Search results for: weighted gradient method
Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 8419

Search results for: weighted gradient method

7579 Periodic Solutions for Some Strongly Nonlinear Oscillators by He's Energy Balance Method

Authors: Meng Hu, Lili Wang

Abstract:

In this paper, applying He-s energy balance method to determine frequency formulation relations of nonlinear oscillators with discontinuous term or fractional potential. By calculation and computer simulations, compared with the exact solutions show that the results obtained are of high accuracy.

Keywords: He's energy balance method, periodic solution, nonlinear oscillator, discontinuous, fractional potential.

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7578 Solving 94-bit ECDLP with 70 Computers in Parallel

Authors: Shunsuke Miyoshi, Yasuyuki Nogami, Takuya Kusaka, Nariyoshi Yamai

Abstract:

Elliptic curve discrete logarithm problem(ECDLP) is one of problems on which the security of pairing-based cryptography is based. This paper considers Pollard’s rho method to evaluate the security of ECDLP on Barreto-Naehrig(BN) curve that is an efficient pairing-friendly curve. Some techniques are proposed to make the rho method efficient. Especially, the group structure on BN curve, distinguished point method, and Montgomery trick are well-known techniques. This paper applies these techniques and shows its optimization. According to the experimental results for which a large-scale parallel system with MySQL is applied, 94-bit ECDLP was solved about 28 hours by parallelizing 71 computers.

Keywords: Pollard’s rho method, BN curve, Montgomery multiplication.

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7577 Physical Properties and Stability of Emulsions as Affected by Native and Modified Yam Starches

Authors: Nor Hayati Ibrahim, Shamini Nair Achudan

Abstract:

This study was conducted in order to determine the physical properties and stability of mayonnaise-like emulsions as affected by modified yam starches. Native yam starch was modified via pre-gelatinization and cross-linking phosphorylation procedures. The emulsions (50% oil dispersed phase) were prepared with 0.3% native potato, native yam, pre-gelatinized yam and cross-linking phosphorylation yam starches. The droplet size of surface weighted mean diameter was found to be significantly (p < 0.05) lower in the sample with cross-linking phosphorylation yam starch as compared to other samples. Moreover, the viscosity of the sample with pregelatinized yam starch was observed to be higher than that of other samples. The phase separation stability was low in the freshly prepared and stored (45 days, 5°C) emulsions containing native yam starch. This study thus generally suggested that modified yam starches were more suitable (i.e. better physical properties and stability) to be used as stabilizers in a similar system i.e. light mayonnaises, rather than a native yam starch.

Keywords: Oil-in-water emulsions, low-fat mayonnaises, modified yam starches, droplet size distribution, viscosity.

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7576 Synchronization Between Two Chaotic Systems: Numerical and Circuit Simulation

Authors: J. H. Park, T. H. Lee, S. M. Lee, H. Y. Jung

Abstract:

In this paper, a generalized synchronization scheme, which is called function synchronization, for chaotic systems is studied. Based on Lyapunov method and active control method, we design the synchronization controller for the system such that the error dynamics between master and slave chaotic systems is asymptotically stable. For verification of our theory, computer and circuit simulations for a specific chaotic system is conducted.

Keywords: Chaotic systems, synchronization, Lyapunov method, simulation.

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7575 Multidimensional Compromise Programming Evaluation of Digital Commerce Websites

Authors: C. Ardil

Abstract:

Multidimensional compromise programming evaluation of digital commerce websites is essential not only to have recommendations for improvement, but also to make comparisons with global business competitors. This research provides a multidimensional decision making model that prioritizes the objective criteria weights of various commerce websites using multidimensional compromise solution. Evaluation of digital commerce website quality can be considered as a complex information system structure including qualitative and quantitative factors for a multicriteria decision making problem. The proposed multicriteria decision making approach mainly consists of three sequential steps for the selection problem. In the first step, three major different evaluation criteria are characterized for website ranking problem. In the second step, identified critical criteria are weighted using the standard deviation procedure. In the third step, the multidimensional compromise programming is applied to rank the digital commerce websites.

Keywords: Standard deviation, commerce website, website evaluation, multicriteria decision making, multicriteria compromise programming, website quality, multidimensional decision analysis.

