Search results for: The Matrix

Authors: A. O. Oladele

Abstract:

The purpose of this study was to find out the effectiveness of neurological impress method and repeated reading technique on reading fluency of children with learning disabilities. Thirty primary four pupils in three public primary schools participated in the study. There were two experimental groups and a control. This research employed a 3 by 2 factorial matrix and the participants were taught for one session. Two hypotheses were formulated to guide the research. T-test was used to analyse the data gathered, and data analysis revealed that pupils exposed to the two treatment strategies had improvement in their reading fluency. It was recommended that the two strategies used in the study can be used to intervene in reading fluency problems in children with learning disabilities.

Keywords: Learning disabilities, neurological impress method, repeated reading, reading fluency.

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Authors: Djalal Eddine Khodja, Aissa Kheldoun

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In this work we present the modelling of the induction machine, taking into consideration the stator defects of the induction machine. It is based on the theory of electromagnetic coupling of electrical circuits. In fact, for the modelling of stationary defects such as short circuit between turns in the same phase, we introduce only in the matrix the coefficients of resistance and inductance of stator and in the mutual inductance stator-rotor. These coefficients take account the number of turns in short-circuit deducted from the total number of turns in the same phase; in this way we obtain the number of useful turns. In addition, all these faults involved, will be used for the creation of the database that will be used to develop an automated system failures of the induction machine.

Keywords: Asynchronous machine, Indicatory Values Statorfaults, Multi-turns Model, Three-phases Model.

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Authors: Phu-Cuong Nguyen, Seung-Eock Kim

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This paper presents nonlinear elastic dynamic analysis of 3-D semi-rigid steel frames including geometric and connection nonlinearities. The geometric nonlinearity is considered by using stability functions and updating geometric stiffness matrix. The nonlinear behavior of the steel beam-to-column connection is considered by using a zero-length independent connection element comprising of six translational and rotational springs. The nonlinear dynamic equilibrium equations are solved by the Newmark numerical integration method. The nonlinear time-history analysis results are compared with those of previous studies and commercial SAP2000 software to verify the accuracy and efficiency of the proposed procedure.

Keywords: Geometric nonlinearity, nonlinear time-historyanalysis, semi-rigid connection, stability functions.

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Authors: Do Phuc, Nguyen Thi Kim Phung

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In this paper, we represent protein structure by using graph. A protein structure database will become a graph database. Each graph is represented by a spectral vector. We use Jacobi rotation algorithm to calculate the eigenvalues of the normalized Laplacian representation of adjacency matrix of graph. To measure the similarity between two graphs, we calculate the Euclidean distance between two graph spectral vectors. To cluster the graphs, we use M-tree with the Euclidean distance to cluster spectral vectors. Besides, M-tree can be used for graph searching in graph database. Our proposal method was tested with graph database of 100 graphs representing 100 protein structures downloaded from Protein Data Bank (PDB) and we compare the result with the SCOP hierarchical structure.

Keywords: Eigenvalues, m-tree, graph database, protein structure, spectra graph theory.

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Authors: B. J. Sandesh, Saurabha Jirgi, S. Vidya, Prakash Eljer, Gowri Srinivasa

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In this paper, we propose a framework to help users to search and retrieve the portions in the lecture video of their interest. This is achieved by temporally segmenting and indexing the lecture video using the topic keywords. We use transcribed text from the video and documents relevant to the video topic extracted from the web for this purpose. The keywords for indexing are found by applying the non-negative matrix factorization (NMF) topic modeling techniques on the web documents. Our proposed technique first creates indices on the transcribed documents using the topic keywords, and these are mapped to the video to find the start and end time of the portions of the video for a particular topic. This time information is stored in the index table along with the topic keyword which is used to retrieve the specific portions of the video for the query provided by the users.

Keywords: Video indexing and retrieval, lecture videos, content based video search, multimodal indexing.

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Authors: Ashutosh Kumar Singh, Anand Mohan

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This paper deals with efficient computation of probability coefficients which offers computational simplicity as compared to spectral coefficients. It eliminates the need of inner product evaluations in determination of signature of a combinational circuit realizing given Boolean function. The method for computation of probability coefficients using transform matrix, fast transform method and using BDD is given. Theoretical relations for achievable computational advantage in terms of required additions in computing all 2n probability coefficients of n variable function have been developed. It is shown that for n ≥ 5, only 50% additions are needed to compute all probability coefficients as compared to spectral coefficients. The fault detection techniques based on spectral signature can be used with probability signature also to offer computational advantage.

