Search results for: non-linear fractional variational problems

Authors: Alina Fedossova, Jose Jorge Sierra Molina

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SIP problems are part of non-classical optimization. There are problems in which the number of variables is finite, and the number of constraints is infinite. These are semi-infinite programming problems. Most algorithms for semi-infinite programming problems reduce the semi-infinite problem to a finite one and solve it by classical methods of linear or nonlinear programming. Typically, any of the constraints or the objective function is nonlinear, so the problem often involves nonlinear programming. An investment portfolio is a set of instruments used to reach the specific purposes of investors. The risk of the entire portfolio may be less than the risks of individual investment of portfolio. For example, we could make an investment of M euros in N shares for a specified period. Let yi> 0, the return on money invested in stock i for each dollar since the end of the period (i = 1, ..., N). The logical goal here is to determine the amount xi to be invested in stock i, i = 1, ..., N, such that we maximize the period at the end of ytx value, where x = (x1, ..., xn) and y = (y1, ..., yn). For us the optimal portfolio means the best portfolio in the ratio "risk-return" to the investor portfolio that meets your goals and risk ways. Therefore, investment goals and risk appetite are the factors that influence the choice of appropriate portfolio of assets. The investment returns are uncertain. Thus we have a semi-infinite programming problem. We solve a semi-infinite optimization problem of portfolio selection using the outer approximations methods. This approach can be considered as a developed Eaves-Zangwill method applying the multi-start technique in all of the iterations for the search of relevant constraints' parameters. The stochastic outer approximations method, successfully applied previously for robotics problems, Chebyshev approximation problems, air pollution and others, is based on the optimal criteria of quasi-optimal functions. As a result we obtain mathematical model and the optimal investment portfolio when yields are not clear from the beginning. Finally, we apply this algorithm to a specific case of a Colombian bank.

Keywords: outer approximation methods, portfolio problem, semi-infinite programming, numerial solution

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Authors: Asma Hanif, A. I. K. Butt, Shabir Ahmad, Rahim Ud Din, Mustafa Inc

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This paper is devoted to a study of the fuzzy fractional mathematical model reviewing the transmission dynamics of the infectious disease Covid-19. The proposed dynamical model consists of susceptible, exposed, symptomatic, asymptomatic, quarantine, hospitalized and recovered compartments. In this study, we deal with the fuzzy fractional model defined in Caputo’s sense. We show the positivity of state variables that all the state variables that represent different compartments of the model are positive. Using Gronwall inequality, we show that the solution of the model is bounded. Using the notion of the next-generation matrix, we find the basic reproduction number of the model. We demonstrate the local and global stability of the equilibrium point by using the concept of Castillo-Chavez and Lyapunov theory with the Lasalle invariant principle, respectively. We present the results that reveal the existence and uniqueness of the solution of the considered model through the fixed point theorem of Schauder and Banach. Using the fuzzy hybrid Laplace method, we acquire the approximate solution of the proposed model. The results are graphically presented via MATLAB-17.

Keywords: Caputo fractional derivative, existence and uniqueness, gronwall inequality, Lyapunov theory

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Authors: Sujeet Kumar Singh, Shiv Prasad Yadav

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This paper develops an approach for solving intuitionistic fuzzy linear fractional programming (IFLFP) problem where the cost of the objective function, the resources, and the technological coefficients are triangular intuitionistic fuzzy numbers. Here, the IFLFP problem is transformed into an equivalent crisp multi-objective linear fractional programming (MOLFP) problem. By using fuzzy mathematical programming approach the transformed MOLFP problem is reduced into a single objective linear programming (LP) problem. The proposed procedure is illustrated through a numerical example.

