Search results for: Impulsive; Stochastic functional differential equation; Strict stability; Razumikhin technique.

Authors: M. Pandi, K. Premalatha

Abstract:

A DNA microarray technology is a collection of microscopic DNA spots attached to a solid surface. Scientists use DNA microarrays to measure the expression levels of large numbers of genes simultaneously or to genotype multiple regions of a genome. Elucidating the patterns hidden in gene expression data offers a tremendous opportunity for an enhanced understanding of functional genomics. However, the large number of genes and the complexity of biological networks greatly increase the challenges of comprehending and interpreting the resulting mass of data, which often consists of millions of measurements. It is handled by clustering which reveals the natural structures and identifying the interesting patterns in the underlying data. In this paper, gene based clustering in gene expression data is proposed using Cuckoo Search with Differential Evolution (CS-DE). The experiment results are analyzed with gene expression benchmark datasets. The results show that CS-DE outperforms CS in benchmark datasets. To find the validation of the clustering results, this work is tested with one internal and one external cluster validation indexes.

Keywords: DNA, Microarray, genomics, Cuckoo Search, Differential Evolution, Gene expression data, Clustering.

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Authors: Liu Ping

Abstract:

In this paper, for an arbitrary multiplicative functional f from the set of all upper triangular fuzzy matrices to the fuzzy algebra, we prove that there exist a multiplicative functional F and a functional G from the fuzzy algebra to the fuzzy algebra such that the image of an upper triangular fuzzy matrix under f can be represented as the product of all the images of its main diagonal elements under F and other elements under G.

Keywords: Multiplicative functional, triangular fuzzy matrix, fuzzy addition operation, fuzzy multiplication operation.

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Authors: George Yungchih Wang

Abstract:

For decades financial economists have been attempted to determine the optimal investment policy by recognizing the option value embedded in irreversible investment whose project value evolves as a geometric Brownian motion (GBM). This paper aims to examine the effects of the optimal investment trigger and of the misspecification of stochastic processes on investment in real options applications. Specifically, the former explores the consequence of adopting optimal investment rules on the distributions of corporate value under the correct assumption of stochastic process while the latter analyzes the influence on the distributions of corporate value as a result of the misspecification of stochastic processes, i.e., mistaking an alternative process as a GBM. It is found that adopting the correct optimal investment policy may increase corporate value by shifting the value distribution rightward, and the misspecification effect may decrease corporate value by shifting the value distribution leftward. The adoption of the optimal investment trigger has a major impact on investment to such an extent that the downside risk of investment is truncated at the project value of zero, thereby moving the value distributions rightward. The analytical framework is also extended to situations where collection lags are in place, and the result indicates that collection lags reduce the effects of investment trigger and misspecification on investment in an opposite way.

Keywords: GBM, real options, investment trigger, misspecification, collection lags

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Authors: Emad K. Jaradat, Ala’a Al-Faqih

Abstract:

Nonlinear Schrödinger equations are regularly experienced in numerous parts of science and designing. Varieties of analytical methods have been proposed for solving these equations. In this work, we construct an approximate solution for the nonlinear Schrodinger equations, with harmonic oscillator potential, by Elzaki Decomposition Method (EDM). To illustrate the effects of harmonic oscillator on the behavior wave function, nonlinear Schrodinger equation in one and two dimensions is provided. The results show that, it is more perfectly convenient and easy to apply the EDM in one- and two-dimensional Schrodinger equation.

Keywords: Non-linear Schrodinger equation, Elzaki decomposition method, harmonic oscillator, one and two- dimensional Schrodinger equation.

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Authors: Ch. Surawut, D. Sukawat

Abstract:

In order to utilize results from global climate models, dynamical and statistical downscaling techniques have been developed. For dynamical downscaling, usually a limited area numerical model is used, with associated high computational cost. This research proposes dynamic equation for specific space-time regional climate downscaling from the Educational Global Climate Model (EdGCM) for Southeast Asia. The equation is for surface air temperature. This equation provides downscaling values of surface air temperature at any specific location and time without running a regional climate model. In the proposed equations, surface air temperature is approximated from ground temperature, sensible heat flux and 2m wind speed. Results from the application of the equation show that the errors from the proposed equations are less than the errors for direct interpolation from EdGCM.

