Search results for: split activation function.

Authors: Refaat M Mohamed, Ayman El-Baz, Aly A. Farag

Abstract:

This paper introduces a novel approach to estimate the clique potentials of Gibbs Markov random field (GMRF) models using the Support Vector Machines (SVM) algorithm and the Mean Field (MF) theory. The proposed approach is based on modeling the potential function associated with each clique shape of the GMRF model as a Gaussian-shaped kernel. In turn, the energy function of the GMRF will be in the form of a weighted sum of Gaussian kernels. This formulation of the GMRF model urges the use of the SVM with the Mean Field theory applied for its learning for estimating the energy function. The approach has been tested on synthetic texture images and is shown to provide satisfactory results in retrieving the synthesizing parameters.

Keywords: Image Modeling, MRF, Parameters Estimation, SVM Learning.

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Authors: Ni Ni Soe , Masahiro Nakagawa

Abstract:

Functional near infrared spectroscopy (fNIRS) is a practical non-invasive optical technique to detect characteristic of hemoglobin density dynamics response during functional activation of the cerebral cortex. In this paper, fNIRS measurements were made in the area of motor cortex from C4 position according to international 10-20 system. Three subjects, aged 23 - 30 years, were participated in the experiment. The aim of this paper was to evaluate the effects of different motor activation tasks of the hemoglobin density dynamics of fNIRS signal. The chaotic concept based on deterministic dynamics is an important feature in biological signal analysis. This paper employs the chaotic properties which is a novel method of nonlinear analysis, to analyze and to quantify the chaotic property in the time series of the hemoglobin dynamics of the various motor imagery tasks of fNIRS signal. Usually, hemoglobin density in the human brain cortex is found to change slowly in time. An inevitable noise caused by various factors is to be included in a signal. So, principle component analysis method (PCA) is utilized to remove high frequency component. The phase pace is reconstructed and evaluated the Lyapunov spectrum, and Lyapunov dimensions. From the experimental results, it can be conclude that the signals measured by fNIRS are chaotic.

Keywords: Chaos, hemoglobin, Lyapunov spectrum, motorimagery, near infrared spectroscopy (NIRS), principal componentanalysis (PCA).

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Authors: S. Yaman, S. Rostami

Abstract:

In this study, a black box modeling of the coupled-tank system is obtained by using fuzzy sets. The derived model is tested via adaptive neuro fuzzy inference system (ANFIS). In order to achieve a better control performance, the parameters of three different controller types, classical proportional integral controller (PID), fuzzy PID and function tuner method, are tuned by one of the evolutionary computation method, genetic algorithm. All tuned controllers are applied to the fuzzy model of the coupled-tank experimental setup and analyzed under the different reference input values. According to the results, it is seen that function tuner method demonstrates better robust control performance and guarantees the closed loop stability.

Keywords: Function tuner method, fuzzy modeling, fuzzy PID controller, genetic algorithm.

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Authors: Nonlapun Meenil, Pisitpong Intarapong, Thitima Wongsheree, Pranchalee Samanpiboon

Abstract:

In Thailand, once pineapples are harvested, they must be classified into two classes based on their sweetness: sweet and unsweet. This paper has studied and developed the assessment of internal quality of pineapples using a low-cost compact spectroscopy sensor according to the spectroscopy approach and Neural Network (NN). During the experiments, Batavia pineapples were utilized, generating 100 samples. The extracted pineapple juice of each sample was used to determine the Soluble Solid Content (SSC) labeling into sweet and unsweet classes. In terms of experimental equipment, the sensor cover was specifically designed to install the sensor and light source to read the reflectance at a five mm depth from pineapple flesh. By using a spectroscopy sensor, data on visible and near-infrared reflectance (Vis-NIR) were collected. The NN was used to classify the pineapple classes. Before the classification step, the preprocessing methods, which are class balancing, data shuffling, and standardization, were applied. The 510 nm and 900 nm reflectance values of the middle parts of pineapples were used as features of the NN. With the sequential model and ReLU activation function, 100% accuracy of the training set and 76.67% accuracy of the test set were achieved. According to the abovementioned information, using a low-cost compact spectroscopy sensor has achieved favorable results in classifying the sweetness of the two classes of pineapples.

Keywords: Spectroscopy, soluble solid content, pineapple, neural network.