Keywords: triangular intuitionistic fuzzy number, linear programming problem, multi objective linear programming problem, fuzzy mathematical programming, membership function

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Authors: Nisa Özge Önal, Kamil Karaçuha, Göksu Hazar Erdinç, Banu Bahar Karaçuha, Ertuğrul Karaçuha

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The study aims to use a mathematical approach with the fractional calculus which is developed to have the ability to continuously analyze the factors related to the children’s height development. Until now, tracking the development of the child is getting more important and meaningful. Knowing and determining the factors related to the physical development of the child any desired time would provide better, reliable and accurate results for childcare. In this frame, 7 groups for height percentile curve (3th, 10th, 25th, 50th, 75th, 90th, and 97th) of Turkey are used. By using discrete height data of 0-18 years old children and the least squares method, a continuous curve is developed valid for any time interval. By doing so, in any desired instant, it is possible to find the percentage and location of the child in Percentage Chart. Here, with the help of the fractional calculus theory, a mathematical model is developed. The outcomes of the proposed approach are quite promising compared to the linear and the polynomial method. The approach also yields to predict the expected values of children in the sense of height.

Keywords: children growth percentile, children physical development, fractional calculus, linear and polynomial model

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Authors: Zhen Cheng, Xinyu Dai, Shujian Huang, Jiajun Chen

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Recently, explanatory natural language inference has attracted much attention for the interpretability of logic relationship prediction, which is also known as explanation generation for Natural Language Inference (NLI). Existing explanation generators based on discriminative Encoder-Decoder architecture have achieved noticeable results. However, we find that these discriminative generators usually generate explanations with correct evidence but incorrect logic semantic. It is due to that logic information is implicitly encoded in the premise-hypothesis pairs and difficult to model. Actually, logic information identically exists between premise-hypothesis pair and explanation. And it is easy to extract logic information that is explicitly contained in the target explanation. Hence we assume that there exists a latent space of logic information while generating explanations. Specifically, we propose a generative model called Variational Explanation Generator (VariationalEG) with a latent variable to model this space. Training with the guide of explicit logic information in target explanations, latent variable in VariationalEG could capture the implicit logic information in premise-hypothesis pairs effectively. Additionally, to tackle the problem of posterior collapse while training VariaztionalEG, we propose a simple yet effective approach called Logic Supervision on the latent variable to force it to encode logic information. Experiments on explanation generation benchmark—explanation-Stanford Natural Language Inference (e-SNLI) demonstrate that the proposed VariationalEG achieves significant improvement compared to previous studies and yields a state-of-the-art result. Furthermore, we perform the analysis of generated explanations to demonstrate the effect of the latent variable.

Keywords: natural language inference, explanation generation, variational auto-encoder, generative model

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Authors: Sadia Arshad, Yifa Tang, Dumitru Baleanu

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A fractional order mathematical model is proposed that incorporate CD8+ cells, natural killer cells, cytokines and tumor cells. The tumor cells growth in the absence of an immune response is modeled by logistic law as it was the simplest form for which predictions also agreed with the experimental data. Natural Killer Cells are our first line of defense. NK cells directly kill tumor cells through several mechanisms, including the release of cytoplasmic granules containing perforin and granzyme, expression of tumor necrosis factor (TNF) family members. The effect of the NK cells on the tumor cell population is expressed with the product term. Rational form is used to describe interaction between CD8+ cells and tumor cells. A number of cytokines are produced by NKs, including tumor necrosis factor TNF, IFN, and interleukin (IL-10). Source term for cytokines is modeled by Michaelis-Menten form to indicate the saturated effects of the immune response. Stability of the equilibrium points is discussed for biologically significant values of bifurcation parameters. We studied the treatment of fractional order system by investigating analytical conditions of tumor eradication. Numerical simulations are presented to illustrate the analytical results.

Keywords: cancer model, fractional calculus, numerical simulations, stability analysis

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Authors: Francis O. Nwawuru

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The convergence analysis of split monotone inclusion problems and fixed point problems of certain nonlinear mappings are investigated in the setting of real Hilbert spaces. Inertial extrapolation term in the spirit of Polyak is incorporated to speed up the rate of convergence. Under standard assumptions, a strong convergence of the proposed algorithm is established without computing the resolvent operator or involving Yosida approximation method. The stepsize involved in the algorithm does not depend on the spectral radius of the linear operator. Furthermore, applications of the proposed algorithm in solving some related optimization problems are also considered. Our result complements and extends numerous results in the literature.

