Search results for: linear parameter varying systems

Authors: Muhammad Zikril Hakim Md. Azizi, Norhafizan Ahmad, Raja Ariffin Raja Ghazilla

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Postural stability, attention level of the archer and particularly the vibrations of the bow itself plays a prominent role in determining the athletes performance. Many techniques and systems had been developing to monitor the parameters of the archers during training. In Malaysia, archery coaches tend to use non-scientific ways that they are familiar with, to evaluate archer performance. An approach that provides more affordable yet accurate systems to the masses and relatively easy system deployment procedure need to be proposed. Hence, this project will address to fulfil the needs. Three area of the archer parameter were included for data monitoring sensors. Attention level can be measured using EEG sensor, centre of mass linked to the postural stability can be measured by foot pressure sensor, and the bow vibrations in three axis will be relayed by the vibrations sensors placed directly on the bow using wireless sensors. Arduino based microcontroller used to relay all the data back to the interfacing systems. Interface systems will be using Python language and C++ framework for user interface and hardware interfacing systems. All sensor data can be observed in real time using the in-house applications, and each sessions can be saved to common files so that coach and the team can have a further discussion and comparisons.

Keywords: archery, graphical user interface, microcontroller, wireless sensor, monitoring system

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Authors: Petr Jizba, Jan Korbel

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We combine an axiomatics of Rényi with the q-deformed version of Khinchin axioms to obtain a measure of information (i.e., entropy) which accounts both for systems with embedded self-similarity and non-extensivity. We show that the entropy thus obtained is uniquely solved in terms of a one-parameter family of information measures. The ensuing maximal-entropy distribution is phrased in terms of a special function known as the Lambert W-function. We analyze the corresponding ‘high’ and ‘low-temperature’ asymptotics and reveal a non-trivial structure of the parameter space.

Keywords: multifractals, Rényi information entropy, THC entropy, MaxEnt, heavy-tailed distributions

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Authors: Tim Farrelly

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In recent years, AI-powered object detection systems have opened the doors for innovative new applications and products, especially those operating in the real world or ‘on edge’ – namely, in sport. This paper investigates the viability of an ultra-low parameter convolutional neural network specially designed for the detection of footballs on ‘on the edge’ devices. The main contribution of this paper is the exploration of integrating new design features (depth-wise separable convolutional blocks and squeezed and excitation modules) into an ultra-low parameter network and demonstrating subsequent improvements in performance. The results show that tracking the ball from Full HD images with negligibly high accu-racy is possible in real-time.

Keywords: deep learning, object detection, machine vision applications, sport, network design

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Authors: Rajib Mia, Badam Singh Kushvah

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In this present paper, we use the photogravitational version of the restricted three body problem (RTBP) in binary systems. In the photogravitational RTBP, an infinitesimal particle (dust grain) is moving under the gravitational attraction and radiation pressure from the two bigger primaries. The third particle does not affect the motion of two bigger primaries. The zero-velocity curves, zero-velocity surfaces and their projections on the plane are studied. We have used existing analytical method to solve the equations of motion. We have obtained the Lagrangian points in some binary stellar systems. It is found that mass reduction factor affects the Lagrangian points. The linear stability of Lagrangian points is studied and found that these points are unstable. Moreover, trajectories of the infinitesimal particle at the triangular points are studied.

Keywords: binary systems, Lagrangian points, linear stability, photogravitational RTBP, trajectories

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Authors: Carolina Senabre, Sergio Valero, Emilio Velasco

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The brake systems of vehicles are tested periodically by a “brake tester” at Ministry of Transport (MOT) stations. This tester measures the effectiveness of vehicle. This parameter is established by the International Committee of Vehicle Inspection (CITA). In this paper, we present an investigation of the influence of the tire size on the measurements of brake force on three MOT brake testers. We performed an analysis of the vehicle braking capacity test at MOT stations. The influence of varying wheel diameter and width on the measurement of braking at MOT stations has been analyzed. Thereby, the MOT brake tester as a verification system for a vehicle has been evaluated.

