Commenced in January 2007
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Edition: International
Paper Count: 2650

Search results for: POI

40 Using Interval Trees for Approximate Indexing of Instances

Authors: Khalil el Hindi

Abstract:

This paper presents a simple and effective method for approximate indexing of instances for instance based learning. The method uses an interval tree to determine a good starting search point for the nearest neighbor. The search stops when an early stopping criterion is met. The method proved to be very effective especially when only the first nearest neighbor is required.

Keywords: Instance based learning, interval trees, the knn algorithm, machine learning.

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39 Locating Center Points for Radial Basis Function Networks Using Instance Reduction Techniques

Authors: Rana Yousef, Khalil el Hindi

Abstract:

The behavior of Radial Basis Function (RBF) Networks greatly depends on how the center points of the basis functions are selected. In this work we investigate the use of instance reduction techniques, originally developed to reduce the storage requirements of instance based learners, for this purpose. Five Instance-Based Reduction Techniques were used to determine the set of center points, and RBF networks were trained using these sets of centers. The performance of the RBF networks is studied in terms of classification accuracy and training time. The results obtained were compared with two Radial Basis Function Networks: RBF networks that use all instances of the training set as center points (RBF-ALL) and Probabilistic Neural Networks (PNN). The former achieves high classification accuracies and the latter requires smaller training time. Results showed that RBF networks trained using sets of centers located by noise-filtering techniques (ALLKNN and ENN) rather than pure reduction techniques produce the best results in terms of classification accuracy. The results show that these networks require smaller training time than that of RBF-ALL and higher classification accuracy than that of PNN. Thus, using ALLKNN and ENN to select center points gives better combination of classification accuracy and training time. Our experiments also show that using the reduced sets to train the networks is beneficial especially in the presence of noise in the original training sets.

Keywords: Radial basis function networks, Instance-based reduction, PNN.

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38 Economy-Based Computing with WebCom

Authors: Adarsh Patil, David A. Power, John P. Morrison

Abstract:

Grid environments consist of the volatile integration of discrete heterogeneous resources. The notion of the Grid is to unite different users and organisations and pool their resources into one large computing platform where they can harness, inter-operate, collaborate and interact. If the Grid Community is to achieve this objective, then participants (Users and Organisations) need to be willing to donate or share their resources and permit other participants to use their resources. Resources do not have to be shared at all times, since it may result in users not having access to their own resource. The idea of reward-based computing was developed to address the sharing problem in a pragmatic manner. Participants are offered a reward to donate their resources to the Grid. A reward may include monetary recompense or a pro rata share of available resources when constrained. This latter point may imply a quality of service, which in turn may require some globally agreed reservation mechanism. This paper presents a platform for economybased computing using the WebCom Grid middleware. Using this middleware, participants can configure their resources at times and priority levels to suit their local usage policy. The WebCom system accounts for processing done on individual participants- resources and rewards them accordingly.

Keywords: WebCom, Economy-based computing, WebComGrid Bank Reward, Condensed Graph, Distributor, Accounting, GridPoint.

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37 A Novel Arabic Text Steganography Method Using Letter Points and Extensions

Authors: Adnan Abdul-Aziz Gutub, Manal Mohammad Fattani

Abstract:

This paper presents a new steganography approach suitable for Arabic texts. It can be classified under steganography feature coding methods. The approach hides secret information bits within the letters benefiting from their inherited points. To note the specific letters holding secret bits, the scheme considers the two features, the existence of the points in the letters and the redundant Arabic extension character. We use the pointed letters with extension to hold the secret bit 'one' and the un-pointed letters with extension to hold 'zero'. This steganography technique is found attractive to other languages having similar texts to Arabic such as Persian and Urdu.

Keywords: Arabic text, Cryptography, Feature coding, Information security, Text steganography, Text watermarking.

