Search results for: Order reduction

Authors: Hossein Kabir, Mojtaba Sadeghi

Abstract:

Finding the linear and nonlinear responses of a typical single-degree-of-freedom system (SDOF) is always being regarded as a time-consuming process. This study attempts to provide modifications in the renowned Newmark method in order to make it more time efficient than it used to be and make it more accurate by modifying the system in its own non-linear state. The efficacy of the presented method is demonstrated by assigning three base excitations such as Tabas 1978, El Centro 1940, and MEXICO CITY/SCT 1985 earthquakes to a SDOF system, that is, SDOF, to compute the strength reduction factor, yield pseudo acceleration, and ductility factor.

Keywords: Single-degree-of-freedom system, linear acceleration method, nonlinear excited system, equivalent displacement method.

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Authors: Sachin Bhalekar

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In this paper we propose a necessary condition for the existence of chaos in delay differential equations of fractional order. To explain the proposed theory, we discuss fractional order Liu system and financial system involving delay.

Keywords: Caputo derivative, delay, stability, chaos.

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Authors: A. Zolfagharian, M.Z. Md. Zain, A. R. AbuBakar, M. Hussein

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Application of flexible structures has been significantly, increased in industry and aerospace missions due to their contributions and unique advantages over the rigid counterparts. In this paper, vibration analysis of a flexible structure i.e., automobile wiper blade is investigated and controlled. The wiper generates unwanted noise and vibration during the wiping the rain and other particles on windshield which may cause annoying noise in different ranges of frequency. A two dimensional analytical modeled wiper blade whose model accuracy is verified by numerical studies in literature is considered in this study. Particle swarm optimization (PSO) is employed in alliance with input shaping (IS) technique in order to control or to attenuate the amplitude level of unwanted noise/vibration of the wiper blade.

Keywords: Input shaping, noise reduction, particle swarmoptimization, wiper blade

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Authors: V. Prasannavenkadesan, A. Elango, S. Chockalingam

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During machining process, chatter is an unavoidable phenomenon. Boring bars possess the cantilever shape and due to this, it is subjected to chatter. The adverse effect of chatter includes the increase in temperature which will leads to excess tool wear. To overcome these problems, in this investigation, Cartridge brass (Cu – 70% and Zn – 30%) is passively fixed on the boring bar and also clearance is provided in order to reduce the displacement, tool wear and cutting temperature. A conventional all geared lathe is attached with vibrometer and pyrometer is used to measure the displacement and temperature. The influence of input parameters such as cutting speed, depth of cut and clearance on temperature, tool wear and displacement are investigated for various cutting conditions. From the result, the optimum conditions to obtain better damping in boring process for chatter reduction is identified.

Keywords: Boring, chatter, mass damping, passive damping.

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Authors: A. M. Sagir

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Discrete linear multistep block method of uniform order for the solution of first order initial value problems (IVP­s­) in ordinary differential equations (ODE­s­) is presented in this paper. The approach of interpolation and collocation approximation are adopted in the derivation of the method which is then applied to first order ordinary differential equations with associated initial conditions. The continuous hybrid formulations enable us to differentiate and evaluate at some grids and off – grid points to obtain four discrete schemes, which were used in block form for parallel or sequential solutions of the problems. Furthermore, a stability analysis and efficiency of the block method are tested on ordinary differential equations, and the results obtained compared favorably with the exact solution.

Keywords: Block Method, First Order Ordinary Differential Equations, Hybrid, Self starting.

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Authors: Takashi Sakai, Itaru Kumisawa

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Because die drawing requires insertion of a die, a plug, or a mandrel, higher precision and efficiency are demanded for drawing equipment for a tube having smaller diameter. Manufacturing of such tubes is also accompanied by problems such as cracking and fracture. We specifically examine dieless drawing, which is less affected by these drawing-related difficulties. This deformation process is governed by a similar principle to that of reduction in diameter when pulling a heated glass tube. We conducted dieless drawing of SUS304 stainless steel microtubes under various conditions with three factor parameters of heating temperature, area reduction, and drawing speed. We used SEM-EBSD to observe the processing condition effects on microstructural elements. As the result of this study, crystallographic orientation of microtube is clear by using SEM-EBSD analysis.

Keywords: Microtube, dieless drawing, IPF, inverse pole figure, GOS, grain orientation spread, crystallographic analysis.

