N. B. Moseley

Abstract:

Many macromolecular structure entries in the Protein Data Bank (PDB) have a range of regional (localized) quality issues, be it derived from X-ray crystallography, Nuclear Magnetic Resonance (NMR) spectroscopy, or other experimental approaches. However, most PDB entries are judged by global quality metrics like R-factor, R-free, and resolution for X-ray crystallography or backbone phi-psi distribution statistics and average restraint violations for NMR. Regional quality is often ignored when PDB entries are re-used for a variety of structurally based analyses. The binding of ligands, especially ligands involved in energy metabolism, is of particular interest in many structurally focused protein studies. Using a regional quality metric that provides chemically interpretable information from electron density maps, a significant number of outliers in regional structural quality was detected across X-ray crystallographic PDB entries for proteins bound to biochemically critical ligands. In this study, a series of analyses was performed to evaluate both specific and general potential factors that could promote these outliers. In particular, these potential factors were the minimum distance to a metal ion, the minimum distance to a crystal contact, and the isotropic atomic b-factor. To evaluate these potential factors, Fisher’s exact tests were performed, using regional quality criteria of outlier (top 1%, 2.5%, 5%, or 10%) versus non-outlier compared to a potential factor metric above versus below a certain outlier cutoff. The results revealed a consistent general effect from region-specific normalized b-factors but no specific effect from metal ion contact distances and only a very weak effect from crystal contact distance as compared to the b-factor results. These findings indicate that no single specific potential factor explains a majority of the outlier ligand-bound regions, implying that human error is likely as important as these other factors. Thus, all factors, including human error, should be considered when regions of low structural quality are detected. Also, the downstream re-use of protein structures for studying ligand-bound conformations should screen the regional quality of the binding sites. Doing so prevents misinterpretation due to the presence of structural uncertainty or flaws in regions of interest.

Keywords: Biomacromolecular structure, coenzyme, electron density discrepancy analysis, X-ray crystallography.

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Authors: Khunsa Fatima, Umar K. Khattak, Allah Bakhsh Kausar

Abstract:

Satellite images interpretation and analysis assist geologists by providing valuable information about geology and minerals of an area to be surveyed. A test site in Fatejang of district Attock has been studied using Landsat ETM+ and ASTER satellite images for lithological mapping. Five different supervised image classification techniques namely maximum likelihood, parallelepiped, minimum distance to mean, mahalanobis distance and spectral angle mapper have been performed upon both satellite data images to find out the suitable classification technique for lithological mapping in the study area. Results of these five image classification techniques were compared with the geological map produced by Geological Survey of Pakistan. Result of maximum likelihood classification technique applied on ASTER satellite image has highest correlation of 0.66 with the geological map. Field observations and XRD spectra of field samples also verified the results. A lithological map was then prepared based on the maximum likelihood classification of ASTER satellite image.

Keywords: ASTER, Landsat-ETM+, Satellite, Image classification.

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Authors: S. Dorbani, M. Badaoui, D. Benouar

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The goal of the paper is to highlight the effect of the building design and epicentral distance on the storey lateral displacements, for several reinforced concrete buildings (6, 9 and 12 stories). These structures are subjected to seismic accelerations from the Boumerdes earthquake (Algeria, May 21st, Mw = 6.8). Using the response spectrum method (modal spectral approach), the analysis is performed in both longitudinal and transverse directions. The building design is expressed through the fundamental period and epicentral distance is used to represent the earthquake effect variation on storey lateral displacements and interstory drift for the considered buildings.

Keywords: Epicentral distance, interstory drift, lateral displacement, natural period, reinforced concrete buildings.

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Authors: A.A.Andaleeb, Rozhan.M.Idrus, Issham Ismail, A.K. Mokaram

Abstract:

This paper reports the findings of a research conducted to evaluate the ownership and usage of technology devices within Distance Education students- according to their age. This research involved 45 Distance Education students from USM Universiti Sains Malaysia (DEUSM) as its respondents. Data was collected through questionnaire that had been developed by the researchers based on some literature review. The data was analyzed to find out the frequencies of respondents agreements towards ownership of technology devices and the use of technology devices. The findings shows that all respondents own mobile phone and majority of them reveal that they use mobile on regular basis. The student in the age 30-39 has the heist ownership of the technology devices.

Keywords: technology devices, mobile phone, distance learners, techno readiness Index, Age

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Authors: Loke Kean Koay, Mani Maran Ratnam

Abstract:

A mechanically-resonant torsional spring scanner was developed in a recent study. Various methods were developed to improve the angular displacement of the scanner while maintaining the scanner frequency. However the effects of rotor magnet radial position on scanner characteristics were not well investigated. In this study, the relationships between the magnet position and the scanner characteristics such as natural frequency, angular displacement and stress level were studied. A finite element model was created and an average deviation of 3.18% was found between the simulation and experimental results, qualifying the simulation results as a guide for further investigations. Three magnet positions on the transverse oscillating suspended plate were investigated by finite element analysis (FEA) and one of the positions were selected as the design position. The magnet position with the longest distance from the twist axis of mirror was selected since it attains minimum stress level, while exceeding the minimum critical flicker frequency and delivering the targeted angular displacement to the scanner.