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Authors: H. Ogras, M. Turk

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In this study, a system of encryption based on chaotic sequences is described. The system is used for encrypting digital image data for the purpose of secure image transmission. An image secure communication scheme based on Logistic map chaotic sequences with a nonlinear function is proposed in this paper. Encryption and decryption keys are obtained by one-dimensional Logistic map that generates secret key for the input of the nonlinear function. Receiver can recover the information using the received signal and identical key sequences through the inverse system technique. The results of computer simulations indicate that the transmitted source image can be correctly and reliably recovered by using proposed scheme even under the noisy channel. The performance of the system will be discussed through evaluating the quality of recovered image with and without channel noise.

Keywords: Digital image, Image encryption, Secure communication

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Authors: I. Alhassan, A. M. Saddiq, A. G. Gashua, K. K. Gwio-Kura

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The main objective of any irrigation program is the development of an efficient water management system to sustain crop growth and development and avoid physiological water stress in the growing plants. Field experiment to evaluate the effects of some soil moisture conservation practices on yield and water use efficiency (WUE) of maize was carried out in three locations (i.e. Mubi and Yola in the northern Guinea Savannah and Ganye in the southern Guinea Savannah of Adamawa State, Nigeria) during the dry seasons of 2013 and 2014. The experiment consisted of three different irrigation levels (7, 10 and 12 day irrigation intervals), two levels of mulch (mulch and un-mulched) and two tillage practices (no tillage and minimum tillage) arranged in a randomized complete block design with split-split plot arrangement and replicated three times. The Blaney-Criddle method was used for measuring crop evapotranspiration. The results indicated that seven-day irrigation intervals and mulched treatment were found to have significant effect (P>0.05) on grain yield and water use efficiency in all the locations. The main effect of tillage was non-significant (P<0.05) on grain yield and WUE. The interaction effects of irrigation and mulch were significant (P>0.05) on grain yield and WUE at Mubi and Yola. Generally, higher grain yield and WUE were recorded on mulched and seven-day irrigation intervals, whereas lower values were recorded on un-mulched with 12-day irrigation intervals. Tillage exerts little influence on the yield and WUE. Results from Ganye were found to be generally higher than those recorded in Mubi and Yola; it also showed that an irrigation interval of 10 days with mulching could be adopted for the Ganye area, while seven days interval is more appropriate for Mubi and Yola.

Keywords: Irrigation, maize, mulching, tillage, guinea savannah.

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Authors: V. Vargas-Alejo, L. E. Montero-Moguel

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Understanding the concept of function has been important in mathematics education for many years. In this study, the models built by a group of five business administration and accounting undergraduate students when carrying out a population growth activity are analyzed. The theoretical framework is the Models and Modeling Perspective. The results show how the students included tables, graphics, and algebraic representations in their models. Using technology was useful to interpret, describe, and predict the situation. The first model, the students built to describe the situation, was linear. After that, they modified and refined their ways of thinking; finally, they created exponential growth. Modeling the activity was useful to deep on mathematical concepts such as covariation, rate of change, and exponential function also to differentiate between linear and exponential growth.

Keywords: Covariation reasoning, exponential function, modeling, representations.

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

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

Keywords: Variable speed refrigeration system, Fuzzy logic control, membership function range, control performance.

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Authors: Jiming Yang

Abstract:

A kind of singularly perturbed boundary value problems is under consideration. In order to obtain its approximation, simple upwind difference discretization is applied. We use a moving mesh iterative algorithm based on equi-distributing of the arc-length function of the current computed piecewise linear solution. First, a maximum norm a posteriori error estimate on an arbitrary mesh is derived using a different method from the one carried out by Chen [Advances in Computational Mathematics, 24(1-4) (2006), 197-212.]. Then, basing on the properties of discrete Green-s function and the presented posteriori error estimate, we theoretically prove that the discrete solutions computed by the algorithm are first-order uniformly convergent with respect to the perturbation parameter ε.

Keywords: Convergence analysis, green's function, singularly perturbed, equi-distribution, moving mesh.