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36 Architectural, Technological and Performance Issues in Enterprise Applications

Authors: Melek Oktay, Ayşe Betül Gülbağcı, Mustafa Sarıöz

Abstract:

Enterprise applications are complex systems that are hard to develop and deploy in organizations. Although software application development tools, frameworks, methodologies and patterns are rapidly developing; many projects fail by causing big costs. There are challenging issues that programmers and designers face with while working on enterprise applications. In this paper, we present the three of the significant issues: Architectural, technological and performance. The important subjects in each issue are pointed out and recommendations are given. In architectural issues the lifecycle, meta-architecture, guidelines are pointed out. .NET and Java EE platforms are presented in technological issues. The importance of performance, measuring performance and profilers are explained in performance issues.

Keywords: Enterprise Applications, Architecture, Technology, Performance.

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35 Image Search by Features of Sorted Gray level Histogram Polynomial Curve

Authors: Awais Adnan, Muhammad Ali, Amir Hanif Dar

Abstract:

Image Searching was always a problem specially when these images are not properly managed or these are distributed over different locations. Currently different techniques are used for image search. On one end, more features of the image are captured and stored to get better results. Storing and management of such features is itself a time consuming job. While on the other extreme if fewer features are stored the accuracy rate is not satisfactory. Same image stored with different visual properties can further reduce the rate of accuracy. In this paper we present a new concept of using polynomials of sorted histogram of the image. This approach need less overhead and can cope with the difference in visual features of image.

Keywords: Sorted Histogram, Polynomial Curves, feature pointsof images, Grayscale, visual properties of image.

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34 Self-Adaptive Differential Evolution Based Power Economic Dispatch of Generators with Valve-Point Effects and Multiple Fuel Options

Authors: R.Balamurugan, S.Subramanian

Abstract:

This paper presents the solution of power economic dispatch (PED) problem of generating units with valve point effects and multiple fuel options using Self-Adaptive Differential Evolution (SDE) algorithm. The global optimal solution by mathematical approaches becomes difficult for the realistic PED problem in power systems. The Differential Evolution (DE) algorithm is found to be a powerful evolutionary algorithm for global optimization in many real problems. In this paper the key parameters of control in DE algorithm such as the crossover constant CR and weight applied to random differential F are self-adapted. The PED problem formulation takes into consideration of nonsmooth fuel cost function due to valve point effects and multi fuel options of generator. The proposed approach has been examined and tested with the numerical results of PED problems with thirteen-generation units including valve-point effects, ten-generation units with multiple fuel options neglecting valve-point effects and ten-generation units including valve-point effects and multiple fuel options. The test results are promising and show the effectiveness of proposed approach for solving PED problems.

Keywords: Multiple fuels, power economic dispatch, selfadaptivedifferential evolution and valve-point effects.

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33 Strip Decomposition Parallelization of Fast Direct Poisson Solver on a 3D Cartesian Staggered Grid

Authors: Minh Vuong Pham, Frédéric Plourde, Son Doan Kim

Abstract:

A strip domain decomposition parallel algorithm for fast direct Poisson solver is presented on a 3D Cartesian staggered grid. The parallel algorithm follows the principles of sequential algorithm for fast direct Poisson solver. Both Dirichlet and Neumann boundary conditions are addressed. Several test cases are likewise addressed in order to shed light on accuracy and efficiency in the strip domain parallelization algorithm. Actually the current implementation shows a very high efficiency when dealing with a large grid mesh up to 3.6 * 109 under massive parallel approach, which explicitly demonstrates that the proposed algorithm is ready for massive parallel computing.

Keywords: Strip-decomposition, parallelization, fast directpoisson solver.

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32 A Comparative Analysis of Activity-Based Costing and Traditional Costing

Authors: Derya Eren Akyol, Gonca Tuncel, G. Mirac Bayhan

Abstract:

Activity-Based Costing (ABC) which has become an important aspect of manufacturing/service organizations can be defined as a methodology that measures the cost and performance of activities, resources and cost objects. It can be considered as an alternative paradigm to traditional cost-based accounting systems. The objective of this paper is to illustrate an application of ABC method and to compare the results of ABC with traditional costing methods. The results of the application highlight the weak points of traditional costing methods and an S-Curve obtained is used to identify the undercosted and overcosted products of the firm.

