Search results for: Quadratic stability theory
Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 2882

Search results for: Quadratic stability theory

1322 Robust Adaptive Control of a Robotic Manipulator with Unknown Dead Zone and Friction Torques

Authors: Ibrahim F. Jasim, Najah F. Jasim

Abstract:

The problem of controlling a two link robotic manipulator, consisting of a rotating and a prismatic links, is addressed. The actuations of both links are assumed to have unknown dead zone nonlinearities and friction torques modeled by LuGre friction model. Because of the existence of the unknown dead zone and friction torque at the actuations, unknown parameters and unmeasured states would appear to be part of the overall system dynamics that need for estimation. Unmeasured states observer, unknown parameters estimators, and robust adaptive control laws have been derived such that closed loop global stability is achieved. Simulation results have been performed to show the efficacy of the suggested approach.

Keywords: Adaptive Robust Control, Dead Zone, Friction Torques, Robotic Manipulators.

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1321 An Integrated Framework for Engaging Stakeholders in the Circular Economy Processes Using Building Information Modeling and Virtual Reality

Authors: Erisasadat Sahebzamani, Núria Forcada, Francisco Lendinez

Abstract:

Global climate change has become increasingly problematic over the past few decades. The construction industry has contributed to greenhouse gas emissions in recent decades. Considering these issues and the high demand for materials in the construction industry, Circular Economy (CE) is considered necessary to keep materials in the loop and extend their useful lives. By providing tangible benefits, Construction 4.0 facilitates the adoption of CE by reducing waste, updating standard work, sharing knowledge, and increasing transparency and stability. This study aims to present a framework for integrating CE and digital tools like Building Information Modeling (BIM) and Virtual Reality (VR) to examine the impact on the construction industry based on stakeholders' perspectives.

Keywords: Circular Economy, Building Information Modeling, Virtual Reality, stakeholder engagement.

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1320 Multiple Positive Periodic Solutions to a Periodic Predator-Prey-Chain Model with Harvesting Terms

Authors: Zhouhong Li, Jiming Yang

Abstract:

In this paper, a class of predator-prey-chain model with harvesting terms are studied. By using Mawhin-s continuation theorem of coincidence degree theory and some skills of inequalities, some sufficient conditions are established for the existence of eight positive periodic solutions. Finally, an example is presented to illustrate the feasibility and effectiveness of the results.

Keywords: Positive periodic solutions, Predator-prey-chain model, coincidence degree, harvesting term.

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1319 State Economic Safety in the Conditions of Innovative Economy Formation

Authors: Zh.Mingaleva, K.Balkova

Abstract:

Innovations and innovative activity get the increasing value for successful financial and economic activity of the countries and regions. The level of innovative sphere development determines place of a country or a region in world economy and forms a basis of steady economic growth. This article is devoted to different aspects of organization of the national economic safety in the conditions of innovative development, its problems, risks and threats. Economy can be considered as aspiring for transition to innovative way only with finding of economic safety: financial independence, power stability and technological progress. There are statistical indicators, defining the level of economic security and factors, threatening economic safety of the state. The research is based on the analysis of factors and indicators in conditions of innovative development. The paper is illustrated by the examples of possible estimated system of the economic safety level.

Keywords: Economic safety, innovative activity, innovative development, innovative economy

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1318 Haar wavelet Method for Solving Initial and Boundary Value Problems of Bratu-type

Authors: S.G.Venkatesh, S.K.Ayyaswamy, G.Hariharan

Abstract:

In this paper, we present a framework to determine Haar solutions of Bratu-type equations that are widely applicable in fuel ignition of the combustion theory and heat transfer. The method is proposed by applying Haar series for the highest derivatives and integrate the series. Several examples are given to confirm the efficiency and the accuracy of the proposed algorithm. The results show that the proposed way is quite reasonable when compared to exact solution.

Keywords: Haar wavelet method, Bratu's problem, boundary value problems, initial value problems, adomain decomposition method.