Keywords: brake tester, ministry of transport facilities, wheel diameter, efficiency

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Authors: Zhi Zhu, Boquan Zhang, Tian Jing, Jingjing Li, Tao Wang

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Data assimilation is a model and data hybrid-driven method that dynamically fuses new observation data with a numerical model to iteratively approach the real system state. It is widely used in state prediction and parameter inference of continuous systems. Because of the discrete event system’s non-linearity and non-Gaussianity, traditional Kalman Filter based on linear and Gaussian assumptions cannot perform data assimilation for such systems, so particle filter has gradually become a technical approach for discrete event simulation data assimilation. Hence, we proposed a particle filter-based discrete event simulation data assimilation method and took the unmanned aerial vehicle (UAV) maintenance service system as a proof of concept to conduct simulation experiments. The experimental results showed that the filtered state data is closer to the real state of the system, which verifies the effectiveness of the proposed method. This research can provide a reference framework for the data assimilation process of other complex nonlinear systems, such as discrete-time and agent simulation.

Keywords: discrete event simulation, data assimilation, particle filter, model and data-driven

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Authors: Rupinder Kaur, Harjot Kaur

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The objective of this paper is to present the study of the effect of varying thickness on rotating composite disks made from Al-SiC_P having different particle sizes. Mathematical modeling is used to calculate the effect of varying thickness with different particle sizes on rotating composite disks in radial as well as tangential directions with thermal gradients. In comparison to various particle sizes with varied thicknesses, long-term deformation occurs. The results are displayed visually, demonstrating how creep deformation decreases with changing particle size and thickness.

Keywords: creep, varying thickness, particle size, stresses and strain rates

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Authors: Bernt O. Myrvold

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Lignosulfonates (LS) find widespread use as dispersants, binders, anti-oxidants, and fillers. In most of these applications LS is used in formulation together with a number of other components. To better understand the interactions between LS and water and possibly other components in a formulation, the Hansen solubility parameters have been determined for some LS. The Hansen solubility parameter splits the total solubility parameter into three components, the dispersive, polar and hydrogen bonding part. The Hansen solubility parameter was determined by comparing the solubility in a number of solvents and solvent mixtures. We have found clear differences in the solubility parameters, with softwood LS being closer to water than hardwood LS.

Keywords: Hansen solubility parameter, lignosulfonate (LS), solubility, solvent

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Authors: Foo Shen Hwang, Thomas Confrey, Stephen Scully, Barry Flannery

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Thermal characterization plays an important role in battery pack design. Lithium-ion batteries have to be maintained between 15-35 °C to operate optimally. Heat is generated (Q) internally within the batteries during both the charging and discharging phases. This can be quantified using several standard methods. The most common method of calculating the batteries heat generation is through the addition of both the joule heating effects and the entropic changes across the battery. In addition, such values can be derived by identifying the open-circuit voltage (OCV), nominal voltage (V), operating current (I), battery temperature (T) and the rate of change of the open-circuit voltage in relation to temperature (dOCV/dT). This paper focuses on experimental characterization and comparative modelling of the heat generation rate (Q) across several current discharge rates (0.5C, 1C, and 1.5C) of a 18650 cell. The analysis is conducted utilizing several non-linear mathematical functions methods, including polynomial, exponential, and power models. Parameter fitting is carried out over the respective function orders; polynomial (n = 3~7), exponential (n = 2) and power function. The generated parameter fitting functions are then used as heat source functions in a 3-D computational fluid dynamics (CFD) solver under natural convection conditions. Generated temperature profiles are analyzed for errors based on experimental discharge tests, conducted at standard room temperature (25°C). Initial experimental results display low deviation between both experimental and CFD temperature plots. As such, the heat generation function formulated could be easier utilized for larger battery applications than other methods available.

Keywords: computational fluid dynamics, curve fitting, lithium-ion battery, voltage drop

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Authors: Debajyoti Choudhuri, Ratan Kumar Giri, Shesadev Pradhan

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The perturbed nonlinear Schrodinger equation involving the p-biharmonic and the p-Laplacian operators involving a real valued parameter and a continuous real valued potential function defined over the N- dimensional Euclidean space has been considered. By the variational technique, an existence result pertaining to a nontrivial solution to this non-linear partial differential equation has been proposed. Further, by the Concentration lemma, the concentration of solutions to the same problem defined on the set consisting of those elements where the potential function vanishes as the real parameter approaches to infinity has been addressed.

Keywords: p-Laplacian, p-biharmonic, elliptic PDEs, Concentration lemma, Sobolev space

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Authors: M. S. H. Chowdhury, Ishak Hashim

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In this paper, linear and non-linear stiff systems of ordinary differential equations are solved by the classical Adomian decomposition method (ADM) and the multi-stage Adomian decomposition method (MADM). The MADM is a technique adapted from the standard Adomian decomposition method (ADM) where standard ADM is converted into a hybrid numeric-analytic method called the multistage ADM (MADM). The MADM is tested for several examples. Comparisons with an explicit Runge-Kutta-type method (RK) and the classical ADM demonstrate the limitations of ADM and promising capability of the MADM for solving stiff initial value problems (IVPs).