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Authors: Areej Almalkawi, Sameer Hamadna, Parviz Soroushian, Nalin Darsana

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The work on indigenous binders in this paper focused on the following indigenous raw materials: red clay, red lava and pumice (as primary aluminosilicate precursors), wood ash and gypsum (as supplementary minerals), and sodium sulfate and lime (as alkali activators). The experimental methods used for evaluation of these indigenous raw materials included laser granulometry, x-ray fluorescence (XRF) spectroscopy, and chemical reactivity. Formulations were devised for transforming these raw materials into alkali aluminosilicate-based hydraulic cements. These formulations were processed into hydraulic cements via simple heating and milling actions to render thermal activation, mechanochemical and size reduction effects. The resulting hydraulic cements were subjected to laser granulometry, heat of hydration and reactivity tests. These cements were also used to prepare mortar mixtures, which were evaluated via performance of compressive strength tests. The measured values of strength were correlated with the reactivity, size distribution and microstructural features of raw materials. Some of the indigenous hydraulic cements produced in this reporting period yielded viable levels of compressive strength. The correlation trends established in this work are being evaluated for development of simple and thorough methods of qualifying indigenous raw materials for use in production of indigenous hydraulic cements.

Keywords: One-part geopolymer cement, aluminosilicate precursors, thermal activation, mechanochemical.

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Authors: Meryem Elif Çelebi Yakartepe, Ayşe Betül Gökarslan

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Sustainable architecture is an approach which accepts the climatic and tomographic data as a necessary preliminary data package by favoring the relationship between human and nature and which strives to use the sources economically. The reflection of sustainable architecture approach to preservation - restoration discipline is including the architectural inheritance to daily life with its unique or new function by restoring it.

The restoration decisions of Yılan Kırkan Fountain in Isparta province of Turkey is a good example of the works of sustainable architecture and the preservation of architectural inheritance. It is aimed that Yılan Kırkan Fountain, which is desolate nowadays with no function, to be restored by the local authorities and university, included in the daily life and continue its function.

Keywords: Conservation and Restoration, Sustainable Architecture, Movement Technique, Isparta Yılan Kırkan Fountain.

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Authors: A. Bakhtiar, B. Purwanggono, N. Metasari

Abstract:

PT XYZ is a bottled drinking water company. To preserve production resources owned by the company so that the resources could be utilized well, it has implemented maintenance management system, which has important role in company's profitability, and is one of the factors influenced overall company's performance. Yet, up to now the company has never measured maintenance activities' contribution to company's performance. Performance evaluation is done according to adapted Balanced Scorecard model fitted to maintenance function context. This model includes six perspectives: innovation and growth, production, maintenance, environment, costumer, and finance. Actual performance measurement is done through Analytic Hierarchy Process and Objective Matrix. From the research done, we can conclude that the company's maintenance function is categorized in moderate performance. But, there are some indicators which has high priority but low performance, which are: costumers' complain rate, work lateness rate, and Return on Investment.

Keywords: Maintenance, performance, balanced scorecard, objective matrix.

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Authors: Christiono Utomo, Arazi Idrus, Isnanto, Annisa Nugraheni, Farida Rahmawati

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A Negotiation Support is required on a value-based decision to enable each stakeholder to evaluate and rank the solution alternatives before engaging into negotiation with the other stakeholders. This study demonstrates a process of negotiation support model for selection of a building system from value-based design perspective. The perspective is based on comparison of function and cost of a building system. Multi criteria decision techniques were applied to determine the relative value of the alternative solutions for performing the function. A satisfying option game theory are applied to the criteria of value-based decision which are LCC (life cycle cost) and function based FAST. The results demonstrate a negotiation process to select priorities of a building system. The support model can be extended to an automated negotiation by combining value based decision method, group decision and negotiation support.

Keywords: NSS, Value-based, Decision, Construction.

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Authors: E. G. Bautista, J. M. Reyes, O. Bautista, J. C. Arcos

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In this work, we analyze the deformation of surface waves in shallow flows conditions, propagating in a channel of slowly varying cross-section. Based on a singular perturbation technique, the main purpose is to predict the motion of waves by using a dimensionless formulation of the governing equations, considering that the longitudinal variation of the transversal section obey a power-law distribution. We show that the spatial distribution of the waves in the varying cross-section is a function of a kinematic parameter,κ , and two geometrical parameters εh and w ε . The above spatial behavior of the surface elevation is modeled by an ordinary differential equation. The use of single formulas to model the varying cross sections or transitions considered in this work can be a useful approximation to natural or artificial geometrical configurations.