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Authors: Kostas Siozios, Dimitrios Soudris, Antonios Thanailakis

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Modern applications realized onto FPGAs exhibit high connectivity demands. Throughout this paper we study the routing constraints of Virtex devices and we propose a systematic methodology for designing a novel general-purpose interconnection network targeting to reconfigurable architectures. This network consists of multiple segment wires and SB patterns, appropriately selected and assigned across the device. The goal of our proposed methodology is to maximize the hardware utilization of fabricated routing resources. The derived interconnection scheme is integrated on a Virtex style FPGA. This device is characterized both for its high-performance, as well as for its low-energy requirements. Due to this, the design criterion that guides our architecture selections was the minimal Energy×Delay Product (EDP). The methodology is fully-supported by three new software tools, which belong to MEANDER Design Framework. Using a typical set of MCNC benchmarks, extensive comparison study in terms of several critical parameters proves the effectiveness of the derived interconnection network. More specifically, we achieve average Energy×Delay Product reduction by 63%, performance increase by 26%, reduction in leakage power by 21%, reduction in total energy consumption by 11%, at the expense of increase of channel width by 20%.

Keywords: Design Methodology, FPGA, Interconnection, Low-Energy, High-Performance, CAD tool.

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Authors: A. Suleng, T. Jelstad Olsen, J. Šindler, P. Bárta

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A dead leg is a typical subsea production system component. CFD is required to model heat transfer within the dead leg. Unfortunately its solution is time demanding and thus not suitable for fast prediction or repeated simulations. Therefore there is a need to create a thermal FEA model, mimicking the heat flows and temperatures seen in CFD cool down simulations. This paper describes the conventional way of tuning and a new automated way using parametric model order reduction (PMOR) together with an optimization algorithm. The tuned FE analyses replicate the steady state CFD parameters within a maximum error in heat flow of 6 % and 3 % using manual and PMOR method respectively. During cool down, the relative error of the tuned FEA models with respect to temperature is below 5% comparing to the CFD. In addition, the PMOR method obtained the correct FEA setup five times faster than the manually tuned FEA.

Keywords: CFD, convective heat, FEA, model tuning, subseaproduction

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Authors: M. Jouki, N. Khazaei , A. Kalbasi , H. Tavakolipour, S. Rajabifar, F. Motamedi. Sedeh, A. Jouki

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Irradiation is considered one of the most efficient technological processes for the reduction of microorganisms in food. It can be used to improve the safety of food products, and to extend their shelf lives. The aim of this study was to evaluate the effects of gamma irradiation for improvement of saffron shelf life. Samples were treated with 0 (none irradiated), 1.0, 2.0, 3.0 and 4.0 kGy of gamma irradiation and held for 2 months. The control and irradiated samples were underwent microbial analysis, chemical characteristics and sensory evaluation at 30 days intervals. Microbial analysis indicated that irradiation had a significant effect (P < 0.05) on the reduction of microbial loads. There was no significant difference in sensory quality and chemical characteristics during storage in saffron.

Keywords: gamma irradiation, saffron, microbes, contamination.

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Authors: Bubaker M. F. Bushofa, Abdel Hafez A. Azab

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This article is based on the technique which is called Discrete Parameter Tracking (DPT). First introduced by A. A. Azab [8] which is applicable for less order reference model. The order of the reference model is (n-l) and n is the number of the adjustable parameters in the physical plant. The technique utilizes a modified gradient method [9] where the knowledge of the exact order of the nonadaptive system is not required, so, as to eliminate the identification problem. The applicability of the mentioned technique (DPT) was examined through the solution of several problems. This article introduces the solution of a third order system with three adjustable parameters, controlled according to second order reference model. The adjustable parameters have great initial error which represent condition. Computer simulations for the solution and analysis are provided to demonstrate the simplicity and feasibility of the technique.

Keywords: Adaptive Control System, Discrete Parameter Tracking, Discrete Time Model.