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7574 Element-Independent Implementation for Method of Lagrange Multipliers

Authors: Gil-Eon Jeong, Sung-Kie Youn, K. C. Park

Abstract:

Treatment for the non-matching interface is an important computational issue. To handle this problem, the method of Lagrange multipliers including classical and localized versions are the most popular technique. It essentially imposes the interface compatibility conditions by introducing Lagrange multipliers. However, the numerical system becomes unstable and inefficient due to the Lagrange multipliers. The interface element-independent formulation that does not include the Lagrange multipliers can be obtained by modifying the independent variables mathematically. Through this modification, more efficient and stable system can be achieved while involving equivalent accuracy comparing with the conventional method. A numerical example is conducted to verify the validity of the presented method.

Keywords: Element-independent formulation, non-matching interface, interface coupling, methods of Lagrange multipliers.

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7573 Work Function Engineering of Functionally Graded ZnO+Ga2O3 Thin Film for Solar Cell and Organic Light Emitting Diodes Applications

Authors: Yong-Taeg Oh, Won Song, Seok-Eui Choi, Bo-Ra Koo, Dong-Chan Shin

Abstract:

ZnO+Ga2O3 functionally graded thin films (FGTFs) were examined for their potential use as Solar cell and organic light emitting diodes (OLEDs). FGTF transparent conducting oxides (TCO) were fabricated by combinatorial RF magnetron sputtering. The composition gradient was controlled up to 10% by changing the plasma power of the two sputter guns. A Ga2O3+ZnO graded region was placed on the top layer of ZnO. The FGTFs showed up to 80% transmittance. Their surface resistances were reduced to < 10% by increasing the Ga2O3: pure ZnO ratio in the TCO. The FGTFs- work functions could be controlled within a range of 0.18 eV. The controlled work function is a very promising technology because it reduces the contact resistance between the anode and Hall transport layers of OLED and solar cell devices.

Keywords: Work Function, TCO, Functionally Graded Thin Films, Resistance, Transmittance.

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7572 Free Vibration Analysis of Non-Uniform Euler Beams on Elastic Foundation via Homotopy Perturbation Method

Authors: U. Mutman, S. B. Coskun

Abstract:

In this study Homotopy Perturbation Method (HPM) is employed to investigate free vibration of an Euler beam with variable stiffness resting on an elastic foundation. HPM is an easy-to-use and very efficient technique for the solution of linear or nonlinear problems. HPM produces analytical approximate expression which is continuous in the solution domain. This work shows that HPM is a promising method for free vibration analysis of nonuniform Euler beams on elastic foundation. Several case problems have been solved by using the technique and solutions have been compared with those available in the literature.

Keywords: Homotopy Perturbation Method, Elastic Foundation, Vibration, Beam

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7571 Sinc-Galerkin Method for the Solution of Problems in Calculus of Variations

Authors: M. Zarebnia, N. Aliniya

Abstract:

In this paper, a numerical solution based on sinc functions is used for finding the solution of boundary value problems which arise from the problems of calculus of variations. This approximation reduce the problems to an explicit system of algebraic equations. Some numerical examples are also given to illustrate the accuracy and applicability of the presented method.

Keywords: Calculus of variation; Sinc functions; Galerkin; Numerical method

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7570 On the Approximate Solution of Continuous Coefficients for Solving Third Order Ordinary Differential Equations

Authors: A. M. Sagir

Abstract:

This paper derived four newly schemes which are combined in order to form an accurate and efficient block method for parallel or sequential solution of third order ordinary differential equations of the form y''' = f(x, y, y', y''), y(α)=y0, y'(α)=β, y''(α)=η with associated initial or boundary conditions. The implementation strategies of the derived method have shown that the block method is found to be consistent, zero stable and hence convergent. The derived schemes were tested on stiff and non – stiff ordinary differential equations, and the numerical results obtained compared favorably with the exact solution.

Keywords: Block Method, Hybrid, Linear Multistep, Self starting, Third Order Ordinary Differential Equations.