Keywords: Activity-based costing, cost drivers, overheads, traditional costing.

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31 A Reconfigurable Processing Element Implementation for Matrix Inversion Using Cholesky Decomposition

Authors: Aki Happonen, Adrian Burian, Erwin Hemming

Abstract:

Fixed-point simulation results are used for the performance measure of inverting matrices using a reconfigurable processing element. Matrices are inverted using the Cholesky decomposition algorithm. The reconfigurable processing element is capable of all required mathematical operations. The fixed-point word length analysis is based on simulations of different condition numbers and different matrix sizes.

Keywords: Cholesky Decomposition, Fixed-point, Matrixinversion, Reconfigurable processing.

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30 A New Pattern for Handwritten Persian/Arabic Digit Recognition

Authors: A. Harifi, A. Aghagolzadeh

Abstract:

The main problem for recognition of handwritten Persian digits using Neural Network is to extract an appropriate feature vector from image matrix. In this research an asymmetrical segmentation pattern is proposed to obtain the feature vector. This pattern can be adjusted as an optimum model thanks to its one degree of freedom as a control point. Since any chosen algorithm depends on digit identity, a Neural Network is used to prevail over this dependence. Inputs of this Network are the moment of inertia and the center of gravity which do not depend on digit identity. Recognizing the digit is carried out using another Neural Network. Simulation results indicate the high recognition rate of 97.6% for new introduced pattern in comparison to the previous models for recognition of digits.

Keywords: Pattern recognition, Persian digits, NeuralNetwork.

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29 Development of Effective Cooling Schemes of Gas Turbine Blades Based on Computer Simulation

Authors: Pasayev, C. Askerov, R. Sadiqov, C. Ardil

Abstract:

In contrast to existing of calculation of temperature field of a profile part a blade with convective cooling which are not taking into account multi connective in a broad sense of this term, we develop mathematical models and highly effective combination (BIEM AND FDM) numerical methods from the point of view of a realization on the PC. The theoretical substantiation of these methods is proved by the appropriate theorems.

Keywords: multi coherent systems, method of the boundary integrated equations, singular operators, gas turbines

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28 BEM Formulations Based on Kirchhoffs Hypoyhesis to Perform Linear Bending Analysis of Plates Reinforced by Beams

Authors: Gabriela R. Fernandes, Renato F. Denadai, Guido J. Denipotti

Abstract:

In this work, are discussed two formulations of the boundary element method - BEM to perform linear bending analysis of plates reinforced by beams. Both formulations are based on the Kirchhoff's hypothesis and they are obtained from the reciprocity theorem applied to zoned plates, where each sub-region defines a beam or a slab. In the first model the problem values are defined along the interfaces and the external boundary. Then, in order to reduce the number of degrees of freedom kinematics hypothesis are assumed along the beam cross section, leading to a second formulation where the collocation points are defined along the beam skeleton, instead of being placed on interfaces. On these formulations no approximation of the generalized forces along the interface is required. Moreover, compatibility and equilibrium conditions along the interface are automatically imposed by the integral equation. Thus, these formulations require less approximation and the total number of the degree s of freedom is reduced. In the numerical examples are discussed the differences between these two BEM formulations, comparing as well the results to a well-known finite element code.

Keywords: Boundary elements, Building floor structures, Platebending.

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27 Robotic Hands: Design Review and Proposal of New Design Process

Authors: Jimmy W. Soto Martell, Giuseppina Gini

Abstract:

In this paper we intend to ascertain the state of the art on multifingered end-effectors, also known as robotic hands or dexterous robot hands, and propose an experimental setup for an innovative task based design approach, involving cutting edge technologies in motion capture. After an initial description of the capabilities and complexity of a human hand when grasping objects, in order to point out the importance of replicating it, we analyze the mechanical and kinematical structure of some important works carried out all around the world in the last three decades and also review the actuators and sensing technologies used. Finally we describe a new design philosophy proposing an experimental setup for the first stage using recent developments in human body motion capture systems that might lead to lighter and always more dexterous robotic hands.