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Authors: Ivandic Kreso, Spiranec Miljenko, Kavur Boris, Strelec Stjepan

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This article discusses the possibility of using dilatometer tests (DMT) together with in situ seismic tests (MASW) in order to get the shape of G-g degradation curve in cohesive soils (clay, silty clay, silt, clayey silt and sandy silt). MASW test provides the small soil stiffness (G_o from v_s) at very small strains and DMT provides the stiffness of the soil at ‘work strains’ (M_DMT). At different test locations, dilatometer shear stiffness of the soil has been determined by the theory of elasticity. Dilatometer shear stiffness has been compared with the theoretical G-g degradation curve in order to determine the typical range of shear deformation for different types of cohesive soil. The analysis also includes factors that influence the shape of the degradation curve (G-g) and dilatometer modulus (M_DMT), such as the overconsolidation ratio (OCR), plasticity index (IP) and the vertical effective stress in the soil (s_vo'). Parametric study in this article defines the range of shear strain g_DMT and G_DMT/G_o relation depending on the classification of a cohesive soil (clay, silty clay, clayey silt, silt and sandy silt), function of density (loose, medium dense and dense) and the stiffness of the soil (soft, medium hard and hard). The article illustrates the potential of using MASW and DMT to obtain G-g degradation curve in cohesive soils.

Keywords: Dilatometer testing, MASW testing, shear wave, soil stiffness, stiffness reduction, shear strain.

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Authors: Anamika Gupta, Naveen Kumar, Vasudha Bhatnagar

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This paper focuses on analyzing medical diagnostic data using classification rules in data mining and context reduction in formal concept analysis. It helps in finding redundancies among the various medical examination tests used in diagnosis of a disease. Classification rules have been derived from positive and negative association rules using the Concept lattice structure of the Formal Concept Analysis. Context reduction technique given in Formal Concept Analysis along with classification rules has been used to find redundancies among the various medical examination tests. Also it finds out whether expensive medical tests can be replaced by some cheaper tests.

Keywords: Data Mining, Formal Concept Analysis, Medical Data, Negative Classification Rules.

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Authors: Ei Ei Phyu, Kyaw Myo Lin, Thin Thin Moe

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The paper presents an approach to improve the reliability and reduce line losses of practical distribution system applying network reconfiguration. The change of the topology redirects the power flow within the distribution network to obtain better performance of the system. Practical distribution network (Pyigyitagon Industrial Zone (I)) is used as the case study network. The detailed calculations of the reliability indices are done by using analytical method and power flow calculation is performed by Newton-Rephason solver. The comparison of various network reconfiguration techniques are described with respect to power loss and reliability index levels. Finally, the optimal reconfigured network is selected among difference cases based on the two factors: the most reliable network and the least loss minimization.

Keywords: Distribution system reliability, loss reduction, network reconfiguration, reliability enhancement, reliability indices.

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Authors: Atisthan Wuttimanop, Suchada Rianmora

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To support design process for launching the product on time, reverse engineering (RE) process has been introduced for quickly generating 3D CAD model from its physical object. The accuracy of the 3D CAD model depends upon the data acquisition technique selected, contact or non-contact methods. In order to reduce times used for acquiring surface and eliminating noises, the automated selective acquisition system has been developed and presented in this research as the alternative channel for non-contact acquisition technique where the data is selectively and locally scanned contour by contour without performing data reduction process. The results present as the organized contour points which are directly used to generate 3D virtual model. The comparison between the proposed technique and another non-contact scanning technique has been presented and discussed.

Keywords: Automated selective acquisition system, Non-contact acquisition, Reverse engineering, 3D scanners.

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Authors: Zita Šereš, Dragana Šoronja Simović, Ljubica Dokić, Lidietta Giorno, Biljana Pajin, Cecilia Hodur, Nikola Maravić

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Membrane technology is convenient for separation of suspended solids, colloids and high molecular weight materials that are present. The idea is that the waste stream from edible oil industry, after the separation of oil by using skimmers is subjected to microfiltration and the obtained permeate can be used again in the production process. The wastewater from edible oil industry was used for the microfiltration. For the microfiltration of this effluent a tubular membrane was used with a pore size of 200 nm at transmembrane pressure in range up to 3 bar and in range of flow rate up to 300 L/h. Box–Behnken design was selected for the experimental work and the responses considered were permeate flux and chemical oxygen demand (COD) reduction. The reduction of the permeate COD was in the range 40-60% according to the feed. The highest permeate flux achieved during the process of microfiltration was 160 L/m²h.

Keywords: Ceramic membrane, edible oil, microfiltration, wastewater.