Keywords: stiff system of ODEs, Runge-Kutta Type Method, Adomian decomposition method, Multistage ADM

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Authors: Ouahab Kadri, Leila Hayet Mouss

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In this paper, we present three diagnostic modules for complex and dynamic systems. These modules are based on three ant colony algorithms, which are AntTreeStoch, Lumer & Faieta and Binary ant colony. We chose these algorithms for their simplicity and their wide application range. However, we cannot use these algorithms in their basement forms as they have several limitations. To use these algorithms in a diagnostic system, we have proposed three variants. We have tested these algorithms on datasets issued from two industrial systems, which are clinkering system and pasteurization system.

Keywords: ant colony algorithms, complex and dynamic systems, diagnosis, classification, optimization

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Authors: M. P. Nanda Kumar, K. Dheeraj

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The design of a feedback controller, so as to minimize a given performance criterion, for a general non-linear dynamical system is difficult; if not impossible. But for a large class of non-linear dynamical systems, the open loop control that minimizes a performance criterion can be obtained using calculus of variations and Pontryagin’s minimum principle. In this paper, the open loop optimal trajectories, that minimizes a given performance measure, is used to train the neural network whose inputs are state variables of non-linear dynamical systems and the open loop optimal control as the desired output. This trained neural network is used as the feedback controller. In other words, attempts are made here to solve the “inverse optimal control problem” by using the state and control trajectories that are optimal in an open loop sense.

Keywords: inverse optimal control, radial basis function, neural network, controller design

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Authors: Mojtaba Hakimi-Moghaddam

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Positive realness as the most important property of driving point impedance of passive electrical networks appears in the control systems stability theory in 1960’s. There are three important subsets of positive real (PR) systems are introduced by researchers, that is, loos-less positive real (LLPR) systems, weakly strictly positive real (WSPR) systems and strictly positive real (SPR) systems. In this paper, definitions, properties, lemmas, and theorems related to family of positive real systems are summarized. Properties in both frequency domain and state space representation of system are explained. Also, several illustrative examples are presented.

Keywords: real rational matrix transfer functions, positive realness property, strictly positive realness property, Hermitian form asymptotic property, pole-zero properties

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Authors: R. Periyasamy, Deepak Joshi, Sneh Anand

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Foot posture assessment is important to evaluate foot type, causing gait and postural defects in all age groups. Although different methods are used for classification of foot arch type in clinical/research examination, there is no clear approach for selecting the most appropriate measurement system. Therefore, the aim of this study was to develop a system for evaluation of foot type as clinical decision-making aids for diagnosis of flat and normal arch based on the Arch Index (AI) and foot pressure distribution parameter - Power Ratio (PR) data. The accuracy of the system was evaluated for 27 subjects with age ranging from 24 to 65 years. Foot area measurements (hind foot, mid foot, and forefoot) were acquired simultaneously from foot pressure intensity image using portable PedoPowerGraph system and analysis of the image in frequency domain to obtain foot pressure distribution parameter - PR data. From our results, we obtain 100% classification accuracy of normal and flat foot by using the linear discriminant analysis method. We observe there is no misclassification of foot types because of incorporating foot pressure distribution data instead of only arch index (AI). We found that the mid-foot pressure distribution ratio data and arch index (AI) value are well correlated to foot arch type based on visual analysis. Therefore, this paper suggests that the proposed system is accurate and easy to determine foot arch type from arch index (AI), as well as incorporating mid-foot pressure distribution ratio data instead of physical area of contact. Hence, such computational tool based system can help the clinicians for assessment of foot structure and cross-check their diagnosis of flat foot from mid-foot pressure distribution.