Keywords: Dynamic Equation, Downscaling, Inverse distance weight interpolation.

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Authors: Zhenying Hong, Guangwei Yuan, Xuedong Fu, Shulin Yang

Abstract:

In this paper, the typical exponential method, diamond difference and modified time discrete scheme is researched for self adaptive time step. The second-order time evolution scheme is applied to time-dependent spherical neutron transport equation by discrete ordinates method. The numerical results show that second-order time evolution scheme associated exponential method has some good properties. The time differential curve about neutron current is more smooth than that of exponential method and diamond difference and modified time discrete scheme.

Keywords: Exponential method, diamond difference, modified time discrete scheme, second-order time evolution scheme.

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Authors: Behzad Mohammadzadeh, Huyk Chun Noh

Abstract:

Plate is one of the popular structural elements used in a wide range of industries and structures. They may be subjected to blast loads during explosion events, missile attacks or aircraft attacks. This study is to investigate dynamic responses of the rectangular plate subjected to explosive loads. The effects of material properties and plate thickness on responses of the plate are to be investigated. The compressive pressure is applied to the surface of the plate. Different amounts of thickness in the range from 1mm to 30mm are considered for the plate to evaluate the changes in responses of the plate with respect to plate thickness. Two different properties are considered for the steel. First, the analysis is performed by considering only the elastic-plastic properties for the steel plate. Later on damping is considered to investigate its effects on the responses of the plate. To do analysis, numerical method using a finite element based package ABAQUS is applied. Finally, dynamic responses and graphs showing the relation between maximum displacement of the plate and aim parameters are provided.

Keywords: Impulsive loaded plates, dynamic analysis, abaqus, material nonlinearity.

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Authors: Enas Elsayed Abutaleb, Mohamed Taher Eldesoky, Shahenda Abd El Rasol

Abstract:

Background: Muscle Energy Techniques (MET) have been widely used by manual therapists over the past years, but still limited research validated its use and there was limited evidence to substantiate the theories used to explain its effects. Objective: To investigate the effect of Muscle Energy Technique (MET) on anterior pelvic tilt in patients with lumbar spondylosis. Design: Randomized controlled trial. Subjects: Thirty patients with anterior pelvic tilt from both sexes were involved, aged between 35 to 50 years old and they were divided into MET and control groups with 15 patients in each. Methods: All patients received 3sessions/week for 4 weeks where the study group received MET, Ultrasound and Infrared, and the control group received U.S and I.R only. Pelvic angle was measured by palpation meter, pain severity by the visual analogue scale and functional disabilities by the Oswestry disability index. Results: Both groups showed significant improvement in all measured variables. The MET group was significantly better than the control group in pelvic angle, pain severity, and functional disability as p-value were (0.001, 0.0001, 0.0001) respectively. Conclusion and implication: the study group fulfilled greater improvement in all measured variables than the control group which implies that application of MET in combination with U.S and I.R were more effective in improving pelvic tilting angle, pain severity and functional disabilities than using electrotherapy only.

Keywords: Anterior pelvic tilt, lumbar spondylosis, muscle energy technique exercise, palpation meter.

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Authors: Mohammad Taghi Darvishi, Mohammad Najafi

Abstract:

This paper considers the (2+1)-dimensional breaking soliton equation in its bilinear form. Some exact solutions to this equation are explicitly derived by the idea of three-wave solution method with the assistance of Maple. We can see that the new idea is very simple and straightforward.

Keywords: Soliton solution, computerized symbolic computation, painleve analysis, (2+1)-dimensional breaking soliton equation, Hirota's bilinear form.

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Authors: Sidhartha Panda, N. P. Padhy

Abstract:

The application of a Static Synchronous Series Compensator (SSSC) controller to improve the transient stability performance of a power system is thoroughly investigated in this paper. The design problem of SSSC controller is formulated as an optimization problem and Particle Swarm Optimization (PSO) Technique is employed to search for optimal controller parameters. By minimizing the time-domain based objective function, in which the deviation in the oscillatory rotor angle of the generator is involved; transient stability performance of the system is improved. The proposed controller is tested on a weakly connected power system subjected to different severe disturbances. The non-linear simulation results are presented to show the effectiveness of the proposed controller and its ability to provide efficient damping of low frequency oscillations. It is also observed that the proposed SSSC controller improves greatly the voltage profile of the system under severe disturbances.