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Authors: Chengjie Wang, Li Yin, Chuanmin Wang

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This paper presents a physics-based model for the high-voltage fast recovery diodes. The model provides a good trade-off between reverse recovery time and forward voltage drop realized through a combination of lifetime control and emitter efficiency reduction techniques. The minority carrier lifetime can be extracted from the reverse recovery transient response and forward characteristics. This paper also shows that decreasing the amount of the excess carriers stored in the drift region will result in softer characteristics which can be achieved using a lower doping level. The developed model is verified by experiment and the measurement data agrees well with the model.

Keywords: Emitter efficiency, lifetime control, P-i-N diode, physics-based model

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Authors: Josefina Barnachea Janier, Radzuan B. Razali, Afza Shafie, Brahim Belhaouari Samir

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In the Enhanced Oil Recovery (EOR) method, use of Carbon dioxide flooding whereby CO₂ is injected into an oil reservoir to increase output when extracting oil resulted significant recovery worldwide. The carbon dioxide function as a pressurizing agent when mixed into the underground crude oil will reduce its viscosity and will enable a rapid oil flow. Despite the CO₂’s advantage in the oil recovery, it may result to asphaltene precipitation a problem that will cause the reduction of oil produced from oil wells. In severe cases, asphaltene precipitation can cause costly blockages in oil pipes and machinery. This paper presents reviews of several studies done on mathematical modeling of asphaltene precipitation. The synthesized result from several researches done on this topic can be used as guide in order to better understand asphaltene precipitation. Likewise, this can be used as initial reference for students, and new researchers doing study on asphaltene precipitation.

Keywords: Asphaltene precipitation, crude oil, carbon dioxide flooding, enhanced oil recovery.

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Authors: M. Tahar Bouzaher

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This numerical study aims to develop a coupled, passive and active control strategy of the flow around a cylinder of diameter D, and Re=4000. The strategy consists to put a cylindrical rod in front of a deforming cylinder. The quasi- elliptical deformation of cylinder follow a sinusoidal law in order to reduce the drag force. To analyze the evolution of unsteady vortices, the Large Eddy Simulation approach is used in this 2D simulation, carried out using ANSYS – Fluent. The movement of deformation is reproduced using an internal subroutine, introduced in the form of a User Defined Function UDF. Two diameters of the rod were tested for a rod placed at a distance L = 3 ×d, with an amplitudes of deformation A = 5%, A = 25% and A = 50% of the cylinder diameter, the frequency of deformation take the values fd = 1fn, 5fn and 8fn, which fn represents the naturel vortex shedding frequency. The results show substantial changes in the flow behavior and for a rod of 6mm (1% D) with amplitude A = 25%, and with a 2fn frequency, drag reduction of 60% was recorded.

Keywords: CFD, Flow separation, control, Boundary layer, rod, Cylinder.

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Authors: S. Boria

Abstract:

In the last years, the crashworthiness of an automotive body structure can be improved, since the beginning of the design stage, thanks to the development of specific optimization tools. It is well known how the finite element codes can help the designer to investigate the crashing performance of structures under dynamic impact. Therefore, by coupling nonlinear mathematical programming procedure and statistical techniques with FE simulations, it is possible to optimize the design with reduced number of analytical evaluations. In engineering applications, many optimization methods which are based on statistical techniques and utilize estimated models, called meta-models, are quickly spreading. A meta-model is an approximation of a detailed simulation model based on a dataset of input, identified by the design of experiments (DOE); the number of simulations needed to build it depends on the number of variables. Among the various types of meta-modeling techniques, Kriging method seems to be excellent in accuracy, robustness and efficiency compared to other ones when applied to crashworthiness optimization. Therefore the application of such meta-model was used in this work, in order to improve the structural optimization of a bumper for a racing car in composite material subjected to frontal impact. The specific energy absorption represents the objective function to maximize and the geometrical parameters subjected to some design constraints are the design variables. LS-DYNA codes were interfaced with LS-OPT tool in order to find the optimized solution, through the use of a domain reduction strategy. With the use of the Kriging meta-model the crashworthiness characteristic of the composite bumper was improved.

Keywords: Composite material, crashworthiness, finite element analysis, optimization.