Keywords: arch index, computational tool, static foot pressure intensity image, foot pressure distribution, linear discriminant analysis

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Authors: Mohammad Parsaeimaram, Fang Congqi

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In this paper, one of the significant seismic design parameter as response modification factor in reinforced concrete (RC) buildings with base isolation system was evaluated. The seismic isolation system is a capable approach to absorbing seismic energy at the base and transfer to the substructure with lower response modification factor as compared to non-isolated structures. A response spectrum method and static nonlinear pushover analysis in according to Uniform Building Code (UBC-97), have been performed on building models involve 5, 8, 12 and 15 stories building with fixed and isolated bases consist of identical moment resisting configurations. The isolation system is composed of lead rubber bearing (LRB) was designed with help UBC-97 parameters. The force-deformation behavior of isolators was modeled as bi-linear hysteretic behavior which can be effectively used to create the isolation systems. The obtained analytical results highlight the response modification factor of considered base isolation system with higher values than recommended in the codes. The response modification factor is used in modern seismic codes to scale down the elastic response of structures.

Keywords: response modification factor, base isolation system, pushover analysis, lead rubber bearing, bi-linear hysteretic

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Authors: Sumit Kumar Vishwakarma

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The aim of the paper is to investigate the influence of inhomogeneity associated with the elastic constants and density of the orthotropic medium. The inhomogeneity is considered as exponential function of depth. The impact of gravity had been discussed. Using the concept of separation of variables, the system of a partial differential equation (equation of motion) has been converted into ordinary differential equation, which is coupled in nature. It further reduces to a biquadratic equation whose roots were found by using MATLAB. A suitable boundary condition is employed to derive the dispersion equation in a closed-form. Numerical simulations had been performed to show the influence of the inhomogeneity parameter. It was observed that as the numerical values of increases, the phase velocity of Rayleigh waves decreases at a particular wavenumber. Graphical illustrations were drawn to visualize the effect of the increasing and decreasing values of the inhomogeneity parameter. It can be concluded that it has a remarkable bearing on the phase velocity as well as damping velocity.

Keywords: Rayleigh waves, orthotropic medium, gravity field, inhomogeneity

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Authors: Yue Hu, Xilu Zhao, Takao Yamaguchi, Manabu Sasajima, Yoshio Koike, Akira Hara

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This study tried optimization of design parameter of a cone loudspeaker unit as an example of the high flexibility of the products design. We developed an acoustic analysis software program that considers the impact of damping caused by air viscosity. In sound reproduction, it is difficult to each design the parameter of the loudspeaker. To overcome the limitation of the design problem in practice, this paper proposes a new an acoustic analysis algorithm to optimize design the parameter of the loudspeaker. The material character of cone paper and the loudspeaker edge was the design parameter, and the vibration displacement of the cone paper was the objective function. The results of the analysis were compared with the predicted value. They had high accuracy to the predicted value. These results suggest that, though the parameter design is difficult by experience and intuition, it can be performed comparatively easily using the optimization design by the developed acoustic analysis software.

Keywords: air viscosity, loudspeaker, cone paper, edge, optimization

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Authors: Vikram Saini, Lillie Dewan

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This paper presents the iterative schemes based on Least square, Hierarchical Least Square and Stochastic Approximation Gradient method for the Identification of Wiener model with parametric structure. A gradient method is presented for the parameter estimation of wiener model with noise conditions based on the stochastic approximation. Simulation results are presented for the Wiener model structure with different static non-linear elements in the presence of colored noise to show the comparative analysis of the iterative methods. The stochastic gradient method shows improvement in the estimation performance and provides fast convergence of the parameters estimates.

Keywords: hard non-linearity, least square, parameter estimation, stochastic approximation gradient, Wiener model

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Authors: Durata Haciu, Berti Manisa, Ozgur Birer

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ZnO nanoparticles have been synthesized by ultrasonic irradiation from simple linear alcohols and water/ethanolic mixtures, at 50 oC. By changing the composition of the solvent, the shape could be altered. While no product was obtained from methanolic solutions, in ethanol, sheet like lamellar structures prevail.n-propanol and n-butanol resulted in needle like structures. The morphology of ZnO could be thus tailored in a simple way, by varying the solvent, under ultrasonic irradiation, in a relatively less time consuming method. Variation of the morphology and size of Zn also provides a means for modulating the band-gap. Although the chemical effects of ultrasound do not come from direct interaction with molecular species, the high energy derived from acoustic cavitation creates a unique interaction of energy and matter with great potential for synthesis.