Keywords: Surface waves, Asymptotic solution, Power law function, Non-dispersive waves.

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Authors: Yi Yu, Lin Ma, Yong Sun, Yuantong Gu

Abstract:

This paper presents a novel method for remaining useful life prediction using the Elliptical Basis Function (EBF) network and a Markov chain. The EBF structure is trained by a modified Expectation-Maximization (EM) algorithm in order to take into account the missing covariate set. No explicit extrapolation is needed for internal covariates while a Markov chain is constructed to represent the evolution of external covariates in the study. The estimated external and the unknown internal covariates constitute an incomplete covariate set which are then used and analyzed by the EBF network to provide survival information of the asset. It is shown in the case study that the method slightly underestimates the remaining useful life of an asset which is a desirable result for early maintenance decision and resource planning.

Keywords: Elliptical Basis Function Network, Markov Chain, Missing Covariates, Remaining Useful Life

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Authors: Phyu Phyu Mar, Khin Maung Latt

Abstract:

S-boxes (Substitution boxes) are keystones of modern symmetric cryptosystems (block ciphers, as well as stream ciphers). S-boxes bring nonlinearity to cryptosystems and strengthen their cryptographic security. They are used for confusion in data security An S-box satisfies the strict avalanche criterion (SAC), if and only if for any single input bit of the S-box, the inversion of it changes each output bit with probability one half. If a function (cryptographic transformation) is complete, then each output bit depends on all of the input bits. Thus, if it were possible to find the simplest Boolean expression for each output bit in terms of the input bits, each of these expressions would have to contain all of the input bits if the function is complete. From some important properties of S-box, the most interesting property SAC (Strict Avalanche Criterion) is presented and to analyze this property three analysis methods are proposed.

Keywords: S-boxes, cryptosystems, strict avalanche criterion, function, analysis methods.

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Authors: Maharavo Randrianarivony

Abstract:

The objective is to split a simply connected polygon into a set of convex quadrilaterals without inserting new boundary nodes. The presented approach consists in repeatedly removing quadrilaterals from the polygon. Theoretical results pertaining to quadrangulation of simply connected polygons are derived from the usual 2-ear theorem. It produces a quadrangulation technique with O(n) number of quadrilaterals. The theoretical methodology is supplemented by practical results and CAD surface segmentation.

Keywords: Quadrangulation, simply connected, two-ear theorem.

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Authors: Tomohiro Hachino, Hitoshi Takata, Shigeru Nakayama, Ichiro Iimura, Seiji Fukushima, Yasutaka Igarashi

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This paper presents a nonparametric identification of continuous-time nonlinear systems by using a Gaussian process (GP) model. The GP prior model is trained by artificial bee colony algorithm. The nonlinear function of the objective system is estimated as the predictive mean function of the GP, and the confidence measure of the estimated nonlinear function is given by the predictive covariance of the GP. The proposed identification method is applied to modeling of a simplified electric power system. Simulation results are shown to demonstrate the effectiveness of the proposed method.

Keywords: Artificial bee colony algorithm, Gaussian process model, identification, nonlinear system, electric power system.

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Authors: Noriko Hanakawa, Masaki Obana

Abstract:

A simulation tool is proposed for big-scale projection mapping events. The tool has four main functions based on Mapbox and Unity utilities. The first function is building three-dimensional models of real cities using Mapbox. The second function is movie projections to some buildings in real cities using Unity. The third is a movie sending function from a PC to a virtual projector. The fourth function is mapping movies with fitting buildings. The simulation tool was adapted to a real projection mapping event held in 2019. The event completed, but it faced a severe problem in the movie projection to the target building. Extra tents were set in front of the target building, and the tents became obstacles to the movie projection. The simulation tool developed herein could reconstruct the problems of the event. Therefore, if the simulation tool was developed before the 2019 projection mapping event, the problem of the tents being obstacles could have been avoided using the tool. Moreover, we confirmed that the simulation tool is useful for planning future projection mapping events to avoid various extra equipment obstacles, such as utility poles, planting trees, and monument towers.

