Journal of Applied Mathematics The latest articles from Hindawi Publishing Corporation © 2016 , Hindawi Publishing Corporation . All rights reserved. Viscosity Solution of Mean-Variance Portfolio Selection of a Jump Markov Process with No-Shorting Constraints Wed, 04 May 2016 11:27:34 +0000 We consider the so-called mean-variance portfolio selection problem in continuous time under the constraint that the short-selling of stocks is prohibited where all the market coefficients are random processes. In this situation the Hamilton-Jacobi-Bellman (HJB) equation of the value function of the auxiliary problem becomes a coupled system of backward stochastic partial differential equation. In fact, the value function often does not have the smoothness properties needed to interpret it as a solution to the dynamic programming partial differential equation in the usual (classical) sense; however, in such cases can be interpreted as a viscosity solution. Here we show the unicity of the viscosity solution and we see that the optimal and the value functions are piecewise linear functions based on some Riccati differential equations. In particular we solve the open problem posed by Li and Zhou and Zhou and Yin. Moussa Kounta Copyright © 2016 Moussa Kounta. All rights reserved. Split Common Fixed Point Problem for a Class of Total Asymptotic Pseudocontractions Thu, 28 Apr 2016 14:03:27 +0000 We study the split common fixed point problem (SCFP) for a class of total asymptotically pseudocontractive mappings. We obtain some important properties of our class of mappings including the demiclosedness property and the closedness and convexity of the fixed point set. We then propose an algorithm and prove weak and strong convergence theorems for the approximation of solutions of the SCFP for certain class of these mappings. E. E. Chima and M. O. Osilike Copyright © 2016 E. E. Chima and M. O. Osilike. All rights reserved. Self-Triggered Model Predictive Control for Linear Systems Based on Transmission of Control Input Sequences Wed, 27 Apr 2016 16:05:28 +0000 A networked control system (NCS) is a control system where components such as plants and controllers are connected through communication networks. Self-triggered control is well known as one of the control methods in NCSs and is a control method that for sampled-data control systems both the control input and the aperiodic sampling interval (i.e., the transmission interval) are computed simultaneously. In this paper, a self-triggered model predictive control (MPC) method for discrete-time linear systems with disturbances is proposed. In the conventional MPC method, the first one of the control input sequence obtained by solving the finite-time optimal control problem is sent and applied to the plant. In the proposed method, the first some elements of the control input sequence obtained are sent to the plant, and each element is sequentially applied to the plant. The number of elements is decided according to the effect of disturbances. In other words, transmission intervals can be controlled. Finally, the effectiveness of the proposed method is shown by numerical simulations. Koichi Kobayashi Copyright © 2016 Koichi Kobayashi. All rights reserved. The Variational Homotopy Perturbation Method for Solving Dimensional Burgers’ Equations Tue, 19 Apr 2016 13:55:34 +0000 The variational homotopy perturbation method VHPM is used for solving -dimensional Burgers’ system. Some examples are examined to validate that the method reduced the calculation size, treating the difficulty of nonlinear term and the accuracy. F. A. Hendi, B. S. Kashkari, and A. A. Alderremy Copyright © 2016 F. A. Hendi et al. All rights reserved. Pricing Strategies in the Remanufacturing Market for the Uncertain Market Size in the Second Period Thu, 31 Mar 2016 17:35:59 +0000 Our main endeavor is to investigate the effect of the uncertain market size in the second period on the pricing strategies in the remanufacturing market. Observing the previous research, we find that the market size in the second period is always supposed to be certain. However, there is substantial empirical and experimental evidence that the market size in reality deviates from this assumption. In fact, though the market size in the first period is definitized, it is difficult to confirm the change of the market scale in the second period, since this change is affected by all kinds of elements, such as the awareness of environmental protection, some consumers’ psychological factors, and the related governments’ policies. Hence, we pay attention to the case in which the change rate of the market scale in the second period is random variable. We suppose that this rate satisfies uniform distribution on . Underlying this assumption, we further provide an insight into the game-playing relationship between original equipment manufacturers (OEMs) and remanufacturers. Moreover, we delicately and subtly incorporate the game theory, stochastic analysis, adversarial risk analysis (ARA), and optimization methods into the pricing strategies in the remanufacturing market. Last but not least, considerable efforts and attempts have been made to subtly test the sensitivity of an optimal solution to the different parameters. Liurui Deng and Shenggang Yang Copyright © 2016 Liurui Deng and Shenggang Yang. All rights reserved. Noise Folding in Completely Perturbed Compressed Sensing Wed, 30 Mar 2016 11:12:55 +0000 This paper first presents a new generally perturbed compressed sensing (CS) model , which incorporated a general nonzero perturbation into sensing matrix and a noise into signal simultaneously based on the standard CS model and is called noise folding in completely perturbed CS model. Our construction mainly will whiten the new proposed CS model and explore in restricted isometry property () and coherence of the new CS model under some conditions. Finally, we use OMP to give a numerical simulation which shows that our model is feasible although the recovered value of signal is not exact compared with original signal because of measurement noise , signal noise , and perturbation involved. Limin Zhou, Xinxin Niu, and Jing Yuan Copyright © 2016 Limin Zhou et al. All rights reserved. Assessing Heterogeneity for Factor Analysis Model with Continuous and Ordinal Outcomes Wed, 30 Mar 2016 08:55:00 +0000 Factor analysis models with continuous and ordinal responses are a useful tool for assessing relations between the latent variables and mixed observed responses. These models have been successfully applied to many different fields, including behavioral, educational, and social-psychological sciences. However, within the Bayesian analysis framework, most developments are constrained within parametric families, of which the particular distributions are specified for the parameters of interest. This leads to difficulty in dealing with outliers and/or distribution deviations. In this paper, we propose a Bayesian semiparametric modeling for factor analysis model with continuous and ordinal variables. A truncated stick-breaking prior is used to model the distributions of the intercept and/or covariance structural parameters. Bayesian posterior analysis is carried out through the simulation-based method. Blocked Gibbs sampler is implemented to draw observations from the complicated posterior. For model selection, the logarithm of pseudomarginal likelihood is developed to compare the competing models. Empirical results are presented to illustrate the application of the methodology. Ye-Mao Xia and Jian-Wei Gou Copyright © 2016 Ye-Mao Xia and Jian-Wei Gou. All rights reserved. Chaotic Convection in a Viscoelastic Fluid Saturated Porous Medium with a Heat Source Thu, 10 Mar 2016 09:11:48 +0000 Chaotic convection in a viscoelastic fluid saturated porous layer, heated from below, is studied by using Oldroyd’s type constituting relation and in the presence of an internal heat source. A modified Darcy law is used in the momentum equation, and a heat source term has been considered in energy equation. An autonomous system of fourth-order differential equations has been deduced by using a truncated Fourier series. Effect of internal heat generation on chaotic convection has been investigated. The asymptotic behavior can be stationary, periodic, or chaotic, depending upon the flow parameters. Construction of four-scroll, or “two-butterfly,” and chaotic attractor has been examined. B. S. Bhadauria Copyright © 2016 B. S. Bhadauria. All rights reserved. Limit Cycles for the Class of -Dimensional Polynomial Differential Systems Mon, 07 Mar 2016 09:52:14 +0000 We perturb the differential system , , and for inside the class of all polynomial differential systems of degree in , and we prove that at most limit cycles can be obtained for the perturbed system using the first-order averaging theory. Zouhair Diab and Amar Makhlouf Copyright © 2016 Zouhair Diab and Amar Makhlouf. All rights reserved. Novel Analytical and Numerical Methods in Heat Transfer Enhancement and Thermal Management Mon, 22 Feb 2016 07:01:51 +0000 Assunta Andreozzi, Guy Lauriat, Qiuwang Wang, Sotirios Karellas, and Yogesh Jaluria Copyright © 2016 Assunta Andreozzi et al. All rights reserved. Numerical Simulation of Bubble Coalescence and Break-Up in Multinozzle Jet Ejector Sun, 21 Feb 2016 11:48:33 +0000 Designing the jet ejector optimally is a challenging task and has a great impact on industrial applications. Three different sets of nozzles (namely, 1, 3, and 5) inside the jet ejector are compared in this study by using numerical simulations. More precisely, dynamics of bubble coalescence and breakup in the multinozzle jet ejectors are studied by means of Computational Fluid Dynamics (CFD). The population