Keywords: Dexterous manipulation, grasp, multifingered endeffector, robotic hand.

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26 Predictive Model of Sensor Readings for a Mobile Robot

Authors: Krzysztof Fujarewicz

Abstract:

This paper presents a predictive model of sensor readings for mobile robot. The model predicts sensor readings for given time horizon based on current sensor readings and velocities of wheels assumed for this horizon. Similar models for such anticipation have been proposed in the literature. The novelty of the model presented in the paper comes from the fact that its structure takes into account physical phenomena and is not just a black box, for example a neural network. From this point of view it may be regarded as a semi-phenomenological model. The model is developed for the Khepera robot, but after certain modifications, it may be applied for any robot with distance sensors such as infrared or ultrasonic sensors.

Keywords: Mobile robot, sensors, prediction, anticipation.

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25 A Method for Controlling of Hand Prosthesis Based on Neural Network

Authors: Fereidoun Nowshiravan Rahatabad, Mohammad Ali Nekoui, Mohammad Reza Hashemi Golpaygani, AliFallah, Mehdi Kazemzadeh Narbat

Abstract:

The people are differed by their capabilities, skills and mental agilities. The evolution of human from childhood when they are completely dependent up to adultness the time they gradually set the dependency free is too complicated, by considering they have all started from almost one point but some become cleverer and some less. The main control command of a cybernetic hand should be posted by remaining healthy organs of disabled Person. These commands can be from several channels, which their recording and detecting are different and need complicated study. In this research, we suppose that, this stage has been done or in the other words, the command has been already sent and detected. So the main goal is to control a long hand, upper elbow hand missing, by an interest angle define by disabled. It means that, the system input is the position desired by disables and the output is the elbow-joint angle variation. Therefore the goal is a suitable control design based on neural network theory in order to meet the given mapping.

Keywords: Control - system design, Upper limb prosthesis, neuralnetwork.

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24 Towards an Extended SQLf: Bipolar Query Language with Preferences

Authors: L. Ludovic, R. Daniel, S-E Tbahriti

Abstract:

Database management systems that integrate user preferences promise better solution for personalization, greater flexibility and higher quality of query responses. This paper presents a tentative work that studies and investigates approaches to express user preferences in queries. We sketch an extend capabilities of SQLf language that uses the fuzzy set theory in order to define the user preferences. For that, two essential points are considered: the first concerns the expression of user preferences in SQLf by so-called fuzzy commensurable predicates set. The second concerns the bipolar way in which these user preferences are expressed on mandatory and/or optional preferences.

Keywords: Flexible query language, relational database, userpreference.

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23 Encryption Efficiency Analysis and Security Evaluation of RC6 Block Cipher for Digital Images

Authors: Hossam El-din H. Ahmed, Hamdy M. Kalash, Osama S. Farag Allah

Abstract:

This paper investigates the encryption efficiency of RC6 block cipher application to digital images, providing a new mathematical measure for encryption efficiency, which we will call the encryption quality instead of visual inspection, The encryption quality of RC6 block cipher is investigated among its several design parameters such as word size, number of rounds, and secret key length and the optimal choices for the best values of such design parameters are given. Also, the security analysis of RC6 block cipher for digital images is investigated from strict cryptographic viewpoint. The security estimations of RC6 block cipher for digital images against brute-force, statistical, and differential attacks are explored. Experiments are made to test the security of RC6 block cipher for digital images against all aforementioned types of attacks. Experiments and results verify and prove that RC6 block cipher is highly secure for real-time image encryption from cryptographic viewpoint. Thorough experimental tests are carried out with detailed analysis, demonstrating the high security of RC6 block cipher algorithm. So, RC6 block cipher can be considered to be a real-time secure symmetric encryption for digital images.