Keywords: avoiding obstacles, projection mapping, projector position, real 3D map

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Authors: Shiva Subhedar, Abhishek Jain, Shivin Mittal

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Icons are not merely decorative elements in enterprise applications but very often used because of their many advantages such as compactness, visual appeal, etc. Despite these potential advantages, icons often cause usability problems when they are designed without consideration for their many potential downsides. The aim of the current study was to examine the effect of articulatory distance – the distance between the physical appearance of an interface element and what it actually means. In other words, will the subject find the association of the function and its appearance on the interface natural or is the icon difficult for them to associate with its function. We have calculated response time and quality of identification by varying icon concreteness, the context of usage and subject experience in the enterprise context. The subjects were asked to associate icons (prepared for study purpose) with given function options in context and out of context mode. Response time and their selection were recorded for analysis.

Keywords: Icons, icon concreteness, icon recognition, HCI.

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Authors: Farhad Hadinejad, Hasan Saberi, Saeed Kazem

Abstract:

Support vector clustering (SVC) is an important kernelbased clustering algorithm in multi applications. It has got two main bottle necks, the high computation price and labeling piece. In this paper, we presented a modified SVC method, named Grid–SVC, to improve the original algorithm computationally. First we normalized and then we parted the interval, where the SVC is processing, using a novel Grid–based clustering algorithm. The algorithm parts the intervals, based on the density function of the data set and then applying the cartesian multiply makes multi-dimensional grids. Eliminating many outliers and noise in the preprocess, we apply an improved SVC method to each parted grid in a parallel way. The experimental results show both improvement in time complexity order and the accuracy.

Keywords: Grid–based clustering, SVC, Density function, Radial basis function.

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Authors: K. R. Balabiyev, A. O. Kaipbayeva

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The article discusses the legal framework of the government’s environmental function and analyzes the role of the national policy in protection of wetlands. The problem is of interest for it deals with the most important branch of economy – utilization of Kazakhstan’s natural resources, protection of health and environmental wellbeing of the population. Development of a longterm environmental program addressing the protection of wetlands represents the final stage of the government’s environmental policy, and is a relatively new function for the public administration system. It appeared due to the environmental measures that require immediate decisions to be taken. It is an integral part of the effort in the field of management of state-owned natural resource, as well as of the measures aimed at efficient management of natural resources to avoid their early depletion or contamination.

Keywords: Environmental focus, government’s environmental function, protection of wetlands.

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Authors: Manoj Singh, Mumtaz Ahmad Khan, Abdul Hakim Khan

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The object of the present paper is to investigate several general families of bilinear and bilateral generating functions with different argument for the Gauss’ hypergeometric polynomials.

Keywords: Appell’s functions, Gauss hypergeometric functions, Heat polynomials, Kampe’ de Fe’riet function, Laguerre polynomials, Lauricella’s function, Saran’s functions.

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Authors: Zeshan Ahmad, Matteo Zoppi, Rezia Molfino

Abstract:

The objective of positioning the fixture elements in the fixture is to make the workpiece stiff, so that geometric errors in the manufacturing process can be reduced. Most of the work for optimal fixture layout used the minimization of the sum of the nodal deflection normal to the surface as objective function. All deflections in other direction have been neglected. We propose a new method for fixture layout optimization in this paper, which uses the element strain energy. The deformations in all the directions have been considered in this way. The objective function in this method is to minimize the sum of square of element strain energy. Strain energy and stiffness are inversely proportional to each other. The optimization problem is solved by the sequential quadratic programming method. Three different kinds of case studies are presented, and results are compared with the method using nodal deflections as objective function to verify the propose method.

Keywords: Fixture layout, optimization, strain energy, quadratic programming.

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Authors: B. Skrabic, N. Tomic-Plazibat

Abstract:

In this paper real money demand function is analyzed within multivariate time-series framework. Cointegration approach is used (Johansen procedure) assuming interdependence between money demand determinants, which are nonstationary variables. This will help us to understand the behavior of money demand in Croatia, revealing the significant influence between endogenous variables in vector autoregrression system (VAR), i.e. vector error correction model (VECM). Exogeneity of the explanatory variables is tested. Long-run money demand function is estimated indicating slow speed of adjustment of removing the disequilibrium. Empirical results provide the evidence that real industrial production and exchange rate explains the most variations of money demand in the long-run, while interest rate is significant only in short-run.

Keywords: Cointegration, Long-run equilibrium, Money demand function, Vector error correction model.