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1317 Cooling Turbine Blades using Exciting Boundary Layer

Authors: Ali Ghobadi, Seyed Mohammad Javadi, Behnam Rahimi

Abstract:

The present study is concerned with the effect of exciting boundary layer on cooling process in a gas-turbine blades. The cooling process is numerically investigated. Observations show cooling the first row of moving or stable blades leads to increase their life-time. Results show that minimum temperature in cooling line with exciting boundary layer is lower than without exciting. Using block in cooling line of turbines' blade causes flow pattern and stability in boundary layer changed that causes increase in heat transfer coefficient. Results show at the location of block, temperature of turbines' blade is significantly decreased. The k-ε turbulence model is used.

Keywords: Cooling, Exciting Boundary Layer, Heat Transfer, Turbine Blade.

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1316 Four Positive Almost Periodic Solutions to an Impulsive Delayed Plankton Allelopathy System with Multiple Exploit (or Harvesting) Terms

Authors: Fengshuo Zhang, Zhouhong Li

Abstract:

In this paper, we obtain sufficient conditions for the existence of at least four positive almost periodic solutions to an impulsive delayed periodic plankton allelopathy system with multiple exploited (or harvesting) terms. This result is obtained through the use of Mawhins continuation theorem of coincidence degree theory along with some properties relating to inequalities.

Keywords: Almost periodic solutions, plankton allelopathy system, coincidence degree, impulse.

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1315 Controller Design for Active Suspension System of ¼ Car with Unknown Mass and Time-Delay

Authors: Ali Al-Zughaibi, Huw Davies

Abstract:

The purpose of this paper is to present a modeling and control of a quarter-car active suspension system with unknown mass, unknown time-delay and road disturbance. The objective of designing the controller is to derive a control law to achieve stability of the system and convergence that can considerably improve ride comfort and road disturbance handling. This is accomplished by using Routh-Hurwitz criterion based on defined parameters. Mathematical proof is given to show the ability of the designed controller to ensure the target of design, implementation with the active suspension system and enhancement dispersion oscillation of the system despite these problems. Simulations were also performed to control quarter car suspension, where the results obtained from these simulations verify the validity of the proposed design.

Keywords: Active suspension system, disturbance rejection, dynamic uncertainty, time-delay.

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1314 Motion Control of a 2-link Revolute Manipulator in an Obstacle-Ridden Workspace

Authors: Avinesh Prasad, Bibhya Sharma, Jito Vanualailai

Abstract:

In this paper, we propose a solution to the motion control problem of a 2-link revolute manipulator arm. We require the end-effector of the arm to move safely to its designated target in a priori known workspace cluttered with fixed circular obstacles of arbitrary position and sizes. Firstly a unique velocity algorithm is used to move the end-effector to its target. Secondly, for obstacle avoidance a turning angle is designed, which when incorporated into the control laws ensures that the entire robot arm avoids any number of fixed obstacles along its path enroute the target. The control laws proposed in this paper also ensure that the equilibrium point of the system is asymptotically stable. Computer simulations of the proposed technique are presented.

Keywords: 2-link revolute manipulator, motion control, obstacle avoidance, asymptotic stability.

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1313 Two Fourth-order Iterative Methods Based on Continued Fraction for Root-finding Problems

Authors: Shengfeng Li, Rujing Wang

Abstract:

In this paper, we present two new one-step iterative methods based on Thiele-s continued fraction for solving nonlinear equations. By applying the truncated Thiele-s continued fraction twice, the iterative methods are obtained respectively. Analysis of convergence shows that the new methods are fourth-order convergent. Numerical tests verifying the theory are given and based on the methods, two new one-step iterations are developed.

Keywords: Iterative method, Fixed-point iteration, Thiele's continued fraction, Order of convergence.

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1312 Modeling Method and Application in Digital Mockup System towards Mechanical Product

Authors: Huaiyu Zhang

Abstract:

The method of modeling is the key technology for digital mockup (DMU). Based upon the developing for mechanical product DMU, the theory, method and approach for virtual environment (VE) and virtual object (VO) were studied. This paper has expounded the design goal and architecture of DMU system, analyzed the method of DMU application, and researched the general process of physics modeling and behavior modeling.