Keywords: Block cipher, Image encryption, Encryption quality, and Security analysis.

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22 Compromise Ratio Method for Decision Making under Fuzzy Environment using Fuzzy Distance Measure

Authors: Debashree Guha, Debjani Chakraborty

Abstract:

The aim of this paper is to adopt a compromise ratio (CR) methodology for fuzzy multi-attribute single-expert decision making proble. In this paper, the rating of each alternative has been described by linguistic terms, which can be expressed as triangular fuzzy numbers. The compromise ratio method for fuzzy multi-attribute single expert decision making has been considered here by taking the ranking index based on the concept that the chosen alternative should be as close as possible to the ideal solution and as far away as possible from the negative-ideal solution simultaneously. From logical point of view, the distance between two triangular fuzzy numbers also is a fuzzy number, not a crisp value. Therefore a fuzzy distance measure, which is itself a fuzzy number, has been used here to calculate the difference between two triangular fuzzy numbers. Now in this paper, with the help of this fuzzy distance measure, it has been shown that the compromise ratio is a fuzzy number and this eases the problem of the decision maker to take the decision. The computation principle and the procedure of the compromise ratio method have been described in detail in this paper. A comparative analysis of the compromise ratio method previously proposed [1] and the newly adopted method have been illustrated with two numerical examples.

Keywords: Compromise ratio method, Fuzzy multi-attributesingle-expert decision making, Fuzzy number, Linguistic variable

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21 Order Reduction by Least-Squares Methods about General Point ''a''

Authors: Integral square error, Least-squares, Markovparameters, Moment matching, Order reduction.

Abstract:

The concept of order reduction by least-squares moment matching and generalised least-squares methods has been extended about a general point ?a?, to obtain the reduced order models for linear, time-invariant dynamic systems. Some heuristic criteria have been employed for selecting the linear shift point ?a?, based upon the means (arithmetic, harmonic and geometric) of real parts of the poles of high order system. It is shown that the resultant model depends critically on the choice of linear shift point ?a?. The validity of the criteria is illustrated by solving a numerical example and the results are compared with the other existing techniques.

Keywords: Integral square error, Least-squares, Markovparameters, Moment matching, Order reduction.

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20 Multiple Job Shop-Scheduling using Hybrid Heuristic Algorithm

Authors: R.A.Mahdavinejad

Abstract:

In this paper, multi-processors job shop scheduling problems are solved by a heuristic algorithm based on the hybrid of priority dispatching rules according to an ant colony optimization algorithm. The objective function is to minimize the makespan, i.e. total completion time, in which a simultanous presence of various kinds of ferons is allowed. By using the suitable hybrid of priority dispatching rules, the process of finding the best solution will be improved. Ant colony optimization algorithm, not only promote the ability of this proposed algorithm, but also decreases the total working time because of decreasing in setup times and modifying the working production line. Thus, the similar work has the same production lines. Other advantage of this algorithm is that the similar machines (not the same) can be considered. So, these machines are able to process a job with different processing and setup times. According to this capability and from this algorithm evaluation point of view, a number of test problems are solved and the associated results are analyzed. The results show a significant decrease in throughput time. It also shows that, this algorithm is able to recognize the bottleneck machine and to schedule jobs in an efficient way.

Keywords: Job shops scheduling, Priority dispatching rules, Makespan, Hybrid heuristic algorithm.