Keywords: Binary Decision Diagrams, Spectral Coefficients, Fault detection

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Authors: E. Hajizadeh, H. Garmabi

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Toughening of polyamide 6 (PA6)/ Nanoclay (NC) nanocomposites with styrene-ethylene/butadiene-styrene copolymer (SEBS) using maleated styrene-ethylene/butadiene-styrene copolymer (mSEBS)/ as a compatibilizer were investigated by blending them in a co-rotating twin-screw extruder. Response surface method of experimental design was used for optimizing the material and processing parameters. Effect of four factors, including SEBS, mSEBS and NC contents as material variables and order of mixing as a processing factor, on toughness of hybrid nanocomposites were studied. All the prepared samples showed ductile behavior and low temperature Izod impact toughness of some of the hybrid nanocomposites demonstrated 900% improvement compared to the PA6 matrix while the modulus showed maximum enhancement of 20% compared to the pristine PA6 resin.

Keywords: Hybrid nanocomposites, PA6, SEBS rubber, toughness.

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Authors: H. M. Mobarak, Helen Wu, Keqin Xiao

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In this work, a three-dimensional finite element model was developed to investigate the crack breathing behavior at different crack locations considering the effect of unbalance force. A two-disk rotor with a crack is simulated using ABAQUS. The duration of each crack status (open, closed and partially open/closed) during a full shaft rotation was examined to analyse the crack breathing behavior. Unbalanced shaft crack breathing behavior was found to be different at different crack locations. The breathing behavior of crack along the shaft length is divided into different regions depending on the unbalance force and crack location. The simulated results in this work can be further utilised to obtain the time-varying stiffness matrix of the cracked shaft element under the influence of unbalance force.

Keywords: Crack breathing, crack location, slant crack, unbalance force, rotating shaft.

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Authors: Lina Fernandes, G. N. Nayak, D. Ilangovan

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Two short sediment cores collected from mangrove areas of Manori and Thane creeks along Mumbai coast were analysed for sediment composition and metals (Fe, Mn, Cu, Pb, Co, Ni, Zn, Cr and V). The statistical analysis of Pearson correlation matrix proved that there is a significant relationship between metal concentration and finer grain size in Manori creek while poor correlation was observed in Thane creek. Based on the enrichment factor, the present metal to background metal ratios clearly reflected maximum enrichment of Cu and Pb in Manori creek and Mn in Thane creek. Geoaccumulation index calculated indicate that the study area is unpolluted with respect to Fe, Mn, Co, Ni, Zn and Cr in both the cores while moderately polluted with Cu and Pb in Manori creek. Based on contamination degree, both the core sediments were found to be considerably contaminated with metals.

Keywords: Creek, Igeo, Mumbai, trace metals

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Authors: B.Tavakoly Sany, A.H .Sulaiman, GH. Monazami, A. Salleh

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Eight heavy metals (Cu, Cr, Zn, Hg, Pb, Cd, Ni and As) were analyzed in sediment samples in the dry and wet seasons from November 2009 to October 2010 in West Port of Peninsular Malaysia. The heavy metal concentrations (mg/kg dry weight) were ranged from 23.4 to 98.3 for Zn, 22.3 to 80 for Pb, 7.4 to 27.6 Cu, 0.244 to 3.53 for Cd, 7.2 to 22.2 for Ni, 20.2 to 162 for As, 0.11 to 0.409 for Hg and 11.5 to 61.5 for Cr. Metals concentrations in dry season were higher than the rainy season except in cupper and chromium. Analysis of variance with Statistical Analysis System (SAS) shows that the mean concentration of metals in the two seasons (α level=0.05) are not significantly different which shows that the metals were held firmly in the matrix of sediment. Also there are significant differences between control point station with other stations. According to the Interim Sediment Quality guidelines (ISQG), the metal concentrations are moderately polluted, except in arsenic which shows the highest level of pollution.

Keywords: Heavy metals, sediment quality guidelines, west port.