Keywords: DMU, VR, virtual environment, virtual object, physics modeling, behavior modeling

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1311 Vibrational Behavior of Cylindrical Shells in Axial Magnetic Field

Authors: Sedrak Vardanyan

Abstract:

The investigation of the vibrational character of magnetic cylindrical shells placed in an axial magnetic field has important practical applications. In this work, we study the vibrational behaviour of such a cylindrical shell by making use of the so-called exact space treatment, which does not assume any hypothesis. We discuss the effects of several practically important boundary conditions on the vibrations of the described setup. We find that, for some cases of boundary conditions, e.g. clamped, simply supported or peripherally earthed, as well as for some values of the wave numbers, the vibrational frequencies of the shell are approximately zero. The theoretical and numerical exploration of this fact confirms that the vibrations are absent or attenuate very rapidly. For all the considered cases, the imaginary part of the frequencies is negative, which implies stability for the vibrational process.

Keywords: Free vibrations, magnetic cylindrical shells, exact space treatment, bending vibrational frequencies.

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1310 The "Project" Approach in Urban: A Response to Uncertainty

Authors: Nedjima Mouhoubi, Souad Sassi Boudemagh

Abstract:

In this paper, we will try to demonstrate the importance of the project approach in the urban to deal with uncertainty, the importance of the involvement of all stakeholders in the urban project process and that the absence of an actor can lead to project failure but also the importance of the urban project management. These points are handled through the following questions: Does the urban adhere to the theory of complexity? Does the project approach bring hope and solution to make urban planning "sustainable"? How converging visions of actors for the same project? Is the management of urban project the solution to support the urban project approach?

Keywords: Strategic planning, project, urban project stakeholders, management.

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1309 Analysis of Catalytic Properties of Ni3Al Thin Foils for the Methanol and Hexane Decomposition

Authors: M. Michalska-Domańska, P. Jóźwik, Z. Bojar

Abstract:

Intermetallic Ni3Al – based alloys belong to a group of advanced materials characterized by good chemical and physical properties (such as structural stability, corrosion resistance) which offer advenced technological applications. The paper presents the study of catalytic properties of Ni3Al foils (thickness approximately 50 &m) in the methanol and hexane decomposition. The egzamined material posses microcrystalline structure without any additional catalysts on the surface. The better catalytic activity of Ni3Al foils with respect to quartz plates in both methanol and hexane decomposition was confirmed. On thin Ni3Al foils the methanol conversion reaches approximately 100% above 480 oC while the hexane conversion reaches approximately 100% (98,5%) at 500 oC. Deposit formed during the methanol decomposition is built up of carbon nanofibers decorated with metal-like nanoparticles.

Keywords: hexane decomposition, methanol decomposition, Ni3Al thin foils, Ni nanoparticles

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1308 Modification and Characterization of Bacterial Cellulose Biopolymer as Proton Conducting Membrane

Authors: C. W. Lin, S.W. Chen

Abstract:

This study describes the preparation of a novel proton conducting membranes based on bacterial cellulose (BC) modified by grafting of 2-acrylamido-2-methyl-1 -propanesulfonic acid (AMPS) through UV-induced graft polymerization. These AMPS-g-BC membranes have been characterized by various techniques including FTIR, SEM and TGA, to find their successful grafting of AMPS on BC, surface morphology and thermal stability, respectively. Physical properties of AMPS-g-BC membranes have been assessed in terms of Lamda value( λ ), ion exchange capacity(IEC) and proton conductivity. The relationship between degree of grafting and AMPS concentration used for grafting has been determined by weight gain method. An optimum proton conductivity equal to 2.89x10-2 S cm-1 and IEC value equal to 1.79 mmol g-1 have been obtained when 20 wt% AMPS concentration is used for grafting (i.e. the corresponding membrane is notated as AMPS20-g-BC).