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7569 Bit Error Rate Monitoring for Automatic Bias Control of Quadrature Amplitude Modulators

Authors: Naji Ali Albakay, Abdulrahman Alothaim, Isa Barshushi

Abstract:

The most common quadrature amplitude modulator (QAM) applies two Mach-Zehnder Modulators (MZM) and one phase shifter to generate high order modulation format. The bias of MZM changes over time due to temperature, vibration, and aging factors. The change in the biasing causes distortion to the generated QAM signal which leads to deterioration of bit error rate (BER) performance. Therefore, it is critical to be able to lock MZM’s Q point to the required operating point for good performance. We propose a technique for automatic bias control (ABC) of QAM transmitter using BER measurements and gradient descent optimization algorithm. The proposed technique is attractive because it uses the pertinent metric, BER, which compensates for bias drifting independently from other system variations such as laser source output power. The proposed scheme performance and its operating principles are simulated using OptiSystem simulation software for 4-QAM and 16-QAM transmitters.

Keywords: Automatic bias control, optical fiber communication, optical modulation, optical devices.

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7568 Laplace Adomian Decomposition Method Applied to a Two-Dimensional Viscous Flow with Shrinking Sheet

Authors: M. A. Koroma, S. Widatalla, A. F. Kamara, C. Zhang

Abstract:

Our aim in this piece of work is to demonstrate the power of the Laplace Adomian decomposition method (LADM) in approximating the solutions of nonlinear differential equations governing the two-dimensional viscous flow induced by a shrinking sheet.

Keywords: Adomian polynomials, Laplace Adomian decomposition method, Padé Approximant, Shrinking sheet.

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7567 Image Analysis of Fine Structures of Supercavitation in the Symmetric Wake of a Cylinder

Authors: Y. Obikane , M.Kaneko, K.Kakioka, K.Ogura

Abstract:

The fine structure of supercavitation in the wake of a symmetrical cylinder is studied with high-speed video cameras. The flow is observed in a cavitation tunnel at the speed of 8m/sec when the sidewall and the wake are partially filled with the massive cavitation bubbles. The present experiment observed that a two-dimensional ripple wave with a wave length of 0.3mm is propagated in a downstream direction, and then abruptly increases to a thicker three-dimensional layer. IR-photography recorded that the wakes originated from the horseshoe vortexes alongside the cylinder. The wake was developed to inside the dead water zone, which absorbed the bubbly wake propelled from the separated vortices at the center of the cylinder. A remote sensing classification technique (maximum most likelihood) determined that the surface porosity was 0.2, and the mean speed in the mixed wake was 7m/sec. To confirm the existence of two-dimensional wave motions in the interface, the experiments were conducted at a very low frequency, and showed similar gravity waves in both the upper and lower interfaces.

Keywords: Supercavitation, density gradient correlation

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7566 A Quantification Method of Attractiveness of Stations and an Estimation Method of Number of Passengers Taking into Consideration the Attractiveness of the Station

Authors: Naoya Ozaki, Takuya Watanabe, Ryosuke Matsumoto, Noriko Fukasawa

Abstract:

In the metropolitan areas in Japan, in many stations, shopping areas are set up, and escalators and elevators are installed to make the stations be barrier-free. Further, many areas around the stations are being redeveloped. Railway business operators want to know how much effect these circumstances have on attractiveness of the station or number of passengers using the station. So, we performed a questionnaire survey of the station users in the metropolitan areas for finding factors to affect the attractiveness of stations. Then, based on the analysis of the survey, we developed a method to quantitatively evaluate attractiveness of the stations. We also developed an estimation method for number of passengers based on combination of attractiveness of the station quantitatively evaluated and the residential and labor population around the station. Then, we derived precise linear regression models estimating the attractiveness of the station and number of passengers of the station.

Keywords: Attractiveness of the station, estimation method, number of passengers of the station, redevelopment around the station, renovation of the station.

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7565 Generalized Method for Estimating Best-Fit Vertical Alignments for Profile Data

Authors: Said M. Easa, Shinya Kikuchi

Abstract:

When the profile information of an existing road is missing or not up-to-date and the parameters of the vertical alignment are needed for engineering analysis, the engineer has to recreate the geometric design features of the road alignment using collected profile data. The profile data may be collected using traditional surveying methods, global positioning systems, or digital imagery. This paper develops a method that estimates the parameters of the geometric features that best characterize the existing vertical alignments in terms of tangents and the expressions of the curve, that may be symmetrical, asymmetrical, reverse, and complex vertical curves. The method is implemented using an Excel-based optimization method that minimizes the differences between the observed profile and the profiles estimated from the equations of the vertical curve. The method uses a 'wireframe' representation of the profile that makes the proposed method applicable to all types of vertical curves. A secondary contribution of this paper is to introduce the properties of the equal-arc asymmetrical curve that has been recently developed in the highway geometric design field.