Keywords: Particle swarm optimization, transient stability, power system oscillations, SSSC.

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Authors: Syeda Maria Ali Shah, Asif Mansoor, Talat Sharafat Rehmani, Safia Mirza

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The stock market of any country acts as a predictor of the economy. The spread of the COVID-19 pandemic has severely impacted the global financial markets. Besides, it has also critically affected the economy of Pakistan. In this study, we consider the role of the Karachi Stock Exchange (KSE) with regard to the Pakistan Stock Exchange and quantify the impact on macroeconomic variables in presence of COVID-19. The suitable macroeconomic variables are used to quantify the impact of COVID-19 by developing the stochastic model. The sufficiency of the computed model is attained by means of available techniques in the literature. The estimated equations are used to forecast the impact of pandemic on macroeconomic variables. The constructed model can help the policymakers take counteractive measures for restricting the influence of viruses on the Karachi Stock Market.

Keywords: COVID-19, Karachi Stock Market, macroeconomic variables, stochastic model, forecasting.

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Authors: Marzieh Dosti, Alireza Nazemi

Abstract:

Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this paper, a numerical solution for the one-dimensional hyperbolic telegraph equation by using the collocation method using the septic splines is proposed. The scheme works in a similar fashion as finite difference methods. Test problems are used to validate our scheme by calculate L2-norm and L∞-norm. The accuracy of the presented method is demonstrated by two test problems. The numerical results are found to be in good agreement with the exact solutions.

Keywords: B-spline, collocation method, second-order hyperbolic telegraph equation, difference schemes.

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Authors: Paola Lecca, Lorenzo Dematte, Corrado Priami

Abstract:

The present models and simulation algorithms of intracellular stochastic kinetics are usually based on the premise that diffusion is so fast that the concentrations of all the involved species are homogeneous in space. However, recents experimental measurements of intracellular diffusion constants indicate that the assumption of a homogeneous well-stirred cytosol is not necessarily valid even for small prokaryotic cells. In this work a mathematical treatment of diffusion that can be incorporated in a stochastic algorithm simulating the dynamics of a reaction-diffusion system is presented. The movement of a molecule A from a region i to a region j of the space is represented as a first order reaction Ai k- ! Aj , where the rate constant k depends on the diffusion coefficient. The diffusion coefficients are modeled as function of the local concentration of the solutes, their intrinsic viscosities, their frictional coefficients and the temperature of the system. The stochastic time evolution of the system is given by the occurrence of diffusion events and chemical reaction events. At each time step an event (reaction or diffusion) is selected from a probability distribution of waiting times determined by the intrinsic reaction kinetics and diffusion dynamics. To demonstrate the method the simulation results of the reaction-diffusion system of chaperoneassisted protein folding in cytoplasm are shown.

Keywords: Reaction-diffusion systems, diffusion coefficient, stochastic simulation algorithm.

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Authors: Shagufta Tabassum, V. P. Pawar, jr., G. N. Shinde

Abstract:

The study of dielectric relaxation properties of polar liquids in the binary mixture has been carried out at 10, 15, 20 and 25 ºC temperatures for 11 different concentrations using time domain reflectometry technique. The dielectric properties of a solute-solvent mixture of polar liquids in the frequency range of 10 MHz to 30 GHz gives the information regarding formation of monomers and multimers and also an interaction between the molecules of the liquid mixture under study. The dielectric parameters have been obtained by the least squares fit method using the Debye equation characterized by a single relaxation time without relaxation time distribution.

Keywords: Excess properties, relaxation time, static dielectric constant, time domain refelectometry technique.

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Authors: S. Panda, S. C. Swain, A. K. Baliarsingh, A. K. Mohanty, C. Ardil

Abstract:

Many real-world optimization problems involve multiple conflicting objectives and the use of evolutionary algorithms to solve the problems has attracted much attention recently. This paper investigates the application of multi-objective optimization technique for the design of a Thyristor Controlled Series Compensator (TCSC)-based controller to enhance the performance of a power system. The design objective is to improve both rotor angle stability and system voltage profile. A Genetic Algorithm (GA) based solution technique is applied to generate a Pareto set of global optimal solutions to the given multi-objective optimisation problem. Further, a fuzzy-based membership value assignment method is employed to choose the best compromise solution from the obtained Pareto solution set. Simulation results are presented to show the effectiveness and robustness of the proposed approach.