Keywords: fixedpoint, hilbertspace, monotonemapping, resolventoperators

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Authors: Boo-Sung Koh, Seung-Eock Kim

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In this paper, a direct design using a nonlinear inelastic analysis is suggested. Also, this paper compares the load carrying capacity obtained by a nonlinear inelastic analysis with experiment results to verify the accuracy of the results. The allowable stress design results of a railroad through a plate girder bridge and the safety factor of the nonlinear inelastic analysis were compared to examine the safety performance. As a result, the load safety factor for the nonlinear inelastic analysis was twice as high as the required safety factor under the allowable stress design standard specified in the civil engineering structure design standards for urban magnetic levitation railways, which further verified the advantages of the proposed direct design method.

Keywords: direct design, nonlinear inelastic analysis, residual stress, initial geometric imperfection

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Authors: Darius Diogo Barreto, Ajeet Kumar, Sushma Santapuri

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We present an axisymmetric variational formulation for coupled extension-torsion-inflation deformation in magnetoelastomeric thin tubes when both azimuthal and axial magnetic fields are applied. The tube's material is assumed to have a preferred magnetization direction which imparts helical magnetic anisotropy to the tube. We have also derived the expressions of the first derivative of free energy per unit tube's undeformed length with respect to various imposed strain parameters. On applying the thin tube limit, the two nonlinear ordinary differential equations to obtain the in-plane radial displacement and radial component of the Lagrangian magnetic field get converted into a set of three simple algebraic equations. This allows us to obtain simple analytical expressions in terms of the applied magnetic field, magnetization direction, and magnetoelastic constants, which tell us how these parameters can be tuned to generate positive/negative Poisson's effect in such tubes. We consider both torsionally constrained and torsionally relaxed stretching of the tube. The study can be useful in designing magnetoelastic tubular actuators.

Keywords: nonlinear magnetoelasticity, extension-torsion coupling, negative Poisson's effect, helical anisotropy, thin tube

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Authors: Haci Mehmet Guzey, Levent Acar

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In this paper, a novel decentralized controller is developed for linear systems with nonlinear unknown interconnections. A model linear decoupled system is assigned for each system. By using the difference actual and model state dynamics, the problem is formulated as inverse problem. Then, the interconnected dynamics are approximated by using Galerkin’s expansion method for inverse problems. Two different sets of orthogonal basis functions are utilized to approximate the interconnected dynamics. Approximated interconnections are utilized in the controller to cancel the interconnections and decouple the systems. Subsequently, the interconnected systems behave as a collection of decoupled systems.

Keywords: decentralized control, inverse problems, large scale systems, nonlinear interconnections, basis functions, system identification

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Authors: Wanhyun Cho, Seongchae Seo, Soonja Kang

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In this work, we propose a variational technique for image contrast enhancement which utilizes global and local information around each pixel. The energy functional is defined by a weighted linear combination of three terms which are called on a local, a global contrast term and dispersion term. The first one is a local contrast term that can lead to improve the contrast of an input image by increasing the grey-level differences between each pixel and its neighboring to utilize contextual information around each pixel. The second one is global contrast term, which can lead to enhance a contrast of image by minimizing the difference between its empirical distribution function and a cumulative distribution function to make the probability distribution of pixel values becoming a symmetric distribution about median. The third one is a dispersion term that controls the departure between new pixel value and pixel value of original image while preserving original image characteristics as well as possible. Second, we derive the Euler-Lagrange equation for true image that can achieve the minimum of a proposed functional by using the fundamental lemma for the calculus of variations. And, we considered the procedure that this equation can be solved by using a gradient decent method, which is one of the dynamic approximation techniques. Finally, by conducting various experiments, we can demonstrate that the proposed method can enhance the contrast of colour images better than existing techniques.

Keywords: color image, contrast enhancement technique, variational approach, Euler-Lagrang equation, dynamic approximation method, EME measure

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Authors: F. J. Adeyeye, D. Eni, K. M. Okedoye

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In this paper we present a Hybrid of Adomian decomposition method (ADM). This is the substitution of a One-step method of Taylor’s series approximation of orders I and II, into the nonlinear part of Adomian decomposition method resulting in a convergent series scheme. This scheme is applied to solve some Logistic problems represented as Abelian differential equation and the results are compared with the actual solution and Runge-kutta of order IV in order to ascertain the accuracy and efficiency of the scheme. The findings shows that the scheme is efficient enough to solve logistic problems considered in this paper.