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Authors: M. E. Ali Mohsin, Agus Arsad, Othman Y. Alothman

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This study explains the influence of secondary filler on the dispersion of carbon nanotube (CNT) reinforced high density polyethylene (HDPE) nanocomposites (CNT/HDPE). In order to understand the mixed-fillers system, Montmorillonite (MMT) was added to CNT/HDPE nanocomposites. It was followed by investigating their effect on the thermal, mechanical and morphological properties of the aforesaid nanocomposite. Incorporation of 3 wt% each of MMT into CNT/HDPE nanocomposite resulted to the increased values for the tensile and flexural strength, as compared to the pure HDPE matrix. The thermal analysis result showed improved thermal stability of the formulated nanocomposites. Transmission electron microscopy (TEM) images revealed that larger aggregates of CNTs were disappeared upon addition of these two components leading to the enhancement of thermo-mechanical properties for such composites.

Keywords: Secondary filler, Montmorillonite, Carbon nanotube, nanocomposite.

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Authors: P. Gnanasekar, A. Nagappan, S. Sharavanan, O. Saravanan, D. Vinodkumar, T. Elayabharathi, G. Karthik

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In this paper we present the deep study about the Bio- Medical Images and tag it with some basic extracting features (e.g. color, pixel value etc). The classification is done by using a nearest neighbor classifier with various distance measures as well as the automatic combination of classifier results. This process selects a subset of relevant features from a group of features of the image. It also helps to acquire better understanding about the image by describing which the important features are. The accuracy can be improved by increasing the number of features selected. Various types of classifications were evolved for the medical images like Support Vector Machine (SVM) which is used for classifying the Bacterial types. Ant Colony Optimization method is used for optimal results. It has high approximation capability and much faster convergence, Texture feature extraction method based on Gabor wavelets etc..

Keywords: ACO Ant Colony Optimization, Correlogram, CCM Co-Occurrence Matrix, RTS Rough-Set theory

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Authors: Sara A.Yones, Ahmed S. Moussa

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Quantitative measurements of tumor in general and tumor volume in particular, become more realistic with the use of Magnetic Resonance imaging, especially when the tumor morphological changes become irregular and difficult to assess by clinical examination. However, tumor volume estimation strongly depends on the image segmentation, which is fuzzy by nature. In this paper a fuzzy approach is presented for tumor volume segmentation based on the fuzzy connectedness algorithm. The fuzzy affinity matrix resulting from segmentation is then used to estimate a fuzzy volume based on a certainty parameter, an Alpha Cut, defined by the user. The proposed method was shown to highly affect treatment decisions. A statistical analysis was performed in this study to validate the results based on a manual method for volume estimation and the importance of using the Alpha Cut is further explained.

Keywords: Alpha Cut, Fuzzy Connectedness, Magnetic Resonance Imaging, Tumor volume estimation.

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Authors: Tan K. B., Norhashidah Hj. M. Ali

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In this article, we aim to discuss the formulation of two explicit group iterative finite difference methods for time-dependent two dimensional Burger-s problem on a variable mesh. For the non-linear problems, the discretization leads to a non-linear system whose Jacobian is a tridiagonal matrix. We discuss the Newton-s explicit group iterative methods for a general Burger-s equation. The proposed explicit group methods are derived from the standard point and rotated point Crank-Nicolson finite difference schemes. Their computational complexity analysis is discussed. Numerical results are given to justify the feasibility of these two proposed iterative methods.

Keywords: Standard point Crank-Nicolson (CN), Rotated point Crank-Nicolson (RCN), Explicit Group (EG), Explicit Decoupled Group (EDG).

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Authors: F. Sarabchi, M. E. Kalantari

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In this paper, Optimum adaptive loading algorithms are applied to multicarrier system with Space-Time Block Coding (STBC) scheme associated with space-time processing based on singular-value decomposition (SVD) of the channel matrix over Rayleigh fading channels. SVD method has been employed in MIMO-OFDM system in order to overcome subchannel interference. Chaw-s and Compello-s algorithms have been implemented to obtain a bit and power allocation for each subcarrier assuming instantaneous channel knowledge. The adaptive loaded SVD-STBC scheme is capable of providing both full-rate and full-diversity for any number of transmit antennas. The effectiveness of these techniques has demonstrated through the simulation of an Adaptive loaded SVDSTBC system, and the comparison shown that the proposed algorithms ensure better performance in the case of MIMO.

Keywords: OFDM, MIMO, SVD, STBC, Adaptive Loading.