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Authors: D. H. Kim, Y. H. Kim, T. Kim

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A new method identifies coupled fluid-structure system with a reduced set of state variables is presented. Assuming that the structural model is known a priori either from an analysis or a test and using linear transformations between structural and aeroelastic states, it is possible to deduce aerodynamic information from sampled time histories of the aeroelastic system. More specifically given a finite set of structural modes the method extracts generalized aerodynamic force matrix corresponding to these mode shapes. Once the aerodynamic forces are known, an aeroelastic reduced-order model can be constructed in discrete-time, state-space format by coupling the structural model and the aerodynamic system. The resulting reduced-order model is suitable for constant Mach, varying density analysis.

Keywords: ROM (Reduced-Order Model), aero elasticity, AGARD 445.6 wing.

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Authors: A.K.M. Khaled Ahsan Talukder, Taibun Nessa, Kaushik Roy

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The weight constrained shortest path problem (WCSPP) is one of most several known basic problems in combinatorial optimization. Because of its importance in many areas of applications such as computer science, engineering and operations research, many researchers have extensively studied the WCSPP. This paper mainly concentrates on the reduction of total search space for finding WCSP using some existing Genetic Algorithm (GA). For this purpose, some controlled schemes of genetic operators are adopted on list chromosome representation. This approach gives a near optimum solution with smaller elapsed generation than classical GA technique. From further analysis on the matter, a new generalized schema theorem is also developed from the philosophy of Holland-s theorem.

Keywords: Genetic Algorithm, Evolutionary Optimization, Multi Objective Optimization, Non-linear Schema Theorem, WCSPP.

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Authors: Fathi N. Mayoof

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Rolling element bearings are widely used in industry, especially where high load capacity is required. The diagnosis of their conditions is essential matter for downtime reduction and saving cost of maintenance. Therefore, an intensive analysis of frequency spectrum of their faults must be carried out in order to determine the main reason of the fault. This paper focus on a beating phenomena observed in the waveform (time domain) of a cylindrical rolling element bearing. The beating frequencies were not related to any sources nearby the machine nor any other malfunctions (unbalance, misalignment ...etc). More investigation on the spike energy and the frequency spectrum indicated a problem with races of the bearing. Multi-harmonics of the fundamental defects frequencies were observed. Two of them were close to each other in magnitude those were the source of the beating phenomena.

Keywords: Bearing, beating, spike energy, vibration.

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Authors: Achuthsankar S. Nair, J. S. Jayasudha

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The ever increasing use of World Wide Web in the existing network, results in poor performance. Several techniques have been developed for reducing web traffic by compressing the size of the file, saving the web pages at the client side, changing the burst nature of traffic into constant rate etc. No single method was adequate enough to access the document instantly through the Internet. In this paper, adaptive hybrid algorithms are developed for reducing web traffic. Intelligent agents are used for monitoring the web traffic. Depending upon the bandwidth usage, user-s preferences, server and browser capabilities, intelligent agents use the best techniques to achieve maximum traffic reduction. Web caching, compression, filtering, optimization of HTML tags, and traffic dispersion are incorporated into this adaptive selection. Using this new hybrid technique, latency is reduced to 20 – 60 % and cache hit ratio is increased 40 – 82 %.

Keywords: Bandwidth, Congestion, Intelligent Agents, Prefetching, Web Caching.

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Authors: A. Guechi, M. Chegaar

Abstract:

This study investigates how AlGaAs/GaAs thin film solar cells perform under varying global solar spectrum due to the changes of environmental parameters such as the air mass and the atmospheric turbidity. The solar irradiance striking the solar cell is simulated using the spectral irradiance model SMARTS2 (Simple Model of the Atmospheric Radiative Transfer of Sunshine) for clear skies on the site of Setif (Algeria). The results show a reduction in the short circuit current due to increasing atmospheric turbidity, it is 63.09% under global radiation. However increasing air mass leads to a reduction in the short circuit current of 81.73%. The efficiency decreases with increasing atmospheric turbidity and air mass.

Keywords: AlGaAs/GaAs, Solar Cells, Environmental parameters, Spectral Variation, SMARTS.

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Authors: Robert Prucek, Aleš Panáček, Jan Filip, Libor Kvítek, Radek Zbořil

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The preparation of Cu nanoparticles (NPs) through the reduction of copper ions by sodium borohydride in the presence of sodium polyacrylate with a molecular weight of 1200 is reported. Cu NPs were synthesized at a concentration of copper salt equal to 2.5, 5, and 10 mM, and at a molar ratio of copper ions and monomeric unit of polyacrylate equal to 1:2. The as-prepared Cu NPs have diameters of about 2.5–3 nm for copper concentrations of 2.5 and 5 mM, and 6 nm for copper concentration of 10 mM. Depending on the copper salt concentration and concentration of additionally added polyacrylate to Cu particle dispersion, primarily formed NPs grow through the process of aggregation and/or coalescence into clusters and/or particles with a diameter between 20–100 nm. The amount of additionally added sodium polyacrylate influences the stability of Cu particles against air oxidation. The catalytic efficiency of the prepared Cu particles for the reduction of 4-nitrophenol is discussed.