Keywords: Computer-aided design, design optimization, torsional scanner.

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Authors: S. A. Mohamed, A. S. Zayed, O. A. Abolaeha

Abstract:

A new Feed-Forward/Feedback Generalized Minimum Variance Pole-placement Controller to incorporate the robustness of classical pole-placement into the flexibility of generalized minimum variance self-tuning controller for Single-Input Single-Output (SISO) has been proposed in this paper. The design, which provides the user with an adaptive mechanism, which ensures that the closed loop poles are, located at their pre-specified positions. In addition, the controller design which has a feed-forward/feedback structure overcomes the certain limitations existing in similar poleplacement control designs whilst retaining the simplicity of adaptation mechanisms used in other designs. It tracks set-point changes with the desired speed of response, penalizes excessive control action, and can be applied to non-minimum phase systems. Besides, at steady state, the controller has the ability to regulate the constant load disturbance to zero. Example simulation results using both simulated and real plant models demonstrate the effectiveness of the proposed controller.

Keywords: Pole-placement, Minimum variance control, self-tuning control and feedforward control.

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Authors: Narendra Singh Yadav, R.P.Yadav

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Mobile ad hoc network is a collection of mobile nodes communicating through wireless channels without any existing network infrastructure or centralized administration. Because of the limited transmission range of wireless network interfaces, multiple "hops" may be needed to exchange data across the network. In order to facilitate communication within the network, a routing protocol is used to discover routes between nodes. The primary goal of such an ad hoc network routing protocol is correct and efficient route establishment between a pair of nodes so that messages may be delivered in a timely manner. Route construction should be done with a minimum of overhead and bandwidth consumption. This paper examines two routing protocols for mobile ad hoc networks– the Destination Sequenced Distance Vector (DSDV), the table- driven protocol and the Ad hoc On- Demand Distance Vector routing (AODV), an On –Demand protocol and evaluates both protocols based on packet delivery fraction, normalized routing load, average delay and throughput while varying number of nodes, speed and pause time.

Keywords: AODV, DSDV, MANET, relative performance

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Authors: Tawiwat V., Tosapolporn P., Kedit J.

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This research proposes the state of art on how to control or find the trajectory paths of the RRP robot when the prismatic joint is malfunction. According to this situation, the minimum energy of the dynamic optimization is applied. The RRP robot or similar systems have been used in many areas such as fire fighter truck, laboratory equipment and military truck for example a rocket launcher. In order to keep on task that assigned, the trajectory paths must be computed. Here, the open loop control is applied and the result of an example show the reasonable solution which can be applied to the controllable system.

Keywords: RRP robot, Optimal Control, Minimum Energy and Under Actuator.

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Authors: Panhathai Buasri

Abstract:

Hydrogen that used as fuel in fuel cell vehicles can be produced from renewable sources such as wind, solar, and hydro technologies. PV-electrolyzer is one of the promising methods to produce hydrogen with zero pollution emission. Hydrogen production from a PV-electrolyzer system depends on the efficiency of the electrolyzer and photovoltaic array, and sun irradiance at that site. In this study, the amount of hydrogen is obtained using mathematical equations for difference driving distance and sun peak hours. The results show that the minimum of 99 PV modules are used to generate 1.75 kgH2 per day for two vehicles.

Keywords: About four key words or phrases in alphabetical order, separated by commas.

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Authors: Minghui Wang

Abstract:

Based on the classical algorithm LSQR for solving (unconstrained) LS problem, an iterative method is proposed for the least-squares like-minimum-norm symmetric solution of AXB+CYD=E. As the application of this algorithm, an iterative method for the least-squares like-minimum-norm biymmetric solution of AXB=E is also obtained. Numerical results are reported that show the efficiency of the proposed methods.

Keywords: Matrix equation, bisymmetric matrix, least squares problem, like-minimum norm, iterative algorithm.

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Authors: Seong-Mo An, Rohit Ramesh, Young-Sook Lee, Wan-Young Chung

Abstract:

The proper assessment of interaxial distance and convergence control are important factors in stereoscopic imaging technology to make an efficient 3D image. To control interaxial distance and convergence for efficient 3D shooting, horizontal 3D camera rig is designed using some hardware components like 'LM Guide', 'Goniometer' and 'Rotation Stage'. The horizontal 3D camera rig system can be properly aligned by moving the two cameras horizontally in same or opposite directions, by adjusting the camera angle and finally considering horizontal swing as well as vertical swing. In this paper, the relationship between interaxial distance and convergence angle control are discussed and intensive experiments are performed in order to demonstrate an easy and effective 3D shooting.