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Authors: F. Rezaie Moghaddam, J. Amani, T. Rezaie Moghaddam

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Many computational techniques were applied to solution of heat conduction problem. Those techniques were the finite difference (FD), finite element (FE) and recently meshless methods. FE is commonly used in solution of equation of heat conduction problem based on the summation of stiffness matrix of elements and the solution of the final system of equations. Because of summation process of finite element, convergence rate was decreased. Hence in the present paper Cellular Automata (CA) approach is presented for the solution of heat conduction problem. Each cell considered as a fixed point in a regular grid lead to the solution of a system of equations is substituted by discrete systems of equations with small dimensions. Results show that CA can be used for solution of heat conduction problem.

Keywords: Heat conduction, Cellular automata, convergencerate, discrete system.

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Authors: Gkolfo I. Smani, Vasileios Megalooikonomou

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Influence and influence diffusion have been studied extensively in social networks. However, most existing literature on this task are limited on static networks, ignoring the fact that the interactions between users change over time. In this paper, the problem of maximizing influence diffusion in dynamic social networks, i.e., the case of networks that change over time is studied. The DM algorithm is an extension of Matrix Influence (MATI) algorithm and solves the Influence Maximization (IM) problem in dynamic networks and is proposed under the Linear Threshold (LT) and Independent Cascade (IC) models. Experimental results show that our proposed algorithm achieves a diffusion performance better by 1.5 times than several state-of-the-art algorithms and comparable results in diffusion scale with the Greedy algorithm. Also, the proposed algorithm is 2.4 times faster than previous methods.

Keywords: Influence maximization, dynamic social networks, diffusion, social influence.

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Authors: Syed Muhammad Aqil Burney, Tahseen Ahmed Jilani, C. Ardil

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Minimization methods for training feed-forward networks with Backpropagation are compared. Feedforward network training is a special case of functional minimization, where no explicit model of the data is assumed. Therefore due to the high dimensionality of the data, linearization of the training problem through use of orthogonal basis functions is not desirable. The focus is functional minimization on any basis. A number of methods based on local gradient and Hessian matrices are discussed. Modifications of many methods of first and second order training methods are considered. Using share rates data, experimentally it is proved that Conjugate gradient and Quasi Newton?s methods outperformed the Gradient Descent methods. In case of the Levenberg-Marquardt algorithm is of special interest in financial forecasting.

Keywords: Backpropagation algorithm, conjugacy condition, line search, matrix perturbation

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Authors: Kumar Neeraj, Ranjan Haldar

Abstract:

Polyaniline is an indispensible component in lightemitting devices (LEDs), televisions, cellular telephones, automotive, corrosion-resistant coatings, actuators etc. The electrical conductivity properties was found be increased by introduction of metal nano particles. In the present study, an attempt has been made to utilize platinum nano particles to achieve the improved electrical properties. Polyaniline and Pt-polyaniline composite are synthesized by electrochemical routes. X-ray diffractometer confirms the amorphous nature of polyaniline. The Bragg’s diffraction peaks correspond to platinum nanoparticles in Pt-polyaniline composite and thermogravimetric analyzer indicates its decomposition at certain temperature. The Scanning Electron Micrographs of colloidal platinum nanoparticles were spherical, uniform shape in the composite. The current-voltage (I-V) characteristics of the PANI and composites were also studied which indicate a significant decreasing resistivity than PANI-Platinum after introduction of pt nanoparticles in the matrix of polyaniline (PANI).

Keywords: Polyaniline, XRD and Platinum Nanoparticles.

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Authors: Kaveh PourAkbar Saffar, Nima JamilPour, Ahmad Raeisi Najafi, Gholamreza Rouhi, Ahmad Reza Arshi, Abdolhossein Fereidoon

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The presence of chemical bonding between functionalized carbon nanotubes and matrix in carbon nanotube reinforced composites is modeled by elastic beam elements representing covalent bonding characteristics. Neglecting other reinforcing mechanisms in the composite such as relatively weak interatomic Van der Waals forces, this model shows close results to the Rule of Mixtures model-s prediction for effective Young-s modulus of a Representative Volume Element of composite for small volume fractions (~1%) and high aspect ratios (L/D>200) of CNTs.

Keywords: Beam Element, Carbon Nanotube Reinforced Composite, Cross-link, Young's modulus.