Keywords: Optimization, parameters, data, reverse, spreadsheet, vertical curves

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7564 An Improved Prediction Model of Ozone Concentration Time Series Based On Chaotic Approach

Authors: N. Z. A. Hamid, M. S. M. Noorani

Abstract:

This study is focused on the development of prediction models of the Ozone concentration time series. Prediction model is built based on chaotic approach. Firstly, the chaotic nature of the time series is detected by means of phase space plot and the Cao method. Then, the prediction model is built and the local linear approximation method is used for the forecasting purposes. Traditional prediction of autoregressive linear model is also built. Moreover, an improvement in local linear approximation method is also performed. Prediction models are applied to the hourly Ozone time series observed at the benchmark station in Malaysia. Comparison of all models through the calculation of mean absolute error, root mean squared error and correlation coefficient shows that the one with improved prediction method is the best. Thus, chaotic approach is a good approach to be used to develop a prediction model for the Ozone concentration time series.

Keywords: Chaotic approach, phase space, Cao method, local linear approximation method.

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7563 An Adaptive Dynamic Fracture for 3D Fatigue Crack Growth Using X-FEM

Authors: S. Lecheb, A. Nour, A. Chellil, A. Basta, D. Belmiloud, H. Kebi

Abstract:

In recent years, a new numerical method has been developed, the extended finite element method (X-FEM). The objective of this work is to exploit the (X-FEM) for the treatment of the fracture mechanics problems on 3D geometries, where we showed the ability of this method to simulate the fatigue crack growth into two cases: edge and central crack. In the results we compared the six first natural frequencies of mode shapes uncracking with the cracking initiation in the structure, and showed the stress intensity factor (SIF) evolution function as crack size propagation into structure, the analytical validation of (SIF) is presented. For to evidence the aspects of this method, all result is compared between FEA and X-FEM.

Keywords: 3D fatigue crack growth, FEA, natural frequencies, stress intensity factor (SIF), X-FEM.

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7562 Improvement of Ride Comfort of Turning Electric Vehicle Using Optimal Speed Control

Authors: Yingyi Zhou, Tohru Kawabe

Abstract:

With the spread of EVs (electric Vehicles), the ride comfort has been gaining a lot of attention. The influence of the lateral acceleration is important for the improvement of ride comfort of EVs as well as the longitudinal acceleration, especially upon turning of the vehicle. Therefore, this paper proposes a practical optimal speed control method to greatly improve the ride comfort in the vehicle turning situation. For consturcting this method, effective criteria that can appropriately evaluate deterioration of ride comfort is derived. The method can reduce the influence of both the longitudinal and the lateral speed changes for providing a confortable ride. From several simulation results, we can see the fact that the method can prevent aggravation of the ride comfort by suppressing the influence of longitudinal speed change in the turning situation. Hence, the effectiveness of the method is recognized.

Keywords: Electric vehicle, speed control, ride comfort, optimal control theory, driving support system.

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7561 Nanomechanical Characterization of Titanium Alloy Modified by Nitrogen Ion Implantation

Authors: Josef Sepitka, Petr Vlcak, Tomas Horazdovsky, Vratislav Perina

Abstract:

An ion implantation technique was used for designing the surface area of a titanium alloy and for irradiation-enhanced hardening of the surface. The Ti6Al4V alloy was treated by nitrogen ion implantation at fluences of 2·1017 and 4·1017 cm-2 and at ion energy 90 keV. The depth distribution of the nitrogen was investigated by Rutherford Backscattering Spectroscopy. The gradient of mechanical properties was investigated by nanoindentation. The continuous measurement mode was used to obtain depth profiles of the indentation hardness and the reduced storage modulus of the modified surface area. The reduced storage modulus and the hardness increase with increasing fluence. Increased fluence shifts the peak of the mechanical properties as well as the peak of nitrogen concentration towards to the surface. This effect suggests a direct relationship between mechanical properties and nitrogen distribution.

Keywords: Nitrogen ion implantation, titanium-based nanolayer, storage modulus, hardness, microstructure.