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19 Automated Feature Points Management for Video Mosaic Construction

Authors: Jing Li, Quan Pan, Stan. Z. Li, Tao Yang

Abstract:

A novel algorithm for construct a seamless video mosaic of the entire panorama continuously by automatically analyzing and managing feature points, including management of quantity and quality, from the sequence is presented. Since a video contains significant redundancy, so that not all consecutive video images are required to create a mosaic. Only some key images need to be selected. Meanwhile, feature-based methods for mosaicing rely on correction of feature points? correspondence deeply, and if the key images have large frame interval, the mosaic will often be interrupted by the scarcity of corresponding feature points. A unique character of the method is its ability to handle all the problems above in video mosaicing. Experiments have been performed under various conditions, the results show that our method could achieve fast and accurate video mosaic construction. Keywords?video mosaic, feature points management, homography estimation.

Keywords: Video mosaic, feature points management, homography estimation.

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18 Simulation Tools for Fixed Point DSP Algorithms and Architectures

Authors: K. B. Cullen, G. C. M. Silvestre, N. J. Hurley

Abstract:

This paper presents software tools that convert the C/Cµ floating point source code for a DSP algorithm into a fixedpoint simulation model that can be used to evaluate the numericalperformance of the algorithm on several different fixed pointplatforms including microprocessors, DSPs and FPGAs. The tools use a novel system for maintaining binary point informationso that the conversion from floating point to fixed point isautomated and the resulting fixed point algorithm achieves maximum possible precision. A configurable architecture is used during the simulation phase so that the algorithm can produce a bit-exact output for several different target devices.

Keywords: DSP devices, DSP algorithm, simulation model, software

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17 High-Speed Pipeline Implementation of Radix-2 DIF Algorithm

Authors: Christos Meletis, Paul Bougas, George Economakos , Paraskevas Kalivas, Kiamal Pekmestzi

Abstract:

In this paper, we propose a new architecture for the implementation of the N-point Fast Fourier Transform (FFT), based on the Radix-2 Decimation in Frequency algorithm. This architecture is based on a pipeline circuit that can process a stream of samples and produce two FFT transform samples every clock cycle. Compared to existing implementations the architecture proposed achieves double processing speed using the same circuit complexity.

Keywords: Digital signal processing, systolic circuits, FFTalgorithm.

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16 Bridging the Mental Gap between Convolution Approach and Compartmental Modeling in Functional Imaging: Typical Embedding of an Open Two-Compartment Model into the Systems Theory Approach of Indicator Dilution Theory

Authors: Gesine Hellwig

Abstract:

Functional imaging procedures for the non-invasive assessment of tissue microcirculation are highly requested, but require a mathematical approach describing the trans- and intercapillary passage of tracer particles. Up to now, two theoretical, for the moment different concepts have been established for tracer kinetic modeling of contrast agent transport in tissues: pharmacokinetic compartment models, which are usually written as coupled differential equations, and the indicator dilution theory, which can be generalized in accordance with the theory of lineartime- invariant (LTI) systems by using a convolution approach. Based on mathematical considerations, it can be shown that also in the case of an open two-compartment model well-known from functional imaging, the concentration-time course in tissue is given by a convolution, which allows a separation of the arterial input function from a system function being the impulse response function, summarizing the available information on tissue microcirculation. Due to this reason, it is possible to integrate the open two-compartment model into the system-theoretic concept of indicator dilution theory (IDT) and thus results known from IDT remain valid for the compartment approach. According to the long number of applications of compartmental analysis, even for a more general context similar solutions of the so-called forward problem can already be found in the extensively available appropriate literature of the seventies and early eighties. Nevertheless, to this day, within the field of biomedical imaging – not from the mathematical point of view – there seems to be a trench between both approaches, which the author would like to get over by exemplary analysis of the well-known model.

Keywords: Functional imaging, Tracer kinetic modeling, LTIsystem, Indicator dilution theory / convolution approach, Two-Compartment model.

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15 Trajectory Planning Design Equations and Control of a 4 - axes Stationary Robotic Arm

Authors: T.C. Manjunath,

Abstract:

This paper features the trajectory planning design of a indigenously developed 4-Axis SCARA robot which is used for doing successful robotic manipulation task in the laboratory. Once, a trajectory is being designed and given as input to the robot, the robot's gripper tip moves along that specified trajectory. Trajectories have to be designed in the work space only. The main idea of this paper is to design a continuous path trajectory model for the indigenously developed SCARA robot arm during its maneuvering from one point to another point (during pick and place operations) in a workspace avoiding all the obstacles in its path of motion.