balance approach is used for the gas phase such that different bubble size groups are included in CFD and the number densities of each of them are predicted in CFD simulations. Here, commercial CFD software ANSYS Fluent 14.0 is used. The realizable - turbulence model is used in CFD code in three-dimensional computational domains. It is clear that Reynolds-Averaged Navier-Stokes (RANS) models have their limitations, but on the other hand, turbulence modeling is not the key issue in this study and we can assume that the RANS models can predict turbulence of the carrying phase accurately enough. In order to validate our numerical predictions, results of one, three, and five nozzles are compared to laboratory experiments data for Cl2-NaOH system. Predicted gas volume fractions, bubble size distributions, and resulting number densities of the different bubble size groups as well as the interfacial area concentrations are in good agreement with experimental results. Dhanesh Patel, Ashvinkumar Chaudhari, Arto Laari, Matti Heiliö, Jari Hämäläinen, and Kishorilal Agrawal Copyright © 2016 Dhanesh Patel et al. All rights reserved. Green’s Functions for Heat Conduction for Unbounded and Bounded Rectangular Spaces: Time and Frequency Domain Solutions Wed, 17 Feb 2016 07:58:04 +0000 This paper presents a set of fully analytical solutions, together with explicit expressions, in the time and frequency domain for the heat conduction response of homogeneous unbounded and of bounded rectangular spaces (three-, two-, and one-dimensional spaces) subjected to point, line, and plane heat diffusion sources. Particular attention is given to the case of spatially sinusoidal, harmonic line sources. In the literature this problem is often referred to as the two-and-a-half-dimensional fundamental solution or 2.5D Green’s functions. These equations are very useful for formulating three-dimensional thermodynamic problems by means of integral transforms methods and/or boundary elements. The image source technique is used to build up different geometries such as half-spaces, corners, rectangular pipes, and parallelepiped boxes. The final expressions are verified here by applying the equations to problems for which the solution is known analytically in the time domain. Inês Simões, António Tadeu, and Nuno Simões Copyright © 2016 Inês Simões et al. All rights reserved. Production Planning of a Failure-Prone Manufacturing System under Different Setup Scenarios Mon, 15 Feb 2016 07:16:22 +0000 This paper presents a control problem for the optimization of the production and setup activities of an industrial system operating in an uncertain environment. This system is subject to random disturbances (breakdowns and repairs). These disturbances can engender stock shortages. The considered industrial system represents a well-known production context in industry and consists of a machine producing two types of products. In order to switch production from one product type to another, a time factor and a reconfiguration cost for the machine are associated with the setup activities. The parts production rates and the setup strategies are the decision variables which influence the inventory and the capacity of the system. The objective of the study is to find the production and setup policies which minimize the setup and inventory costs, as well as those associated with shortages. A modeling approach based on stochastic optimal control theory and a numerical algorithm used to solve the obtained optimality conditions are presented. The contribution of the paper, for industrial systems not studied in the literature, is illustrated through a numerical example and a comparative study. Guy-Richard Kibouka, Donatien Nganga-Kouya, Jean-Pierre Kenne, Victor Songmene, and Vladimir Polotski Copyright © 2016 Guy-Richard Kibouka et al. All rights reserved. A Method to Construct Generalized Fibonacci Sequences Sun, 14 Feb 2016 12:15:10 +0000 The main purpose of this paper is to study the convergence properties of Generalized Fibonacci Sequences and the series of partial sums associated with them. When the proper values of an real matrix are real and different, we give a necessary and sufficient condition for the convergence of the matrix sequence to a matrix . Adalberto García-Máynez and Adolfo Pimienta Acosta Copyright © 2016 Adalberto García-Máynez and Adolfo Pimienta Acosta. All rights reserved. Steady Flow of Couple-Stress Fluid in Constricted Tapered Artery: Effects of Transverse Magnetic Field, Moving Catheter, and Slip Velocity Wed, 03 Feb 2016 07:29:21 +0000 Steady flow of a couple-stress fluid in constricted tapered artery has been studied under the effects of transverse magnetic field, moving catheter, and slip velocity. With the help of Bessel’s functions, analytic expressions for axial velocity, flow rate, impedance, and wall shear stress have been