Keywords: Multi-objective optimisation, thyristor controlled series compensator, power system stability, genetic algorithm, pareto solution set, fuzzy ranking.

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Authors: Hafiz Muhammad Obaid, Umais Tayyab, Shabbir Majeed Ch.

Abstract:

In this paper, a CMOS differential operational transresistance amplifier (OTRA) is presented. The amplifier is designed and implemented in a standard umc90-nm CMOS technology. The differential OTRA provides wider bandwidth at high gain. It also shows much better rise and fall time and exhibits a very good input current dynamic range of 50 to 50 μA. The OTRA can be used in many analog VLSI applications. The presented amplifier has high gain bandwidth product of 617.6 THz Ω. The total power dissipation of the presented amplifier is also very low and it is 0.21 mW.

Keywords: CMOS, differential, operational transresistance amplifier, OTRA, 90 nm, VLSI.

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Authors: T. H. Young, S. J. Huang, Y. S. Chiu

Abstract:

This paper investigates the parametric stability of an axially moving web subjected to non-uniform in-plane edge excitations on two opposite, simply-supported edges. The web is modeled as a viscoelastic plate whose constitutive relation obeys the Kelvin-Voigt model, and the in-plane edge excitations are expressed as the sum of a static tension and a periodical perturbation. Due to the in-plane edge excitations, the moving plate may bring about parametric instability under certain situations. First, the in-plane stresses of the plate due to the non-uniform edge excitations are determined by solving the in-plane forced vibration problem. Then, the dependence on the spatial coordinates in the equation of transverse motion is eliminated by the generalized Galerkin method, which results in a set of discretized system equations in time. Finally, the method of multiple scales is utilized to solve the set of system equations analytically if the periodical perturbation of the in-plane edge excitations is much smaller as compared with the static tension of the plate, from which the stability boundaries of the moving plate are obtained. Numerical results reveal that only combination resonances of the summed-type appear under the in-plane edge excitations considered in this work.

Keywords: Axially moving viscoelastic plate, in-plane periodic excitation, non-uniformly distributed edge tension, dynamic stability.

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Authors: N. C. Sarkar, A. Bhaskar, Z. Zheng

Abstract:

The driving behavior in area-based (i.e., non-lane based) traffic is induced by the presence of other individuals in the choice space from the driver’s visual perception area. The driving behavior of a subject vehicle is constrained by the potential leaders and leaders are frequently changed over time. This paper is to determine a stochastic model for a parameter of modified intelligent driver model (MIDM) in area-based traffic (as in developing countries). The parametric and non-parametric distributions are presented to fit the parameters of MIDM. The goodness of fit for each parameter is measured in two different ways such as graphically and statistically. The quantile-quantile (Q-Q) plot is used for a graphical representation of a theoretical distribution to model a parameter and the Kolmogorov-Smirnov (K-S) test is used for a statistical measure of fitness for a parameter with a theoretical distribution. The distributions are performed on a set of estimated parameters of MIDM. The parameters are estimated on the real vehicle trajectory data from India. The fitness of each parameter with a stochastic model is well represented. The results support the applicability of the proposed modeling for parameters of MIDM in area-based traffic flow simulation.

Keywords: Area-based traffic, car-following model, micro-simulation, stochastic modeling.

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Authors: Srilekha Mukherjee, Subha Ash, Goutam Sanyal

Abstract:

Information security plays a major role in uplifting the standard of secured communications via global media. In this paper, we have suggested a technique of encryption followed by insertion before transmission. Here, we have implemented two different concepts to carry out the above-specified tasks. We have used a two-point crossover technique of the genetic algorithm to facilitate the encryption process. For each of the uniquely identified rows of pixels, different mathematical methodologies are applied for several conditions checking, in order to figure out all the parent pixels on which we perform the crossover operation. This is done by selecting two crossover points within the pixels thereby producing the newly encrypted child pixels, and hence the encrypted cover image. In the next lap, the first and second order derivative operators are evaluated to increase the security and robustness. The last lap further ensures reapplication of the crossover procedure to form the final stego-image. The complexity of this system as a whole is huge, thereby dissuading the third party interferences. Also, the embedding capacity is very high. Therefore, a larger amount of secret image information can be hidden. The imperceptible vision of the obtained stego-image clearly proves the proficiency of this approach.