Keywords: ultrasound, ZnO, linear alcohols, morphology

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Authors: Habib Akbarzadeh Bengar, Mohammad Asadi Kiadehi, Ali Rameeh

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The result of the past earthquakes has shown that insufficient amount of stirrups and brittle behavior of concrete lead to the shear and flexural failure in reinforced concrete (RC) members. In this paper, an analytical model proposed to predict the nonlinear behavior of RC and SFRC elements and frames. In this model, some important parameter such as shear effect, varying axial load, and longitudinal bar buckling are considered. The results of analytical model were verified with experimental tests. The results of verification have shown that the proposed analytical model can predict the nonlinear behavior of RC and SFRC members and also frames accurately. In addition, the results have shown that use of steel fibers increased bearing capacity and ductility of RC frame. Due to this enhancement in shear strength and ductility, insufficient amount of stirrups, which resulted in shear failure, can be offset with usage of the steel fibers. In addition to the steps taken, to analyze the effects of fibers percentages on the bearing capacity and ductility of frames parametric studies have been performed to investigate of these effects.

Keywords: nonlinear analysis, SFRC frame, shear failure, varying an axial load

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Authors: Ayman Baklizi

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Based on type II censored data, we consider interval estimation of the quantiles of the two-parameter exponential distribution and the difference between the quantiles of two independent two-parameter exponential distributions. We derive asymptotic intervals, Bayesian, as well as intervals based on the generalized pivot variable. We also include some bootstrap intervals in our comparisons. The performance of these intervals is investigated in terms of their coverage probabilities and expected lengths.

Keywords: asymptotic intervals, Bayes intervals, bootstrap, generalized pivot variables, two-parameter exponential distribution, quantiles

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Authors: Tomas Menard

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The synthesis of output-feedback control law has been investigated by many researchers since the last century. While many results exist for the case of Linear Time Invariant systems whose measurements are continuously available, nowadays, control laws are usually implemented on micro-controller, then the measurements are discrete-time by nature. This fact has to be taken into account explicitly in order to obtain a satisfactory behavior of the closed-loop system. One considers here a general class of systems corresponding to an observability normal form and which is subject to uncertainties in the dynamics and sampling of the output. Indeed, in practice, the modeling of the system is never perfect, this results in unknown uncertainties in the dynamics of the model. We propose here an output feedback algorithm which is based on a linear state feedback and a continuous-discrete time observer. The main feature of the proposed control law is that only discrete-time measurements of the output are needed. Furthermore, it is formally proven that the state of the closed loop system exponentially converges toward the origin despite the unknown uncertainties. Finally, the performances of this control scheme are illustrated with simulations.

Keywords: dynamical systems, output feedback control law, sampling, uncertain systems

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Authors: R. M. Chandima Ratnayake, Daniel Dyakov

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Super steel materials play vital role in construction and fabrication of structural, piping and pipeline components. They enable to minimize the life cycle costs in assuring the integrity of onshore and offshore operating systems. In this context, Duplex stainless steel (DSS) material related welding on constructions and fabrications play a significant role in maintaining and assuring integrity at an optimal expenditure over the life cycle of production and process systems as well as associated structures. In DSS welding, the factors such as gap geometry, shielding gas supply rate, welding current, and type of the welding process play a vital role on the final joint performance. Hence, an experimental investigation has been performed using engineering robust design approach (ERDA) to investigate the optimal settings that generate optimal super DSS (i.e. UNS S32750) joint performance. This manuscript illustrates the mathematical approach and experimental design, optimal parameter settings and results of verification experiment.

Keywords: duplex stainless steel welding, engineering robust design, mathematical framework, optimal parameter settings

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Authors: Bandari Shankar, Yohannes Yirga

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In this paper, the problem of heat and mass transfer in unsteady MHD boundary-layer flow of nanofluids over stretching sheet with a non uniform heat source/sink is considered. The unsteadiness in the flow and temperature is caused by the time-dependent stretching velocity and surface temperature. The unsteady boundary layer equations are transformed to a system of non-linear ordinary differential equations and solved numerically using Keller box method. The velocity, temperature, and concentration profiles were obtained and utilized to compute the skin-friction coefficient, local Nusselt number, and local Sherwood number for different values of the governing parameters viz. solid volume fraction parameter, unsteadiness parameter, magnetic field parameter, Schmidt number, space-dependent and temperature-dependent parameters for heat source/sink. A comparison of the numerical results of the present study with previously published data revealed an excellent agreement

Keywords: unsteady, heat and mass transfer, manetohydrodynamics, nanofluid, non-uniform heat source/sink, stretching sheet

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Authors: Ali Afaghi, Sehraneh Ghaemi