obtained. It is of interest to note that these solutions can be used for different types of fluid flow in tubes and not only the case of blood. The effects of various geometric parameters, the parameters arising out of the fluid considered and the magnetic field, are discussed by considering the slip velocity, the catheter velocity, and tapering angle. The study of the above model is very important as it has direct applications in the treatment of cardiovascular diseases. Hamzah Bakhti and Lahcen Azrar Copyright © 2016 Hamzah Bakhti and Lahcen Azrar. All rights reserved. Chemical Entropy Generation and MHD Effects on the Unsteady Heat and Fluid Flow through a Porous Medium Wed, 20 Jan 2016 09:42:04 +0000 Chemical entropy generation and magnetohydrodynamic effects on the unsteady heat and fluid flow through a porous medium have been numerically investigated. The entropy generation due to the use of a magnetic field and porous medium effects on heat transfer, fluid friction, and mass transfer have been analyzed numerically. Using a similarity transformation, the governing equations of continuity, momentum, and energy and concentration equations, of nonlinear system, were reduced to a set of ordinary differential equations and solved numerically. The effects of unsteadiness parameter, magnetic field parameter, porosity parameter, heat generation/absorption parameter, Lewis number, chemical reaction parameter, and Brinkman number parameter on the velocity, the temperature, the concentration, and the entropy generation rates profiles were investigated and the results were presented graphically. Gamal M. Abdel-Rahman Rashed Copyright © 2016 Gamal M. Abdel-Rahman Rashed. All rights reserved. Linear Programming Problem with Interval Type 2 Fuzzy Coefficients and an Interpretation for Its Constraints Wed, 06 Jan 2016 11:21:02 +0000 Interval type 2 fuzzy numbers are a special kind of type 2 fuzzy numbers. These numbers can be described by triangular and trapezoidal shapes. In this paper, first, perfectly normal interval type 2 trapezoidal fuzzy numbers with their left-hand and right-hand spreads and their core have been introduced, which are normal and convex; then a new type of fuzzy arithmetic operations for perfectly normal interval type 2 trapezoidal fuzzy numbers has been proposed based on the extension principle of normal type 1 trapezoidal fuzzy numbers. Moreover, in this proposal, linear programming problems with resources and technology coefficients are perfectly normal interval type 2 fuzzy numbers. To solve this kind of fuzzy linear programming problems, a method based on the degree of satisfaction (or possibility degree) of the constraints has been introduced. In this method the fulfillment of the constraints can be measured with the help of ranking method of fuzzy numbers. Optimal solution is obtained at different degree of satisfaction by using Barnes algorithm with the help of MATLAB. Finally, the optimal solution procedure is illustrated with numerical example. A. Srinivasan and G. Geetharamani Copyright © 2016 A. Srinivasan and G. Geetharamani. All rights reserved. The Dynamics of Epidemic Model with Two Types of Infectious Diseases and Vertical Transmission Tue, 05 Jan 2016 13:40:59 +0000 An epidemic model that describes the dynamics of the spread of infectious diseases is proposed. Two different types of infectious diseases that spread through both horizontal and vertical transmission in the host population are considered. The basic reproduction number is determined. The local and the global stability of all possible equilibrium points are achieved. The local bifurcation analysis and Hopf bifurcation analysis for the four-dimensional epidemic model are studied. Numerical simulations are used to confirm our obtained analytical results. Raid Kamel Naji and Reem Mudar Hussien Copyright © 2016 Raid Kamel Naji and Reem Mudar Hussien. All rights reserved. The Optimal Insurance Policy for the General Fixed Cost of Handling an Indemnity under Rank-Dependent Expected Utility Thu, 31 Dec 2015 05:51:42 +0000 Based on Bernard et al.’s research, we focus on the Pareto optimal insurance design with the insured’s Rank-Dependent Expected Utility (RDEU). Compared with their previous work, our novelties are the more general fixed cost function of the insurer and the discussion of adverse selection and moral hazard. In particular, Bernard et al. only consider the case in which the fixed cost function of handling an indemnity is the linear function. However, the fixed cost function is not just linear functions in real insurance market. So, we explore the more general fixed cost function including both the linear and nonlinear functions. On the other hand, we consider adverse selection and moral hazard which are involved by Bernard et al. Leading adverse selection and moral hazard into our research makes our results