Keywords: Adomian decomposition method, nonlinear part, one-step method, Taylor series approximation, hybrid of Adomian polynomial, logistic problem, Malthusian parameter, Verhulst Model

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Authors: Mohammadreza Akbari, Pooya Soleimani Besheli, Reza Khalili, Davood Domiri Danji

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In this conference, our aims are accuracy, capabilities and power at solving of the complicated non-linear partial differential. Our purpose is to enhance the ability to solve the mentioned nonlinear differential equations at basic science and engineering field and similar issues with a simple and innovative approach. As we know most of engineering system behavior in practical are nonlinear process (especially basic science and engineering field, etc.) and analytical solving (no numeric) these problems are difficult, complex, and sometimes impossible like (Fluids and Gas wave, these problems can't solve with numeric method, because of no have boundary condition) accordingly in this symposium we are going to exposure an innovative approach which we have named it Akbari-Ganji's Method or AGM in engineering, that can solve sets of coupled nonlinear differential equations (ODE, PDE) with high accuracy and simple solution and so this issue will emerge after comparing the achieved solutions by Numerical method (Runge-Kutta 4th). Eventually, AGM method will be proved that could be created huge evolution for researchers, professors and students in whole over the world, because of AGM coding system, so by using this software we can analytically solve all complicated linear and nonlinear partial differential equations, with help of that there is no difficulty for solving all nonlinear differential equations. Advantages and ability of this method (AGM) as follow: (a) Non-linear Differential equations (ODE, PDE) are directly solvable by this method. (b) In this method (AGM), most of the time, without any dimensionless procedure, we can solve equation(s) by any boundary or initial condition number. (c) AGM method always is convergent in boundary or initial condition. (d) Parameters of exponential, Trigonometric and Logarithmic of the existent in the non-linear differential equation with AGM method no needs Taylor expand which are caused high solve precision. (e) AGM method is very flexible in the coding system, and can solve easily varieties of the non-linear differential equation at high acceptable accuracy. (f) One of the important advantages of this method is analytical solving with high accuracy such as partial differential equation in vibration in solids, waves in water and gas, with minimum initial and boundary condition capable to solve problem. (g) It is very important to present a general and simple approach for solving most problems of the differential equations with high non-linearity in engineering sciences especially at civil engineering, and compare output with numerical method (Runge-Kutta 4th) and Exact solutions.

Keywords: new approach, AGM, sets of coupled nonlinear differential equation, exact solutions, numerical

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Authors: H. Triki, Y. Hamaizi, A. El-Akrmi

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We consider the higher order nonlinear Schrödinger equation model with fourth-order dispersion, cubic-quintic terms, and self-steepening. This equation governs the propagation of fem to second pulses in optical fibers. We present new bright and dark solitary wave type solutions for such a model under certain parametric conditions. This kind of solution may be useful to explain some physical phenomena related to wave propagation in a nonlinear optical fiber systems supporting high-order nonlinear and dispersive effects.

Keywords: nonlinear Schrödinger equation, high-order effects, soliton solution

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Authors: Jorge Gonzalez Camus, Carlos Lizama

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In this work is proved the existence of at least one globally attractive mild solution to the Cauchy problem, for fractional evolution equation of neutral type, involving the fractional derivate in Caputo sense. An almost sectorial operator on a Banach space X and a kernel belonging to a large class appears in the equation, which covers many relevant cases from physics applications, in particular, the important case of time - fractional evolution equations of neutral type. The main tool used in this work was the Hausdorff measure of noncompactness and fixed point theorems, specifically Darbo-type. Initially, the equation is a Cauchy problem, involving a fractional derivate in Caputo sense. Then, is formulated the equivalent integral version, and defining a convenient functional, using the analytic integral resolvent operator, and verifying the hypothesis of the fixed point theorem of Darbo type, give us the existence of mild solution for the initial problem. Furthermore, each mild solution is globally attractive, a property that is desired in asymptotic behavior for that solution.