Keywords: Steganography, Crossover, Differential Calculus, Peak Signal to Noise Ratio, Cross-correlation Coefficient.

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Authors: Farai Nyabadza

Abstract:

The functional response of an infective is the relationship between an infected individual-s infection rate and the abundance of the number of susceptibles that one can potentially be infected. In this paper, we consider defensive attitudes for HIV prevention (primary prevention) while at the same time emphasizing on offensive attitudes that reduce infection for those infected (secondary prevention). We look at how defenses can protect an uninfected individual in the case where high risk groups such as commercial sex workers and those who deliberately go out to look for partners. We propose an infection cycle that begins with a search, then an encounter, a proposal and contact. The infection cycle illustrates the various steps an infected individual goes through to successfully infect a susceptible. For heterogeneous transmission of HIV, there will be no infection unless there is contact. The ability to avoid an encounter, detection, proposal and contact constitute defense.

Keywords: Functional response, Infection cycle, Prevention, Defences, SSS equation.

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Authors: Walid S. Alfuhaid, Saud A. Alghamdi, John M. Watkins, M. Edwin Sawan

Abstract:

Designing a controller for stochastic decentralized interconnected large scale systems usually involves a high degree of complexity and computation ability. Noise, observability, and controllability of all system states, connectivity, and channel bandwidth are other constraints to design procedures for distributed large scale systems. The quasi-steady state model investigated in this paper is a reduced order model of the original system using singular perturbation techniques. This paper results in an optimal control synthesis to design an observer based feedback controller by standard stochastic control theory techniques using Linear Quadratic Gaussian (LQG) approach and Kalman filter design with less complexity and computation requirements. Numerical example is given at the end to demonstrate the efficiency of the proposed method.

Keywords: Decentralized, optimal control, output, singular perturb.

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Authors: Naiqin Feng, Yuhui Qiu, Yingshan Zhang, Fang Wang

Abstract:

A logic model for analyzing complex systems- stability is very useful to many areas of sciences. In the real world, we are enlightened from some natural phenomena such as “biosphere", “food chain", “ecological balance" etc. By research and practice, and taking advantage of the orthogonality and symmetry defined by the theory of multilateral matrices, we put forward a logic analysis model of stability of complex systems with three relations, and prove it by means of mathematics. This logic model is usually successful in analyzing stability of a complex system. The structure of the logic model is not only clear and simple, but also can be easily used to research and solve many stability problems of complex systems. As an application, some examples are given.

Keywords: Complex system, logic model, relation, stability.

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Authors: B. Chetti, W. A. Crosby

Abstract:

The work described in this paper is an investigation of the static and dynamic characteristics of two-lobe journal bearings taking into consideration the thermal effects. A thermo-hydrodynamic solution of a finite two-lobe journal bearing is performed by solving the generalized form Reynolds equation with the energy equation, taking into consideration viscosity variation across the film thickness. The static and dynamic characteristics were numerically obtained. The results are evaluated for different values of viscosity-temperature coefficient and Peclet number. The results show that considering the thermal effects in the solution of the two-lobe journal bearing has a marked on the study of its stability.

Keywords: Two-lobe bearing, thermal effect, static and dynamic characteristics.

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Authors: Li Hongwei

Abstract:

In this paper, a predator-prey model with time delay and habitat complexity is investigated. By analyzing the characteristic equations, the local stability of each feasible equilibria of the system is discussed and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By choosing the sum of two delays as a bifurcation parameter, we show that Hopf bifurcations can occur as  crosses some critical values. By deriving the equation describing the flow on the center manifold, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main theoretical results.

Keywords: Predator-prey system, delay, habitat complexity, HOPF bifurcation.

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Authors: R.Marsalek

Abstract:

This paper present a new way to find the aerodynamic characteristic equation of missile for the numerical trajectories prediction more accurate. The goal is to obtain the polynomial equation based on two missile characteristic parameters, angle of attack (α ) and flight speed (ν ). First, the understudied missile is modeled and used for flow computational model to compute aerodynamic force and moment. Assume that performance range of understudied missile where range -10< α <10 and 0< ν <200. After completely obtained results of all cases, the data are fit by polynomial interpolation to create equation of each case and then combine all equations to form aerodynamic characteristic equation, which will be used for trajectories simulation.