more practical and meaningful. Moreover, we provide an insight into the sensitivity of an optimal solution for the insured’s initial wealth and the parameters related to the fixed cost function of handling an indemnity. We further compare the two different utility functions of the insured in terms of influence of optimal policy analysis. Liurui Deng Copyright © 2015 Liurui Deng. All rights reserved. Extremal Trees with respect to Number of -Edge Colourings Mon, 14 Dec 2015 06:51:11 +0000 We determine the smallest and the largest number of -edge colourings in trees. We prove that the star is a unique tree that maximizes the number of all of the -edge colourings and that the path is a unique tree that minimizes it. Krzysztof Piejko Copyright © 2015 Krzysztof Piejko. All rights reserved. Compensating Operator and Weak Convergence of Semi-Markov Process to the Diffusion Process without Balance Condition Sun, 06 Dec 2015 11:25:46 +0000 Weak convergence of semi-Markov processes in the diffusive approximation scheme is studied in the paper. This problem is not new and it is studied in many papers, using convergence of random processes. Unlike other studies, we used in this paper concept of the compensating operator. It enables getting sufficient conditions of weak convergence under the conditions on the local characteristics of output semi-Markov process. Igor V. Malyk Copyright © 2015 Igor V. Malyk. All rights reserved. Numerical Solution of Pantograph-Type Delay Differential Equations Using Perturbation-Iteration Algorithms Wed, 02 Dec 2015 14:19:52 +0000 The pantograph equation is a special type of functional differential equations with proportional delay. The present study introduces a compound technique incorporating the perturbation method with an iteration algorithm to solve numerically the delay differential equations of pantograph type. We put forward two types of algorithms, depending upon the order of derivatives in the Taylor series expansion. The crucial convenience of this method when compared with other perturbation methods is that this method does not require a small perturbation parameter. Furthermore, a relatively fast convergence of the iterations to the exact solutions and more accurate results can be achieved. Several illustrative examples are given to demonstrate the efficiency and reliability of the technique, even for nonlinear cases. M. Mustafa Bahşi and Mehmet Çevik Copyright © 2015 M. Mustafa Bahşi and Mehmet Çevik. All rights reserved. Analytical Solutions of Ionic Diffusion and Heat Conduction in Multilayered Porous Media Sun, 29 Nov 2015 09:09:47 +0000 Ionic diffusion and heat conduction in a multiple layered porous medium have many important engineering applications. One of the examples is the chloride ions from deicers penetrating into concrete structures such as bridge decks. Different overlays can be placed on top of concrete surface to slowdown the chloride penetration. In this paper, the chloride ion diffusion equations were established for concrete structures with multiple layers of protective system. By using Laplace transformation, an analytical solution was developed first for chloride concentration profiles in two-layered system and then extended to multiple layered systems with nonconstant boundary conditions, including the constant boundary and linear boundary conditions. Because ionic diffusion in saturated media and heat conduction are governed by the same form of partial differential equations with different materials parameters, the analytical solution was further extended to handle heat conduction in a multiple layered system under nonconstant boundary conditions. The numerical results were compared with available test data. The basic trends of the analytical solution and the test data agreed quite well. Yu Bai, Ali Harajli, and Yunping Xi Copyright © 2015 Yu Bai et al. All rights reserved. Explicit Solutions for the Solomon-Wilson-Alexiades’s Mushy Zone Model with Convective or Heat Flux Boundary Conditions Mon, 23 Nov 2015 11:08:08 +0000 We complete the Solomon-Wilson-Alexiades’s mushy zone model (Solomon, 1982) for the one-phase Lamé-Clapeyron-Stefan problem by obtaining explicit solutions when a convective or heat flux boundary condition is imposed on the fixed face for a semi-infinite material. We also obtain the necessary and sufficient condition on data in order to get the explicit solutions for both cases which is new with respect to the original model. Moreover, when these conditions are satisfied, the two phase-change problems are equivalent to the same problem with a temperature boundary condition on the fixed face and therefore an inequality for the coefficient which characterized one of the two free interfaces of the model is also obtained. Domingo A. Tarzia Copyright © 2015 Domingo A. Tarzia. All rights reserved. Motion of Bishop Frenet Offsets of