Keywords: Bacterial cellulose, 2-acrylamido-2-methyl-1-propanesulfonic acid, Proton conducting membrane, Self diffusioncoefficient, Fuel cell

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1307 Cubic Trigonometric B-spline Approach to Numerical Solution of Wave Equation

Authors: Shazalina Mat Zin, Ahmad Abd. Majid, Ahmad Izani Md. Ismail, Muhammad Abbas

Abstract:

The generalized wave equation models various problems in sciences and engineering. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline for the approximate solution of wave equation is developed. The usual finite difference approach is used to discretize the time derivative while cubic trigonometric B-spline is applied as an interpolating function in the space dimension. Von Neumann stability analysis is used to analyze the proposed method. Two problems are discussed to exhibit the feasibility and capability of the method. The absolute errors and maximum error are computed to assess the performance of the proposed method. The results were found to be in good agreement with known solutions and with existing schemes in literature.

Keywords: Collocation method, Cubic trigonometric B-spline, Finite difference, Wave equation.

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1306 Design of Adaptive Sliding Mode Controller for Robotic Manipulators Tracking Control

Authors: T. C. Kuo, Y. J. Huang, B. W. Hong

Abstract:

This paper proposes an adaptive sliding mode controller which combines adaptive control and sliding mode control to control a nonlinear robotic manipulator with uncertain parameters. We use an adaptive algorithm based on the concept of sliding mode control to alleviate the chattering phenomenon of control input. Adaptive laws are developed to obtain the gain of switching input and the boundary layer parameters. The stability and convergence of the robotic manipulator control system are guaranteed by applying the Lyapunov theorem. Simulation results demonstrate that the chattering of control input can be alleviated effectively. The proposed controller scheme can assure robustness against a large class of uncertainties and achieve good trajectory tracking performance.

Keywords: Robotic manipulators, sliding mode control, adaptive law, Lyapunov theorem, robustness.

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1305 Preparation of Homogeneous Dense Composite of Zirconia and Alumina (ZTA)using Colloidal Filtration

Authors: H. Wakily, M. Mehrali, H. S. C. Metselaar

Abstract:

Homogeneous composites of alumina and zirconia with a small amount of MgO (<1 wt %) were prepared by colloidal filtration. The object of using ZrO2 (15wt %) was to provide zirconia toughened alumina (ZTA). Suspensions of alumina and Zirconia with various solid loadings and various concentrations of Dolapix CE64 as surfactant were studied. The stability of these suspensions was investigated using rheological measurements. The optimum amount of using Dolapix was 0.8wt% for ZTA containing MgO suspension which gave low apparent viscosity in basic area (100 mPa s at shear rate of 50 s-1). The satisfactory mixtures were made into sample pallets using colloidal filtration. The process was completed with pressureless sintering in suitable temperature. Phase, grain size and qualitative compositional analysis were done using X-ray diffraction (XRD) and scanning electron microscopy (SEM) images. ZTA containing 0.05 wt% MgO shows the lowest grain size for alumina around 0.5 μm. Densification studies show that near full densities (>99%) were obtained for ZTA ceramic containing 0.05 wt% MgO in 1500 °C.

Keywords: Colloidal filtration, Dolapix, MgO, Zirconiatoughened alumina.

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1304 A Family of Zero Stable Block Integrator for the Solutions of Ordinary Differential Equations

Authors: A. M. Sagir

Abstract:

In this paper, linear multistep technique using power series as the basis function is used to develop the block methods which are suitable for generating direct solution of the special second order ordinary differential equations with associated initial or boundary conditions. The continuous hybrid formulations enable us to differentiate and evaluate at some grids and off – grid points to obtain two different four discrete schemes, each of order (5,5,5,5)T, which were used in block form for parallel or sequential solutions of the problems. The computational burden and computer time wastage involved in the usual reduction of second order problem into system of first order equations are avoided by this approach. Furthermore, a stability analysis and efficiency of the block methods are tested on linear and non-linear ordinary differential equations and the results obtained compared favorably with the exact solution.