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7560 Embedded Singly Diagonally Implicit Runge-Kutta –Nystrom Method Order 5(4) for the Integration of Special Second Order ODEs

Authors: Fudziah Ismail

Abstract:

In this paper a new embedded Singly Diagonally Implicit Runge-Kutta Nystrom fourth order in fifth order method for solving special second order initial value problems is derived. A standard set of test problems are tested upon and comparisons on the numerical results are made when the same set of test problems are reduced to first order systems and solved using the existing embedded diagonally implicit Runge-Kutta method. The results suggests the superiority of the new method.

Keywords: Runge-Kutta Nystrom, Special second orderproblems.

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7559 Assessment of the Energy Balance Method in the Case of Masonry Domes

Authors: M. M. Sadeghi, S. Vahdani

Abstract:

Masonry dome structures had been widely used for covering large spans in the past. The seismic assessment of these historical structures is very complicated due to the nonlinear behavior of the material, their rigidness, and special stability configuration. The assessment method based on energy balance concept, as well as the standard pushover analysis, is used to evaluate the effectiveness of these methods in the case of masonry dome structures. The Soltanieh dome building is used as an example to which two methods are applied. The performance points are given from superimposing the capacity, and demand curves in Acceleration Displacement Response Spectra (ADRS) and energy coordination are compared with the nonlinear time history analysis as the exact result. The results show a good agreement between the dynamic analysis and the energy balance method, but standard pushover method does not provide an acceptable estimation.

Keywords: Energy balance method, pushover analysis, time history analysis, masonry dome.

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7558 Vibration of Functionally Graded Cylindrical Shells under Free-Free Boundary Conditions

Authors: A.R.Tahmasebi Birgani, M.Hosseinjani Zamenjani, M.R.Isvandzibaei

Abstract:

In the present work, study of the vibration of thin cylindrical shells made of a functionally gradient material (FGM) composed of stainless steel and nickel is presented. Material properties are graded in the thickness direction of the shell according to volume fraction power law distribution. The objective is to study the natural frequencies, the influence of constituent volume fractions and the effects of boundary conditions on the natural frequencies of the FG cylindrical shell. The study is carried out using third order shear deformation shell theory. The governing equations of motion of FG cylindrical shells are derived based on shear deformation theory. Results are presented on the frequency characteristics, influence of constituent volume fractions and the effects of free-free boundary conditions.

Keywords: Vibration, FGM, Cylindrical shell.

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7557 Fuzzy Scan Method to Detect Clusters

Authors: Laureano Rodríguez, Gladys Casas, Ricardo Grau, Yailen Martínez

Abstract:

The classical temporal scan statistic is often used to identify disease clusters. In recent years, this method has become as a very popular technique and its field of application has been notably increased. Many bioinformatic problems have been solved with this technique. In this paper a new scan fuzzy method is proposed. The behaviors of classic and fuzzy scan techniques are studied with simulated data. ROC curves are calculated, being demonstrated the superiority of the fuzzy scan technique.

Keywords: Scan statistic, fuzzy scan, simulating study

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7556 Approximate Confidence Interval for Effect Size Base on Bootstrap Resampling Method

Authors: S. Phanyaem

Abstract:

This paper presents the confidence intervals for the effect size base on bootstrap resampling method. The meta-analytic confidence interval for effect size is proposed that are easy to compute. A Monte Carlo simulation study was conducted to compare the performance of the proposed confidence intervals with the existing confidence intervals. The best confidence interval method will have a coverage probability close to 0.95. Simulation results have shown that our proposed confidence intervals perform well in terms of coverage probability and expected length.

Keywords: Effect size, confidence interval, Bootstrap Method.

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7555 ISME: Integrated Style Motion Editor for 3D Humanoid Character

Authors: Ismahafezi Ismail, Mohd Shahrizal Sunar

Abstract:

The motion of a realistic 3D humanoid character is very important especially for the industries developing computer animations and games. However, this type of motion is seen with a very complex dimensional data as well as body position, orientation, and joint rotation. Integrated Style Motion Editor (ISME), on the other hand, is a method used to alter the 3D humanoid motion capture data utilised in computer animation and games development. Therefore, this study was carried out with the purpose of demonstrating a method that is able to manipulate and deform different motion styles by integrating Key Pose Deformation Technique and Trajectory Control Technique. This motion editing method allows the user to generate new motions from the original motion capture data using a simple interface control. Unlike the previous method, our method produces a realistic humanoid motion style in real time.