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Authors: Marzuki Khalid, RubiyahYusof, AnisSalwaMohdKhairuddin

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An automated wood recognition system is designed to classify tropical wood species.The wood features are extracted based on two feature extractors: Basic Grey Level Aura Matrix (BGLAM) technique and statistical properties of pores distribution (SPPD) technique. Due to the nonlinearity of the tropical wood species separation boundaries, a pre classification stage is proposed which consists ofKmeans clusteringand kernel discriminant analysis (KDA). Finally, Linear Discriminant Analysis (LDA) classifier and KNearest Neighbour (KNN) are implemented for comparison purposes. The study involves comparison of the system with and without pre classification using KNN classifier and LDA classifier.The results show that the inclusion of the pre classification stage has improved the accuracy of both the LDA and KNN classifiers by more than 12%.

Keywords: Tropical wood species, nonlinear data, featureextractors, classification

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Authors: Chatwarin Poochai, Puttichai Pae-on, Thirawudh Pongpayoon

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The commercial white tyres are usually used for forklifts in food and medicine industries. Conventionally, silica is used as reinforcement in the tyres. However, the adhesion between silica particles and rubber is remarkably poor. To improve the problem of adhesion and hence enhance wear resistance, modification of silica surface is one of the solutions. In this work, the natural rubber compound blending with polyisoprene-coated silica prepared by admicellar polymerization technique was studied to compare with the natural rubber compound of unmodified silica. The surface characterization of modified silica was also examined by SEM, FTIR, and TGA. The results show that polyisoprene-coated silica/natural rubber compound gave better overall mechanical properties, especially wear resistance with the improvement of the adhesion between silica and natural rubber matrix that can be seen in the SEM micrograph.

Keywords: White tyre, admicellar polymerization, modified silica, wear resistance.

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Authors: Debabrata Pradhan, Dong J. Kim, Jong G. Ahn, Seoung W. Lee

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Spent petroleum catalyst from Korean petrochemical industry contains trace amount of metals such as Ni, V and Mo. Therefore an attempt was made to recover those trace metal using bioleaching process. Different leaching parameters such as Fe(II) concentration, pulp density, pH, temperature and particle size of spent catalyst particle were studied to evaluate their effects on the leaching efficiency. All the three metal ions like Ni, V and Mo followed dual kinetics, i.e., initial faster followed by slower rate. The percentage of leaching efficiency of Ni and V were higher than Mo. The leaching process followed a diffusion controlled model and the product layer was observed to be impervious due to formation of ammonium jarosite (NH4)Fe3(SO4)2(OH)6. In addition, the lower leaching efficiency of Mo was observed due to a hydrophobic coating of elemental sulfur over Mo matrix in the spent catalyst.

Keywords: Bioleaching, diffusion control, shrinking core, spentpetroleum catalyst.

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Authors: Ryszard Matysiak, Grzegorz Kamieniarz

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The deterministic quantum transfer-matrix (QTM) technique and its mathematical background are presented. This important tool in computational physics can be applied to a class of the real physical low-dimensional magnetic systems described by the Heisenberg hamiltonian which includes the macroscopic molecularbased spin chains, small size magnetic clusters embedded in some supramolecules and other interesting compounds. Using QTM, the spin degrees of freedom are accurately taken into account, yielding the thermodynamical functions at finite temperatures. In order to test the application for the susceptibility calculations to run in the parallel environment, the speed-up and efficiency of parallelization are analyzed on our platform SGI Origin 3800 with p = 128 processor units. Using Message Parallel Interface (MPI) system libraries we find the efficiency of the code of 94% for p = 128 that makes our application highly scalable.

Keywords: Deterministic simulations, low-dimensional magnets, modeling of complex systems, parallelization.

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Authors: Marque Adrien, Delahaye Daniel, Marechal Pierre, Berry Isabelle

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Wind direction and uncertainty are crucial in aircraft or unmanned aerial vehicle trajectories. By computing wind covariance matrices on each spatial grid point, these spatial grids can be defined as images with symmetric positive definite matrix elements. A data pre-processing step, a specific convolution, a specific max-pooling, and specific flatten layers are implemented to process such images. Then, the neural network is applied to spatial grids, whose elements are wind covariance matrices, to solve classification problems related to the feasibility of unmanned aerial vehicles based on wind direction and wind uncertainty.