Keywords: attractive mild solutions, integral Volterra equations, neutral type equations, non-local in time equations

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Authors: Ali Motalebi Saraji, Reza Zarei Lamuki

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Speed control and behavior improvement of permanent magnet synchronous motors (PMSM) that have reliable performance, low loss, and high power density, especially in industrial drives, are of great importance for researchers. Because of its importance in this paper, coefficients optimization of proportional-integrator-derivative fractional order controller is presented using Particle Swarm Optimization (PSO) algorithm in order to improve the behavior of PMSM in its speed control loop. This improvement is simulated in MATLAB software for the proposed optimized proportional-integrator-derivative fractional order controller with a Genetic algorithm and compared with a full order controller with a classic optimization method. Simulation results show the performance improvement of the proposed controller with respect to two other controllers in terms of rising time, overshoot, and settling time.

Keywords: speed control loop of permanent magnet synchronous motor, fractional and full order proportional-integrator-derivative controller, coefficients optimization, particle swarm optimization, improvement of behavior

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Authors: Takashi Shimizu, Tomoaki Hashimoto

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A class of implicit systems is known as a more generalized class of systems than a class of explicit systems. To establish a control method for such a generalized class of systems, we adopt model predictive control method which is a kind of optimal feedback control with a performance index that has a moving initial time and terminal time. However, model predictive control method is inapplicable to systems whose all state variables are not exactly known. In other words, model predictive control method is inapplicable to systems with limited measurable states. In fact, it is usual that the state variables of systems are measured through outputs, hence, only limited parts of them can be used directly. It is also usual that output signals are disturbed by process and sensor noises. Hence, it is important to establish a state estimation method for nonlinear implicit systems with taking the process noise and sensor noise into consideration. To this purpose, we apply the model predictive control method and unscented Kalman filter for solving the optimization and estimation problems of nonlinear implicit systems, respectively. The objective of this study is to establish a model predictive control with unscented Kalman filter for nonlinear implicit systems.

Keywords: optimal control, nonlinear systems, state estimation, Kalman filter

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Authors: Ahmad Alhawarat, Mustafa Mamat, Mohd Rivaie, Ismail Mohd

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Conjugate gradient method has been enormously used to solve large scale unconstrained optimization problems due to the number of iteration, memory, CPU time, and convergence property, in this paper we find a new class of nonlinear conjugate gradient coefficient with global convergence properties proved by exact line search. The numerical results for our new βK give a good result when it compared with well-known formulas.

Keywords: conjugate gradient method, conjugate gradient coefficient, global convergence

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Authors: Wataru Nakamura, Tomoaki Hashimoto, Liang-Kuang Chen

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This paper provides a state estimation method for automatic control systems of nonlinear vehicle dynamics. A nonlinear tire model is employed to represent the realistic behavior of a vehicle. In general, all the state variables of control systems are not precisedly known, because those variables are observed through output sensors and limited parts of them might be only measurable. Hence, automatic control systems must incorporate some type of state estimation. It is needed to establish a state estimation method for nonlinear vehicle dynamics with restricted measurable state variables. For this purpose, unscented Kalman filter method is applied in this study for estimating the state variables of nonlinear vehicle dynamics. The objective of this paper is to propose a state estimation method using unscented Kalman filter for nonlinear vehicle dynamics. The effectiveness of the proposed method is verified by numerical simulations.

Keywords: state estimation, control systems, observer systems, nonlinear systems

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Authors: Prashant Pandey

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In the present article, an effective Laguerre collocation method is used to obtain the approximate solution of a system of coupled fractional-order non-linear reaction-advection-diffusion equation with prescribed initial and boundary conditions. In the proposed scheme, Laguerre polynomials are used together with an operational matrix and collocation method to obtain approximate solutions of the coupled system, so that our proposed model is converted into a system of algebraic equations which can be solved employing the Newton method. The solution profiles of the coupled system are presented graphically for different particular cases. The salient feature of the present article is finding the stability analysis of the proposed method and also the demonstration of the lower variation of solute concentrations with respect to the column length in the fractional-order system compared to the integer-order system. To show the higher efficiency, reliability, and accuracy of the proposed scheme, a comparison between the numerical results of Burger’s coupled system and its existing analytical result is reported. There are high compatibility and consistency between the approximate solution and its exact solution to a higher order of accuracy. The exhibition of error analysis for each case through tables and graphs confirms the super-linearly convergence rate of the proposed method.