Keywords: Computer animation, humanoid motion, motion capture, motion editing.

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7554 Bifurcation Method for Solving Positive Solutions to a Class of Semilinear Elliptic Equations and Stability Analysis of Solutions

Authors: Hailong Zhu, Zhaoxiang Li

Abstract:

Semilinear elliptic equations are ubiquitous in natural sciences. They give rise to a variety of important phenomena in quantum mechanics, nonlinear optics, astrophysics, etc because they have rich multiple solutions. But the nontrivial solutions of semilinear equations are hard to be solved for the lack of stabilities, such as Lane-Emden equation, Henon equation and Chandrasekhar equation. In this paper, bifurcation method is applied to solving semilinear elliptic equations which are with homogeneous Dirichlet boundary conditions in 2D. Using this method, nontrivial numerical solutions will be computed and visualized in many different domains (such as square, disk, annulus, dumbbell, etc).

Keywords: Semilinear elliptic equations, positive solutions, bifurcation method, isotropy subgroups.

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7553 Frequency-Variation Based Method for Parameter Estimation of Transistor Amplifier

Authors: Akash Rathee, Harish Parthasarathy

Abstract:

In this paper, a frequency-variation based method has been proposed for transistor parameter estimation in a commonemitter transistor amplifier circuit. We design an algorithm to estimate the transistor parameters, based on noisy measurements of the output voltage when the input voltage is a sine wave of variable frequency and constant amplitude. The common emitter amplifier circuit has been modelled using the transistor Ebers-Moll equations and the perturbation technique has been used for separating the linear and nonlinear parts of the Ebers-Moll equations. This model of the amplifier has been used to determine the amplitude of the output sinusoid as a function of the frequency and the parameter vector. Then, applying the proposed method to the frequency components, the transistor parameters have been estimated. As compared to the conventional time-domain least squares method, the proposed method requires much less data storage and it results in more accurate parameter estimation, as it exploits the information in the time and frequency domain, simultaneously. The proposed method can be utilized for parameter estimation of an analog device in its operating range of frequencies, as it uses data collected from different frequencies output signals for parameter estimation.

Keywords: Perturbation Technique, Parameter estimation, frequency-variation based method.

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7552 A Dictionary Learning Method Based On EMD for Audio Sparse Representation

Authors: Yueming Wang, Zenghui Zhang, Rendong Ying, Peilin Liu

Abstract:

Sparse representation has long been studied and several dictionary learning methods have been proposed. The dictionary learning methods are widely used because they are adaptive. In this paper, a new dictionary learning method for audio is proposed. Signals are at first decomposed into different degrees of Intrinsic Mode Functions (IMF) using Empirical Mode Decomposition (EMD) technique. Then these IMFs form a learned dictionary. To reduce the size of the dictionary, the K-means method is applied to the dictionary to generate a K-EMD dictionary. Compared to K-SVD algorithm, the K-EMD dictionary decomposes audio signals into structured components, thus the sparsity of the representation is increased by 34.4% and the SNR of the recovered audio signals is increased by 20.9%.

Keywords: Dictionary Learning, EMD, K-means Method, Sparse Representation.

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7551 Traveling Wave Solutions for Shallow Water Wave Equation by (G'/G)-Expansion Method

Authors: Anjali Verma, Ram Jiwari, Jitender Kumar

Abstract:

This paper presents a new function expansion method for finding traveling wave solution of a non-linear equation and calls it the (G'/G)-expansion method. The shallow water wave equation is reduced to a non linear ordinary differential equation by using a simple transformation. As a result the traveling wave solutions of shallow water wave equation are expressed in three forms: hyperbolic solutions, trigonometric solutions and rational solutions.

Keywords: Shallow water wave equation, Exact solutions, (G'/G) expansion method.

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7550 Non-Polynomial Spline Method for the Solution of Problems in Calculus of Variations

Authors: M. Zarebnia, M. Hoshyar, M. Sedaghati

Abstract:

In this paper, a numerical solution based on nonpolynomial cubic spline functions is used for finding the solution of boundary value problems which arise from the problems of calculus of variations. This approximation reduce the problems to an explicit system of algebraic equations. Some numerical examples are also given to illustrate the accuracy and applicability of the presented method.

Keywords: Calculus of variation; Non-polynomial spline functions; Numerical method

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