Keywords: Block Method, Hybrid, Linear Multistep Method, Self – starting, Special Second Order.

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1303 Iron Doped Biomaterial Calcium Borate: Synthesis and Characterization

Authors: G. Çelik Gül, F. Kurtuluş

Abstract:

Colemanite is the most common borate mineral, and the main source of the boron required by plants, human, and earth. Transition metals exhibit optical and physical properties such as; non-linear optical character, structural diversity, thermal stability, long cycle life and luminescent radiation. The doping of colemanite with a transition metal, bring it very interesting and attractive properties which make them applicable in industry. Iron doped calcium borate was synthesized by conventional solid state method at 1200 °C for 12 h with a systematic pathway. X-ray diffraction (XRD), Fourier transform infrared (FTIR) spectroscopy, scanning electron microscopy/energy dispersive analyze (SEM/EDS) were used to characterize structural and morphological properties. Also, thermal properties were recorded by thermogravimetric-differential thermal analysis (TG/DTA). 

Keywords: Colemanite, conventional synthesis, powder x-ray diffraction, borates.

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1302 Applications of Entropy Measures in Field of Queuing Theory

Authors: R.K.Tuli

Abstract:

In the present communication, we have studied different variations in the entropy measures in the different states of queueing processes. In case of steady state queuing process, it has been shown that as the arrival rate increases, the uncertainty increases whereas in the case of non-steady birth-death process, it is shown that the uncertainty varies differently. In this pattern, it first increases and attains its maximum value and then with the passage of time, it decreases and attains its minimum value.

Keywords: Entropy, Birth-death process, M/G/1 system, G/M/1system, Steady state, Non-steady state

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1301 Black Box Model and Evolutionary Fuzzy Control Methods of Coupled-Tank System

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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1300 Computer Aided Assembly Attributes Retrieval Methods for Automated Assembly Sequence Generation

Authors: M. V. A. Raju Bahubalendruni, Bibhuti Bhusan Biswal, B. B. V. L. Deepak

Abstract:

Achieving an appropriate assembly sequence needs deep verification for its physical feasibility. For this purpose, industrial engineers use several assembly predicates; namely, liaison, geometric feasibility, stability and mechanical feasibility. However, testing an assembly sequence for these predicates requires huge assembly information. Extracting such assembly information from an assembled product is a time consuming and highly skillful task with complex reasoning methods. In this paper, computer aided methods are proposed to extract all the necessary assembly information from computer aided design (CAD) environment in order to perform the assembly sequence planning efficiently. These methods use preliminary capabilities of three-dimensional solid modelling and assembly modelling methods used in CAD software considering equilibrium laws of physical bodies.

Keywords: Assembly automation, assembly attributes, assembly sequence generation, computer aided design.

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1299 The Performance of Disbursement Procedure of Public Works in Thailand

Authors: Israngkura Na Ayudhya B, Kunishima M.

Abstract:

This paper analysis performance of disbursement procedure of public works project in Thailand. The results of research were summarised based on contracts, submitted invoice, inspection dated, copies of disbursement dated between client and their main contractor and interviewed with persons involved in central and local government projects during 1994-2008 in Thailand. The data collection was to investigate the disbursement procedure related to performance in disbursement during construction period (Planned duration of contract against Actual execution date in each month). A graphical presentation of a duration analysis of the projects illustrated significant disbursement formation in each project. It was established that the shortage of staff, the financial stability of clients, bureaucratic, method of disbursement and economics situation has play major role on performance of disbursement to their main contractors.