Keywords: fractional coupled PDE, stability and convergence analysis, diffusion equation, Laguerre polynomials, spectral method

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Authors: Hector Carcel Luis Alberiko Gil-Alana

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This paper examines the time series behavior of three variables (GDP, Price level of Consumption and Population) in the eight countries that belong to the West African Economic and Monetary Union (WAEMU), which are Benin, Burkina Faso, Côte d’Ivoire, Guinea-Bissau, Mali, Niger, Senegal and Togo. The reason for carrying out this study lies in the considerable heterogeneity that can be perceived in the data from these countries. We conduct a long memory and fractional integration modeling framework and we also identify potential breaks in the data. The aim of the study is to perceive up to which degree the eight West African countries that belong to the same monetary union follow the same economic patterns of stability. Testing for mean reversion, we only found strong evidence of it in the case of Senegal for the Price level of Consumption, and in the cases of Benin, Burkina Faso and Senegal for GDP.

Keywords: West Africa, Monetary Union, fractional integration, economic patterns

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Authors: Graziano Chesi

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The topological entropy plays a key role in linear dynamical systems, allowing one to establish the existence of stabilizing feedback controllers for linear systems in the presence of communications constraints. This paper addresses the determination of a robust value of the topological entropy in nonlinear dynamical systems, specifically the largest value of the topological entropy over all linearized models in a region of interest of the state space. It is shown that a sufficient condition for establishing upper bounds of the sought robust value of the topological entropy can be given in terms of a semidefinite program (SDP), which belongs to the class of convex optimization problems.

Keywords: non-linear system, communication constraint, topological entropy

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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: Bello Yusuf Idi

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The Gongola arm of the Upper Banue Trough, Northeastern Nigeria is predominantly covered by the outcrops of Limestone-bearing rocks in form of Sandstone with intercalation of carbonate clay, shale, basaltic, felsphatic and migmatide rocks at subpixel dimension. In this work, subpixel classification algorithm was used to classify the data acquired from landsat 7 Enhance Thematic Mapper (ETM+) satellite system with the aim of producing fractional distribution image for three most economically important solid minerals of the area: Limestone, Basalt and Migmatide. Linear Spectral Unmixing (LSU) algorithm was used to produce fractional distribution image of abundance of the three mineral resources within a 100Km2 portion of the area. The results show that the minerals occur at different proportion all over the area. The fractional map could therefore serve as a guide to the ongoing reconnaissance for the economic potentiality of the formation.

Keywords: linear spectral un-mixing, upper benue trough, gongola arm, geological engineering

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Authors: Sergio Haram Sarmiento, Arcady Ponosov

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Based on appropriate multivariate statistical methodology, we suggest a generic framework for efficient parameter estimation for ordinary differential equations and the corresponding nonlinear models. In this framework classical linear regression strategies is refined into a nonlinear regression by a locally linear modelling technique (known as metamodelling). The approach identifies those latent variables of the given model that accumulate most information about it among all approximations of the same dimension. The method is applied to several benchmark problems, in particular, to the so-called ”power-law systems”, being non-linear differential equations typically used in Biochemical System Theory.

Keywords: principal component analysis, generalized law of mass action, parameter estimation, metamodels

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Authors: Aquil Ahmed, Srikant Gupta, Irfan Ali

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In this paper, we study a multi-objective capacitated transportation problem (MOCTP) with mixed constraints. This paper is comprised of the modelling and optimisation of an MOCTP in a fuzzy environment in which some goals are fractional and some are linear. In real life application of the fuzzy goal programming (FGP) problem with multiple objectives, it is difficult for the decision maker(s) to determine the goal value of each objective precisely as the goal values are imprecise or uncertain. Also, we developed the concept of linearization of fractional goal for solving the MOCTP. In this paper, imprecision of the parameter is handled by the concept of fuzzy set theory by considering these parameters as a trapezoidal fuzzy number. α-cut approach is used to get the crisp value of the parameters. Numerical examples are used to illustrate the method for solving MOCTP.