Keywords: Wind direction, uncertainty level, Unmanned Aerial Vehicle, convolution neural network, SPD matrices.

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Authors: Paulo Gomes, Adelaide Figueiredo

Abstract:

We consider n individuals described by p standardized variables, represented by points of the surface of the unit hypersphere S_n-1. For a previous choice of n individuals we suppose that the set of observables variables comes from a mixture of bipolar Watson distribution defined on the hypersphere. EM and Dynamic Clusters algorithms are used for identification of such mixture. We obtain estimates of parameters for each Watson component and then a partition of the set of variables into homogeneous groups of variables. Additionally we will present a factor analysis model where unobservable factors are just the maximum likelihood estimators of Watson directional parameters, exactly the first principal component of data matrix associated to each group previously identified. Such alternative model it will yield us to directly interpretable solutions (simple structure), avoiding factors rotations.

Keywords: Dynamic Clusters algorithm, EM algorithm, Factor analysis model, Hierarchical Clustering, Watson distribution.

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Authors: O. Hassoon, M. Tarfoui, A. El Malk

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Fiber Bragg optic sensor is embedded in composite material to detect and monitor the damage that occurs in composite structures. In this paper, we deal with the mode-Ι delamination to determine the material strength to crack propagation, using the coupling mode theory and T-matrix method to simulate the FBGs spectrum for both uniform and non-uniform strain distribution. The double cantilever beam test is modeled in FEM to determine the longitudinal strain. Two models are implemented, the first is the global half model, and the second is the sub-model to represent the FBGs with higher refined mesh. This method can simulate damage in composite structures and converting strain to a wavelength shifting in the FBG spectrum.

Keywords: Fiber Bragg grating, Delamination detection, DCB, FBG spectrum, Structure health monitoring.

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Authors: N. Mahmoudi

Abstract:

Composite materials have important assets compared to traditional materials. They bring many functional advantages: lightness, mechanical resistance and chemical, etc. In the present study we examine the effect of a circular central notch and a precrack on the tensile fracture of two woven composite materials. The tensile tests were applied to a standardized specimen, notched and a precarcked (orientation of the crack 0°, 45° and 90°). These tensile tests were elaborated according to an experimental planning design of the type 23.31 requiring 24 experiments with three repetitions. By the analysis of regression, we obtained a mathematical model describing the maximum load according to the influential parameters (hole diameter, precrack length, angle of a precrack orientation). The specimens precracked at 90° have a better behavior than those having a precrack at 45° and still better than those having of the precracks oriented at 0°. In addition the maximum load is inversely proportional to the notch size.

Keywords: Polymer matrix, Glasses, Fracture.

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Authors: K. Khelil, H. Ammar, K. Saouchi

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Fiber Bragg Grating (FBG) structure is an periodically modulated optical fiber. It acts as a selective filter of wavelength whose reflected peak is called Bragg wavelength and it depends on the period of the fiber and the refractive index. The simulation of FBG is based on solving the Coupled Mode Theory equation by using the Transfer Matrix Method which is carried out using MATLAB. It is found that spectral reflectivity is shifted when the change of temperature and strain is uniform. Under non-uniform temperature or strain perturbation, the spectrum is both shifted and destroyed. In case of transverse loading, reflectivity spectrum is split into two peaks, the first is specific to X axis, and the second belongs to Y axis. FBGs are used in civil engineering to detect perturbations applied to buildings.

Keywords: Bragg wavelength, coupled mode theory, optical fiber, temperature measurement.

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Authors: Athanasios Theoharidis, Thomas Kamalakis, Ioannis Neokosmidis, Thomas Sphicopoulos

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In this paper, the application of the Mode Matching (MM) method in the case of photonic crystal waveguide discontinuities is presented. The structure under consideration is divided into a number of cells, which supports a number of guided and evanescent modes. These modes can be calculated numerically by an alternative formulation of the plane wave expansion method for each frequency. A matrix equation is then formed relating the modal amplitudes at the beginning and at the end of the structure. The theory is highly efficient and accurate and can be applied to study the transmission sensitivity of photonic crystal devices due to fabrication tolerances. The accuracy of the MM method is compared to the Finite Difference Frequency Domain (FDFD) and the Adjoint Variable Method (AVM) and good agreement is observed.

Keywords: Optical Communications, Integrated Optics, Photonic Crystals, Optical Waveguide Discontinuities.

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