Keywords: SCARA, Trajectory, Planning.

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14 Frictionless Contact Problem Between Two Orthotropic Elastic Layers

Authors: V. Kahya, A. Birinci, R. Erdol

Abstract:

A frictionless contact problem for a two-layer orthotropic elastic medium loaded through a rigid flat stamp is considered. It is assumed that tensile tractions are not allowed and only compressive tractions can be transmitted across the interface. In the solution, effect of gravity is taken into consideration. If the external load on the rigid stamp is less than or equal to a critical value, continuous contact between the layers is maintained. The problem is expressed in terms of a singular integral equation by using the theory of elasticity and the Fourier transforms. Numerical results for initial separation point, critical separation load and contact stress distribution are presented.

Keywords: Frictionless contact, Initial separation, Orthotropicmaterial, Singular integral equation.

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13 The Number of Rational Points on Conics Cp,k : x2 − ky2 = 1 over Finite Fields Fp

Authors: Ahmet Tekcan

Abstract:

Let p be a prime number, Fp be a finite field, and let k ∈ F*p. In this paper, we consider the number of rational points onconics Cp,k: x2 − ky2 = 1 over Fp. We proved that the order of Cp,k over Fp is p-1 if k is a quadratic residue mod p and is p + 1 if k is not a quadratic residue mod p. Later we derive some resultsconcerning the sums ΣC[x]p,k(Fp) and ΣC[y]p,k(Fp), the sum of x- and y-coordinates of all points (x, y) on Cp,k, respectively.

Keywords: Elliptic curve, conic, rational points.

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12 The Elliptic Curves y2 = x3 - t2x over Fp

Authors: Ahmet Tekcan

Abstract:

Let p be a prime number, Fp be a finite field and t ∈ F*p= Fp- {0}. In this paper we obtain some properties of ellipticcurves Ep,t: y2= y2= x3- t2x over Fp. In the first sectionwe give some notations and preliminaries from elliptic curves. In the second section we consider the rational points (x, y) on Ep,t. Wegive a formula for the number of rational points on Ep,t over Fnp for an integer n ≥ 1. We also give some formulas for the sum of x?andy?coordinates of the points (x, y) on Ep,t. In the third section weconsider the rank of Et: y2= x3- t2x and its 2-isogenous curve Et over Q. We proved that the rank of Etand Etis 2 over Q. In the last section we obtain some formulas for the sums Σt∈F?panp,t for an integer n ≥ 1, where ap,t denote the trace of Frobenius.

Keywords: Elliptic curves over finite fields, rational points onelliptic curves, rank, trace of Frobenius.

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11 The Number of Rational Points on Elliptic Curves y2 = x3 + b2 Over Finite Fields

Authors: Betül Gezer, Hacer Özden, Ahmet Tekcan, Osman Bizim

Abstract:

Let p be a prime number, Fpbe a finite field and let Qpdenote the set of quadratic residues in Fp. In the first section we givesome notations and preliminaries from elliptic curves. In the secondsection, we consider some properties of rational points on ellipticcurves Ep,b: y2= x3+ b2 over Fp, where b ∈ F*p. Recall that theorder of Ep,bover Fpis p + 1 if p ≡ 5(mod 6). We generalize thisresult to any field Fnp for an integer n≥ 2. Further we obtain someresults concerning the sum Σ[x]Ep,b(Fp) and Σ[y]Ep,b(Fp), thesum of x- and y- coordinates of all points (x, y) on Ep,b, and alsothe the sum Σ(x,0)Ep,b(Fp), the sum of points (x, 0) on Ep,b.

Keywords: Elliptic curves over finite fields, rational points on elliptic curves.

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