Keywords: Interaxial, Convergence, Stereoscopic, Horizontal 3D Camera Rig

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Authors: Tarasankar Pramanik, Sukumar Mondal, Madhumangal Pal

Abstract:

In an interval graph G = (V,E) the distance between two vertices u, v is de£ned as the smallest number of edges in a path joining u and v. The eccentricity of a vertex v is the maximum among distances from all other vertices of V . The diameter (δ) and radius (ρ) of the graph G is respectively the maximum and minimum among all the eccentricities of G. The center of the graph G is the set C(G) of vertices with eccentricity ρ. In this context our aim is to establish the relation ρ = δ 2  for an interval graph and to determine the center of it.

Keywords: Interval graph, interval tree, radius, center.

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Authors: Abdussamad U. Jibia, Momoh-Jimoh E. Salami

Abstract:

Eigenvector methods are gaining increasing acceptance in the area of spectrum estimation. This paper presents a successful attempt at testing and evaluating the performance of two of the most popular types of subspace techniques in determining the parameters of multiexponential signals with real decay constants buried in noise. In particular, MUSIC (Multiple Signal Classification) and minimum-norm techniques are examined. It is shown that these methods perform almost equally well on multiexponential signals with MUSIC displaying better defined peaks.

Keywords: Eigenvector, minimum norm, multiexponential, subspace.

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Authors: Yiqiong Yuan, Jun Sun, Dongmei Zhou, Jianan Sun

Abstract:

In this paper, we presented a Multi-Objective Random Drift Particle Swarm Optimization algorithm (MORDPSO-CD) based on RDPSO and crowding distance sorting to improve the convergence and distribution with less computation cost. MORDPSO-CD makes the most of RDPSO to approach the true Pareto optimal solutions fast. We adopt the crowding distance sorting technique to update and maintain the archived optimal solutions. Introducing the crowding distance technique into MORDPSO can make the leader particles find the true Pareto solution ultimately. The simulation results reveal that the proposed algorithm has better convergence and distribution.

Keywords: Multi-objective optimization, random drift particle swarm optimization, crowding distance, Pareto optimal solution.

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Authors: Obeidat I., Bsoul M., Khasawneh A., Kilani Y.

Abstract:

This paper simulates the ad-hoc mesh network in rural areas, where such networks receive great attention due to their cost, since installing the infrastructure for regular networks in these areas is not possible due to the high cost. The distance between the communicating nodes is the most obstacles that the ad-hoc mesh network will face. For example, in Terranet technology, two nodes can communicate if they are only one kilometer far from each other. However, if the distance between them is more than one kilometer, then each node in the ad-hoc mesh networks has to act as a router that forwards the data it receives to other nodes. In this paper, we try to find the critical number of nodes which makes the network fully connected in a particular area, and then propose a method to enhance the intermediate node to accept to be a router to forward the data from the sender to the receiver. Much work was done on technological changes on peer to peer networks, but the focus of this paper will be on another feature which is to find the minimum number of nodes needed for a particular area to be fully connected and then to enhance the users to switch on their phones and accept to work as a router for other nodes. Our method raises the successful calls to 81.5% out of 100% attempt calls.

Keywords: Adjacency matrix, Ad-hoc mesh network, Connectedness, Terranet technology

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Authors: Mohammad Shoyaib, M. Abdullah-Al-Wadud, Oksam Chae

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In this paper an effective approach for segmenting human skin regions in images taken at different environment is proposed. The proposed method uses a color distance map that is flexible enough to reliably detect the skin regions even if the illumination conditions of the image vary. Local image conditions is also focused, which help the technique to adaptively detect differently illuminated skin regions of an image. Moreover, usage of local information also helps the skin detection process to get rid of picking up much noisy pixels.

Keywords: Color Distance map, Reference skin color, Regiongrowing, Skin segmentation.

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Authors: S. Balaji, V. Swaminathan, K. Kannan

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The Minimum Vertex Cover (MVC) problem is a classic graph optimization NP - complete problem. In this paper a competent algorithm, called Vertex Support Algorithm (VSA), is designed to find the smallest vertex cover of a graph. The VSA is tested on a large number of random graphs and DIMACS benchmark graphs. Comparative study of this algorithm with the other existing methods has been carried out. Extensive simulation results show that the VSA can yield better solutions than other existing algorithms found in the literature for solving the minimum vertex cover problem.

Keywords: vertex cover, vertex support, approximation algorithms, NP - complete problem.

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