Ruled Surfaces in Thu, 19 Nov 2015 14:17:24 +0000 The main goal of this paper is to study the motion of two associated ruled surfaces in Euclidean 3-space . In particular, the motion of Bishop Frenet offsets of ruled surfaces is investigated. Additionally, the characteristic properties for such ruled surfaces are given. Finally, an application is presented and plotted using computer aided geometric design. H. N. Abd-Ellah Copyright © 2015 H. N. Abd-Ellah. All rights reserved. Algebrization of Nonautonomous Differential Equations Mon, 09 Nov 2015 09:53:18 +0000 Given a planar system of nonautonomous ordinary differential equations, , conditions are given for the existence of an associative commutative unital algebra with unit and a function on an open set such that and the maps and are Lorch differentiable with respect to for all , where and represent variables in . Under these conditions the solutions of the differential equation over define solutions of the planar system. María Aracelia Alcorta-García, Martín Eduardo Frías-Armenta, María Esther Grimaldo-Reyna, and Elifalet López-González Copyright © 2015 María Aracelia Alcorta-García et al. All rights reserved. Total Graph Interpretation of the Numbers of the Fibonacci Type Mon, 19 Oct 2015 13:53:33 +0000 We give a total graph interpretation of the numbers of the Fibonacci type. This graph interpretation relates to an edge colouring by monochromatic paths in graphs. We will show that it works for almost all numbers of the Fibonacci type. Moreover, we give the lower bound and the upper bound for the number of all -edge colourings in trees. Urszula Bednarz, Iwona Włoch, and Małgorzata Wołowiec-Musiał Copyright © 2015 Urszula Bednarz et al. All rights reserved. Nature-Inspired Algorithms for Real-World Optimization Problems Wed, 07 Oct 2015 10:35:16 +0000 Wei Fang, Xiaodong Li, Mengjie Zhang, and Mengqi Hu Copyright © 2015 Wei Fang et al. All rights reserved. Numerical Study on Similarity of Plume’s Infrared Radiation from Reduced Scaling Solid Rocket Mon, 05 Oct 2015 12:52:26 +0000 Similarity of plume radiation between reduced scaling solid rocket models and full scale ones in ground conditions has been taken for investigation. Flow and radiation of plume from solid rockets with scaling ratio from 0.1 to 1 have been computed. The radiative transfer equation (RTE) is solved by the finite volume method (FVM) in infrared band 2~6 μm. The spectral characteristics of plume gases have been calculated with the weighted-sum-of-gray-gas (WSGG) model, and those of the Al2O3 particles have been solved by the Mie scattering model. Our research shows that, with the decreasing scaling ratio of the rocket engine, the radiation intensity of the plume decreases with 1.5~2.5 power of the scaling ratio. The infrared radiation of the plume gases shows a strong spectral dependency, while that of the Al2O3 particles shows grey property. Spectral radiation intensity of the high temperature core of the solid rocket plume increases greatly in the peak absorption spectrum of plume gases. Al2O3 particle is the major radiation composition in the rocket plume, whose scattering coefficient is much larger than its absorption coefficient. There is good similarity between spectral variations of plumes from different scaling solid rockets. The directional plume radiation rises with the increasing azimuth angle. Xiaoying Zhang and Rui Li Copyright © 2015 Xiaoying Zhang and Rui Li. All rights reserved. Metaheuristic Approaches for Hydropower System Scheduling Mon, 05 Oct 2015 07:04:19 +0000 This paper deals with the short-term scheduling problem of hydropower systems. The objective is to meet the daily energy demand in an economic and safe way. The individuality of the generating units and the nonlinearity of their efficiency curves are taken into account. The mathematical model is formulated as a dynamic, mixed integer, nonlinear, nonconvex, combinatorial, and multiobjective optimization problem. We propose two solution methods using metaheuristic approaches. They combine Genetic Algorithm with Strength Pareto Evolutionary Algorithm and Ant Colony Optimization. Both approaches are divided into two phases. In the first one, to maximize the plant’s net generation, the problem is solved for each hour of the day (static dispatch). In the second phase, to minimize the units’ switching on-off, the day is considered as a whole (dynamic dispatch). The proposed methodology is applied to two Brazilian hydroelectric plants, in cascade, that belong to the national interconnected system. The nondominated solutions from both approaches are presented. All of them meet demand respecting the physical, electrical, and hydraulic constraints. Ieda G. Hidalgo, Regiane S. de Barros, Jéssica P. T. Fernandes, João Paulo F. Estrócio, and Paulo B. Correia Copyright © 2015 Ieda G. Hidalgo et al. All rights reserved.