Keywords: Construction disbursement, Payment procedure, Public works

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1298 A Review on Hydraulic and Morphological Characteristics in River Channels Due to Spurs

Authors: M. Alauddin, M. M. Hossain, M. N. Uddin, M. E. Haque

Abstract:

An optimal design of a spur is the first requirement to make it sustainable and function properly. In view of that, a thorough understanding to the hydro- and morpho-dynamics due to spurs is essential. This paper presents a literature review on the effect of spurs to obtain the most recent design criteria. Perpendicular and upstream aligned impermeable spurs have large disturbances to flow and less stability because of strong vortices and associated scour. Downstream aligned spurs minimize scour holes, but there is a chance of strong return current which could be controlled allowing flow through them. A series arrangement of spurs is important to have the desired results with a special care for the first one. Several equations have been presented in the paper for predicting the scour depth. But, they have to be used carefully. Different flow environments developed by spurs are favorable for various aquatic species. However, it is important to maintain almost a stable flow condition providing stable spurs.

Keywords: Bed topography, flow pattern, scour, spur.

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1297 Numerical Study of a Class of Nonlinear Partial Differential Equations

Authors: Kholod M. Abu-Alnaja

Abstract:

In this work, we derive two numerical schemes for solving a class of nonlinear partial differential equations. The first method is of second order accuracy in space and time directions, the scheme is unconditionally stable using Von Neumann stability analysis, the scheme produced a nonlinear block system where Newton-s method is used to solve it. The second method is of fourth order accuracy in space and second order in time. The method is unconditionally stable and Newton's method is used to solve the nonlinear block system obtained. The exact single soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness of the schemes. The interaction of two solitary waves for different parameters are also discussed.

Keywords: Crank-Nicolson Scheme, Douglas Scheme, Partial Differential Equations

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1296 Chattering-free Sliding Mode Control for an Active Magnetic Bearing System

Authors: Abdul Rashid Husain, Mohamad Noh Ahmad, Abdul Halim Mohd Yatim

Abstract:

In this paper, a few chattering-free Sliding Mode Controllers (SMC) are proposed to stabilize an Active Magnetic Bearing (AMB) system with gyroscopic effect that is proportional to the rotor speed. The improved switching terms of the controller inherited from the saturation-type function and boundary layer control technique is shown to be able to achieve bounded and asymptotic stability, respectively, while the chattering effect in the input is attenuated. This is proven to be advantageous for AMB system since minimization of chattering results in optimized control effort. The performance of each controller is demonstrated via result of simulation in which the measurement of the total consumed energy and maximum control magnitude of each controller illustrates the effectiveness of the proposed controllers.

Keywords: Active Magnetic Bearing (AMB), Sliding Mode Control (SMC), chattering-free SMC.

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1295 Polyacrylate Modified Copper Nanoparticles with Controlled Size

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

Abstract:

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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1294 Mathematical Modeling for Dengue Transmission with the Effect of Season

Authors: R. Kongnuy., P. Pongsumpun

Abstract:

Mathematical models can be used to describe the transmission of disease. Dengue disease is the most significant mosquito-borne viral disease of human. It now a leading cause of childhood deaths and hospitalizations in many countries. Variations in environmental conditions, especially seasonal climatic parameters, effect to the transmission of dengue viruses the dengue viruses and their principal mosquito vector, Aedes aegypti. A transmission model for dengue disease is discussed in this paper. We assume that the human and vector populations are constant. We showed that the local stability is completely determined by the threshold parameter, 0 B . If 0 B is less than one, the disease free equilibrium state is stable. If 0 B is more than one, a unique endemic equilibrium state exists and is stable. The numerical results are shown for the different values of the transmission probability from vector to human populations.

Keywords: Dengue disease, mathematical model, season, threshold parameters.

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1293 Method of Moments Applied to a Cuboidal Cavity Resonator: Effect of Gravitational Field Produced by a Black Hole

Authors: Arti Vaish, Harish Parthasarathy

Abstract:

This paper deals with the formulation of Maxwell-s equations in a cavity resonator in the presence of the gravitational field produced by a blackhole. The metric of space-time due to the blackhole is the Schwarzchild metric. Conventionally, this is expressed in spherical polar coordinates. In order to adapt this metric to our problem, we have considered this metric in a small region close to the blackhole and expressed this metric in a cartesian system locally.

Keywords: Method of moments, General theory of relativity, Electromagnetism, Metric tensor, schwarzchild metric, Wave Equation.

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