Keywords: Copper, nanoparticles, sodium polyacrylate, catalyst, 4-nitrophenol.

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Authors: Fatima Zohra Bissaad, Farid Bounaceur, Nassima Behidj, Nadjiba Chebouti, Fatma Halouane, Bahia Doumandji-Mitiche

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Effect of Beauveria bassiana and Metarhizium anisopliae var. acridum on the 5^th instar nymphs of Schistocerca gregaria was studied in the laboratory. Infection by these both entomopathogenic fungi caused reduction in the hemolymph total protein. The average amounts of total proteins were 2.3, 2.07, 2.09 µg/100 ml of haemolymph in the control and M. anisopliae var. acridum, and B. bassiana based-treatments, respectively. Three types of haemocytes were recognized and identified as prohaemocytes, plasmatocytes and granulocytes. The treatment caused significant reduction in the total haemocyte count and in each haemocyte type on the 9^th day after its application.

Keywords: Beauveria bassiana, haemolymph picture, haemolymph protein, Metarhizium anisopliae var. acridum, Schistocerca gregaria.

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Authors: Mayank Shakya, Soundra Pandian. K. K

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A linear feedback shift register (LFSR) is proposed which targets to reduce the power consumption from within. It reduces the power consumption during testing of a Circuit Under Test (CUT) at two stages. At first stage, Control Logic (CL) makes the clocks of the switching units of the register inactive for a time period when output from them is going to be same as previous one and thus reducing unnecessary switching of the flip-flops. And at second stage, the LFSR reorders the test vectors by interchanging the bit with its next and closest neighbor bit. It keeps fault coverage capacity of the vectors unchanged but reduces the Total Hamming Distance (THD) so that there is reduction in power while shifting operation.

Keywords: Linear Feedback Shift Register, Total Hamming Distance, Fault Coverage, Control Logic

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Authors: L. Rozumová, J. Seidlerová

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The aim of this paper was to study the sorption properties of a blast furnace sludge used as the sorbent. The sorbent was utilized for reduction of content of lead and zinc ions. Sorbent utilized in this work was obtained from metallurgical industry from process of wet gas treatment in iron production. The blast furnace sludge was characterized by X-Ray diffraction, scanning electron microscopy, and XRFS spectroscopy. Sorption experiments were conducted in batch mode. The sorption of metal ions in the sludge was determined by correlation of adsorption isotherm models. The adsorption of lead and zinc ions was best fitted with Langmuir adsorption isotherms. The adsorption capacity of lead and zinc ions was 53.8 mg.g^-1 and 10.7 mg.g^-1, respectively. The results indicated that blast furnace sludge could be effectively used as secondary material and could be also employed as a low-cost alternative for the removal of heavy metals ions from wastewater.

Keywords: Blast furnace sludge, lead, zinc, sorption.

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Authors: H. Bouadi, M. Tadjine

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In this paper; we are interested in dynamic modelling of quadrotor while taking into account the high-order nonholonomic constraints as well as the various physical phenomena, which can influence the dynamics of a flying structure. These permit us to introduce a new state-space representation and new control scheme. We present after the development and the synthesis of a stabilizing control laws design based on sliding mode in order to perform best tracking results. It ensures locally asymptotic stability and desired tracking trajectories. Nonlinear observer is then synthesized in order to estimate the unmeasured states and the effects of the external disturbances such as wind and noise. Finally simulation results are also provided in order to illustrate the performances of the proposed controllers.

Keywords: Dynamic modelling, nonholonomic constraints, sliding mode, nonlinear observer.

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Authors: Benshi Zhu

Abstract:

In this paper, the existence of multiple positive solutions for a class of third-order three-point discrete boundary value problem is studied by applying algebraic topology method.

Keywords: Positive solutions, Discrete boundary value problem, Third-order, Three-point, Algebraic topology

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