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Authors: Min Sun, Zhenyu Liu

Abstract:

In this paper, a new descent-projection method with a new search direction for monotone structured variational inequalities is proposed. The method is simple, which needs only projections and some function evaluations, so its computational load is very tiny. Under mild conditions on the problem-s data, the method is proved to converges globally. Some preliminary computational results are also reported to illustrate the efficiency of the method.

Keywords: variational inequalities, monotone function, global convergence.

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Authors: R. Fatehi, M.A. Fayazbakhsh, M.T. Manzari

Abstract:

Discretization of spatial derivatives is an important issue in meshfree methods especially when the derivative terms contain non-linear coefficients. In this paper, various methods used for discretization of second-order spatial derivatives are investigated in the context of Smoothed Particle Hydrodynamics. Three popular forms (i.e. "double summation", "second-order kernel derivation", and "difference scheme") are studied using one-dimensional unsteady heat conduction equation. To assess these schemes, transient response to a step function initial condition is considered. Due to parabolic nature of the heat equation, one can expect smooth and monotone solutions. It is shown, however in this paper, that regardless of the type of kernel function used and the size of smoothing radius, the double summation discretization form leads to non-physical oscillations which persist in the solution. Also, results show that when a second-order kernel derivative is used, a high-order kernel function shall be employed in such a way that the distance of inflection point from origin in the kernel function be less than the nearest particle distance. Otherwise, solutions may exhibit oscillations near discontinuities unlike the "difference scheme" which unconditionally produces monotone results.

Keywords: Heat conduction, Meshfree methods, Smoothed ParticleHydrodynamics (SPH), Second-order derivatives.

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Authors: Jiann-Ming Wu, Hsiao-Chang Chen, Chun-Chang Wu, Pei-Hsun Hsu

Abstract:

This work explores blind image deconvolution by recursive function approximation based on supervised learning of neural networks, under the assumption that a degraded image is linear convolution of an original source image through a linear shift-invariant (LSI) blurring matrix. Supervised learning of neural networks of radial basis functions (RBF) is employed to construct an embedded recursive function within a blurring image, try to extract non-deterministic component of an original source image, and use them to estimate hyper parameters of a linear image degradation model. Based on the estimated blurring matrix, reconstruction of an original source image from a blurred image is further resolved by an annealed Hopfield neural network. By numerical simulations, the proposed novel method is shown effective for faithful estimation of an unknown blurring matrix and restoration of an original source image.

Keywords: Blind image deconvolution, linear shift-invariant(LSI), linear image degradation model, radial basis functions (rbf), recursive function, annealed Hopfield neural networks.

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Authors: Nasser Mohamed Ramli, Mohamad Syafiq Mohamad

Abstract:

Many types of controllers were applied on the continuous stirred tank reactor (CSTR) unit to control the temperature. In this research paper, Proportional-Integral-Derivative (PID) controller are compared with Fuzzy Logic controller for temperature control of CSTR. The control system for temperature non-isothermal of a CSTR will produce a stable response curve to its set point temperature. A mathematical model of a CSTR using the most general operating condition was developed through a set of differential equations into S-function using MATLAB. The reactor model and S-function are developed using m.file. After developing the S-function of CSTR model, User-Defined functions are used to link to SIMULINK file. Results that are obtained from simulation and temperature control were better when using Fuzzy logic control compared to PID control.

Keywords: CSTR, temperature, PID, fuzzy logic.

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Authors: Rana Khalid Naeem, Waseem Ahmed Khan, Muhammad Akhtar, Asif Mansoor

Abstract:

The Navier Stokes Equations (NSE) for an incompressible fluid of variable viscosity in the presence of an unknown external force in Von-Mises system x,\ are transformed, and some new exact solutions for a class of flows characterized by equation y f x a\b for an arbitrary state equation are determined, where f x is a function, \ the stream function, a z 0 and b are the arbitrary constants. In three, out of four cases, the function f x is arbitrary, and the solutions are the solutions of the flow equations for all the flows characterized by the equationy f x a\b. Streamline patterns for some forms of f x in unbounded and bounded regions are given.

Keywords: Bounded and unbounded region, Exact solution, Navier Stokes equations, Streamline pattern, Variable viscosity, Von- Mises system

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