Keywords: ferro-precipitate, adsorption, SDS, zeta potential

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Authors: Abeer Amayri, Akif A. Bulgak

Abstract:

Due to today’s globalization as well as outsourcing practices of the companies, the Supply Chain (SC) performances have become more dependent on the efficient movement of material among places that are geographically dispersed, where there is more chance for disruptions. One such disruption is the quality and delivery uncertainties of outsourcing. These uncertainties could lead the products to be unsafe and, as is the case in a number of recent examples, companies may have to end up in recalling their products. As a result of these problems, there is a need to develop a methodology for selecting suppliers globally in view of risks associated with low quality and late delivery. Accordingly, we developed a two-stage stochastic model that captures the risks associated with uncertainty in quality and delivery as well as a solution procedure for the model. The stochastic model developed simultaneously optimizes supplier selection and purchase quantities under price discounts over a time horizon. In particular, our target is the study of global organizations with multiple sites and multiple overseas suppliers, where the pricing is offered in suppliers’ local currencies. Our proposed methodology is applied to a case study for a US automotive company having two assembly plants and four potential global suppliers to illustrate how the proposed model works in practice.

Keywords: Global supply chains, quality, stochastic programming, supplier selection.

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Authors: Attapon Charoenpon, Ekkarach Pankeaw

Abstract:

This paper present a new way to find the aerodynamic characteristic equation of missile for the numerical trajectories prediction more accurate. The goal is to obtain the polynomial equation based on two missile characteristic parameters, angle of attack (α ) and flight speed (╬¢ ). First, the understudied missile is modeled and used for flow computational model to compute aerodynamic force and moment. Assume that performance range of understudied missile where range -10< α <10 and 0< ╬¢ <200. After completely obtained results of all cases, the data are fit by polynomial interpolation to create equation of each case and then combine all equations to form aerodynamic characteristic equation, which will be used for trajectories simulation.

Keywords: Aerodynamic, Characteristic Equation, Angle ofAttack, Polynomial interpolation, Trajectories

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Authors: T. Yilmaz, N. Kirac

Abstract:

Beam-column elements are defined as structural members subjected to a combination of axial and bending forces. Lateral torsional buckling is one of the major failure modes in which beam-columns that are bent about its strong axis may buckle out of the plane by deflecting laterally and twisting. This study presents a compact closed-form equation that it can be used for calculating critical lateral torsional-buckling load of beam-columns with monosymmetric sections in the presence of a known axial load. Lateral-torsional buckling behavior of beam-columns subjected to constant axial force and various transverse load cases are investigated by using Ritz method in order to establish proposed equation. Lateral-torsional buckling loads calculated by presented formula are compared to finite element model results. ABAQUS software is utilized to generate finite element models of beam-columns. It is found out that lateral-torsional buckling load of beam-columns with monosymmetric sections can be determined by proposed equation and can be safely used in design.

Keywords: Lateral-torsional buckling, stability, beam-column, monosymmetric section.

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Authors: Jonathan H. Perez, Fumihiko Tanaka, Daisuke Hamanaka, Toshitaka Uchino

Abstract:

A finite element analysis was conducted to determine the effect of moisture diffusion and hygroscopic swelling in rice. A parallel simple stochastic modeling was performed to predict the number of grains cracked as a result of moisture absorption and hygroscopic swelling. Rice grains were soaked in thermally (25 oC) controlled water and then tested for compressive stress. The destructive compressive stress tests revealed through compressive stress calculation that the peak force required to cause cracking in grains soaked in water reduced with time as soaking duration was extended. Results of the experiment showed that several grains had their value of the predicted compressive stress below the von Mises stress and were interpreted as grains which become cracked and/or broke during soaking. The technique developed in this experiment will facilitate the approximation of the number of grains which will crack during soaking.

Keywords: Cracking, Finite element analysis, hygroscopic swelling, moisture diffusion, von Mises stress.

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Authors: Nasrin Eghbali

Abstract:

In this paper we consider the approximate Jordan maps and boundedness of these maps. Also we investigate the stability of approximate Jordan maps and prove some stability properties for approximate Jordan maps.

Keywords: Approximate Jordan map, stability.

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