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The coordination of the multi-agent systems has been one of the interesting topic in recent years, because of its potential applications in many branches of science and engineering such as sensor networks, flocking, underwater vehicles and etc. In the most of the related studies, it is assumed that the dynamics of the multi-agent systems are integer-order and linear and the multi-agent systems with the fractional-order nonlinear dynamics are rarely considered. However many phenomena in nature cannot be described within integer-order and linear characteristics. This paper investigates the leader-following consensus problem for a class of nonlinear fractional-order multi-agent systems based on observer-based cooperative control. In the system, the dynamics of each follower and leader are nonlinear. For a multi-agent system with fixed directed topology firstly, an observer-based consensus protocol is proposed based on the relative observer states of neighboring agents. Secondly, based on the property of the stability theory of fractional-order system, some sufficient conditions are presented for the asymptotical stability of the observer-based fractional-order control systems. The proposed method is applied on a five-agent system with the fractional-order nonlinear dynamics and unavailable states. The simulation example shows that the proposed scenario results in the good performance and can be used in many practical applications.

Keywords: fractional-order multi-agent systems, leader-following consensus, nonlinear dynamics, directed graphs

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Authors: Dong Sang Yoo

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We consider nonlinear uncertain systems such that a priori information of the uncertainties is not available. For such systems, we assume that the upper bound of the uncertainties is represented as a Fredholm integral equation of the first kind and we propose an adaptation law that is capable of estimating the upper bound and design a continuous robust control which renders nonlinear uncertain systems ultimately bounded.

Keywords: adaptive control, estimation, Fredholm integral, uncertain system

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Authors: Motahar Reza, Rajni Chahal, Neha Sharma

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This article addresses the boundary layer flow and heat transfer of Casson fluid over a nonlinearly permeable stretching surface with chemical reaction in the presence of variable magnetic field. The effect of thermal radiation is considered to control the rate of heat transfer at the surface. Using similarity transformations, the governing partial differential equations of this problem are reduced into a set of non-linear ordinary differential equations which are solved by finite difference method. It is observed that the velocity at fixed point decreases with increasing the nonlinear stretching parameter but the temperature increases with nonlinear stretching parameter.

Keywords: boundary layer flow, nonlinear stretching, Casson fluid, heat transfer, radiation

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Authors: Kiurski S. Jelena, Kecić S. Vesna, Aksentijević M. Snežana

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Samples of tap and wastewater were collected in three offset printing facilities in Novi Sad, Serbia. Ten physicochemical parameters were analyzed within all collected samples: pH, conductivity, m - alkalinity, p - alkalinity, acidity, carbonate concentration, hydrogen carbonate concentration, active oxygen content, chloride concentration and total alkali content. All measurements were conducted using the standard analytical and instrumental methods. Comparing the obtained results for tap water and wastewater, a clear quality difference was noticeable, since all physicochemical parameters were significantly higher within wastewater samples. The study also involves the application of simple linear regression analysis on the obtained dataset. By using software package ORIGIN 5 the pH value was mutually correlated with other physicochemical parameters. Based on the obtained values of Pearson coefficient of determination a strong positive correlation between chloride concentration and pH (r = -0.943), as well as between acidity and pH (r = -0.855) was determined. In addition, statistically significant difference was obtained only between acidity and chloride concentration with pH values, since the values of parameter F (247.634 and 182.536) were higher than F_critical (5.59). In this way, results of statistical analysis highlighted the most influential parameter of water contamination in offset printing, in the form of acidity and chloride concentration. The results showed that variable dependence could be represented by the general regression model: y = a₀ + a₁x+ k, which further resulted with matching graphic regressions.

Keywords: pollution, printing industry, simple linear regression analysis, wastewater

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Authors: Mary Chriselda A

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This paper deals with the fact that the Hubble's parameter is not constant and tends to vary stochastically with time. This premise has been proven by converting it to a stochastic differential equation using the Ornstein-Uhlenbeck process. The formulated stochastic differential equation is further solved analytically using the Euler and the Kolmogorov Forward equations, thereby obtaining the probability density function using the Fourier transformation, thereby proving that the Hubble's parameter varies stochastically. This is further corroborated by simulating the observations using Python and R-software for validation of the premise postulated. We can further draw conclusion that the randomness in forces affecting the white noise can eventually affect the Hubble’s Parameter leading to scale invariance and thereby causing stochastic fluctuations in the density and the rate of expansion of the Universe.

Keywords: Chapman Kolmogorov forward differential equations, fourier transformation, hubble's parameter, ornstein-uhlenbeck process , stochastic differential equations

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