Keywords: capacitated transportation problem, multi objective linear programming, multi-objective fractional programming, fuzzy goal programming, fuzzy sets, trapezoidal fuzzy number

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Authors: R. Rama Kishore, Sunesh Malik

Abstract:

Development in the usage of internet for different purposes in recent times creates great threat for the copy right protection of the digital images. Digital watermarking is the best way to rescue from the said problem. This paper presents detailed review of the different watermarking techniques, latest trends in the field and categorized like spatial and transform domain, blind and non-blind methods, visible and non visible techniques etc. It also discusses the different optimization techniques used in the field of watermarking in order to improve the robustness and imperceptibility of the method. Different measures are discussed to evaluate the performance of the watermarking algorithm. At the end, this paper proposes a watermarking algorithm using (k.n) shares of halftone visual cryptography (HVC) instead of (2, 2) share cryptography. (k,n) shares visual cryptography improves the security of the watermark. As halftone is a method of reprographic, it helps in improving the visual quality of watermark image. The proposed method uses fractional transformation to improve the robustness of the copyright protection of the method.

Keywords: digital watermarking, fractional transform, halftone, visual cryptography

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Authors: A. Keshavarz, Z. Roosta

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In this paper, propagation of cos-Gaussian beam in strongly nonlocal nonlinear media has been stimulated by using paraxial group transformation. At first, cos-Gaussian beam, nonlocal nonlinear media, critical power, transfer matrix, and paraxial group transformation are introduced. Then, the propagation of the cos-Gaussian beam in strongly nonlocal nonlinear media is simulated. Results show that beam propagation has periodic structure during self-focusing effect in this case. However, this simple method can be used for investigation of propagation of kinds of beams in ABCD optical media.

Keywords: paraxial group transformation, nonlocal nonlinear media, cos-Gaussian beam, ABCD law

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Authors: A. Giniatoulline

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A nonlinear model of the mathematical fluid dynamics which describes the motion of an incompressible viscous rotating fluid in a homogeneous gravitational field is considered. The model is a generalization of the known Navier-Stokes system with the addition of the Coriolis parameter and the equations for changeable density. An explicit algorithm for the solution is constructed, and the proof of the existence and uniqueness theorems for the strong solution of the nonlinear problem is given. For the linear case, the localization and the structure of the spectrum of inner waves are also investigated.

Keywords: Galerkin method, Navier-Stokes equations, nonlinear partial differential equations, Sobolev spaces, stratified fluid

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Authors: Juliana A. Knocikova

Abstract:

Methods of nonlinear signal analysis are based on finding that random behavior can arise in deterministic nonlinear systems with a few degrees of freedom. Considering the dynamical systems, entropy is usually understood as a rate of information production. Changes in temporal dynamics of physiological data are indicating evolving of system in time, thus a level of new signal pattern generation. During last decades, many algorithms were introduced to assess some patterns of physiological responses to external stimulus. However, the reflex responses are usually characterized by short periods of time. This characteristic represents a great limitation for usual methods of nonlinear analysis. To solve the problems of short recordings, parameter of approximate entropy has been introduced as a measure of system complexity. Low value of this parameter is reflecting regularity and predictability in analyzed time series. On the other side, increasing of this parameter means unpredictability and a random behavior, hence a higher system complexity. Reduced neurophysiological data complexity has been observed repeatedly when analyzing electroneurogram and electromyogram activities during defence reflex responses. Quantitative phrenic neurogram changes are also obvious during severe hypoxia, as well as during airway reflex episodes. Concluding, the approximate entropy parameter serves as a convenient tool for analysis of reflex behavior characterized by short lasting time series.

Keywords: approximate entropy, neurophysiological data, nonlinear dynamics, reflex

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