Search results for: Convection-dominated diffusion equation

Authors: Chin-Jung Lin, Yi-Hsien Yu, Szu-Ying Chen, Ya-Hsuan Liou

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Highly ordered arrays of TiO2 nanotubes (TiNTs) were grown vertically on Ti foil by electrochemical anodization. We controlled the lengths of these TiNTs from 2.4 to 26.8 ¶üÇóμm while varying the water contents (1, 3, and 6 wt%) of the electrolyte in ethylene glycol in the presence of 0.5 wt% NH4F with anodization for various applied voltages (20–80 V), periods (10–240 min) and temperatures (10–30 oC). For vertically aligned TiNT arrays, not only the increase in their tube lengths, but also their geometric (wall thickness and surface roughness) and crystalline structure lead to a significant influence on photocatalytic activity. The length optimization for methylene blue (MB) photodegradation was 18 μm. Further extending the TiNT length yielded lower photocatalytic activity presumably related to the limited MB diffusion and light-penetration depth into the TiNT arrays. The results indicated that a maximum MB photodegradation rate was obtained for the discrete anatase TiO2 nanotubes with thick and rough walls.

Keywords: Anodic oxidation, nanotube, photocatalytic, TiO2.

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Authors: M. Karami Khorramabadi, A. R. Nezamabadi

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Stability of functionally graded beams with piezoelectric layers subjected to axial compressive load that is simply supported at both ends is studied in this paper. The displacement field of beam is assumed based on first order shear deformation beam theory. Applying the Hamilton's principle, the governing equation is established. The influences of applied voltage, dimensionless geometrical parameter, functionally graded index and piezoelectric thickness on the critical buckling load of beam are presented. To investigate the accuracy of the present analysis, a compression study is carried out with a known data.

Keywords: Stability, Functionally graded beam, First order shear deformation theory, Piezoelectric layer.

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Authors: M. Reza Bagerzadeh Karimi, M. Mahdi Bagerzadeh Karimi

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In the present paper, active control system is used in different heights of the building and the most effective part was studied where the active control system is applied. The mathematical model of the building is established in MATLAB and in order to active control the system FLC method was used. Three different locations of the building are chosen to apply active control system, namely at the lowest story, the middle height of the building, and at the highest point of the building with TMD system. The equation of motion was written for high rise building and it was solved by statespace method. Also passive control was used with Tuned Mass Damper (TMD) at the top floor of the building to show the robustness of FLC method when compared with passive control system.

Keywords: Fuzzy Logic Controller (FLC), Tuned Mass Damper(TMD), Active control, passive control

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Authors: Olusheye Akinfenwa, Samuel Jator, Nianmin Yoa

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A block backward differentiation formula of uniform order eight is proposed for solving first order stiff initial value problems (IVPs). The conventional 8-step Backward Differentiation Formula (BDF) and additional methods are obtained from the same continuous scheme and assembled into a block matrix equation which is applied to provide the solutions of IVPs on non-overlapping intervals. The stability analysis of the method indicates that the method is L0-stable. Numerical results obtained using the proposed new block form show that it is attractive for solutions of stiff problems and compares favourably with existing ones.

Keywords: Stiff IVPs, System of ODEs, Backward differentiationformulas, Block methods, Stability.

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Authors: Nisha Goyal, R.K. Gupta

Abstract:

Einstein vacuum equations, that is a system of nonlinear partial differential equations (PDEs) are derived from Weyl metric by using relation between Einstein tensor and metric tensor. The symmetries of Einstein vacuum equations for static axisymmetric gravitational fields are obtained using the Lie classical method. We have examined the optimal system of vector fields which is further used to reduce nonlinear PDE to nonlinear ordinary differential equation (ODE). Some exact solutions of Einstein vacuum equations in general relativity are also obtained.

Keywords: Gravitational fields, Lie Classical method, Exact solutions.

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Authors: Noor Atinah Ahmad, Shazia Javed

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We present a discussion of three adaptive filtering algorithms well known for their one-step termination property, in terms of their relationship with the minimal residual method. These algorithms are the normalized least mean square (NLMS), Affine Projection algorithm (APA) and the recursive least squares algorithm (RLS). The NLMS is shown to be a result of the orthogonality condition imposed on the instantaneous approximation of the Wiener equation, while APA and RLS algorithm result from orthogonality condition in multi-dimensional minimal residual formulation. Further analysis of the minimal residual formulation for the RLS leads to a triangular system which also possesses the one-step termination property (in exact arithmetic)

Keywords: Adaptive filtering, minimal residual method, projection method.

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Authors: Saeed Sarabadan, Kamal Rashedi

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This article presents a numerical method to find the heat flux in an inhomogeneous inverse heat conduction problem with linear boundary conditions and an extra specification at the terminal. The method is based upon applying the satisfier function along with the Ritz-Galerkin technique to reduce the approximate solution of the inverse problem to the solution of a system of algebraic equations. The instability of the problem is resolved by taking advantage of the Landweber’s iterations as an admissible regularization strategy. In computations, we find the stable and low-cost results which demonstrate the efficiency of the technique.

Keywords: Inverse problem, parabolic equations, heat equation, Ritz-Galerkin method, Landweber iterations.

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Authors: D. H. Kim, Y. H. Kim, T. Kim

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A new method identifies coupled fluid-structure system with a reduced set of state variables is presented. Assuming that the structural model is known a priori either from an analysis or a test and using linear transformations between structural and aeroelastic states, it is possible to deduce aerodynamic information from sampled time histories of the aeroelastic system. More specifically given a finite set of structural modes the method extracts generalized aerodynamic force matrix corresponding to these mode shapes. Once the aerodynamic forces are known, an aeroelastic reduced-order model can be constructed in discrete-time, state-space format by coupling the structural model and the aerodynamic system. The resulting reduced-order model is suitable for constant Mach, varying density analysis.

Keywords: ROM (Reduced-Order Model), aero elasticity, AGARD 445.6 wing.

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Authors: Shan-Jen Cheng, Jr-Ming Miao, Sheng-Ju Wu

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The purpose of this paper is applied Taguchi method on the optimization for PEMFC performance, and a representative Computational Fluid Dynamics (CFD) model is selectively performed for statistical analysis. The studied factors in this paper are pressure of fuel cell, operating temperature, the relative humidity of anode and cathode, porosity of gas diffusion electrode (GDE) and conductivity of GDE. The optimal combination for maximum power density is gained by using a three-level statistical method. The results confirmed that the robustness of the optimum design parameters influencing the performance of fuel cell are founded by pressure of fuel cell, 3atm; operating temperature, 353K; the relative humidity of anode, 50%; conductivity of GDE, 1000 S/m, but the relative humidity of cathode and porosity of GDE are pooled as error due to a small sum of squares. The present simulation results give designers the ideas ratify the effectiveness of the proposed robust design methodology for the performance of fuel cell.

Keywords: PEMFC, numerical simulation, optimization, Taguchi method.

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Authors: S. Nagheli, N. Samani, D. A. Barry

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In this paper, the velocity potential and stream function of capture zone for a well field in an aquifer bounded by two parallel streams with or without a uniform regional flow of any directions are presented. The well field includes any number of extraction or injection wells or a combination of both types with any pumping rates. To delineate the capture envelope, the potential and streamlines equations are derived by conformal mapping method. This method can help us to release constrains of other methods. The equations can be applied as useful tools to design in-situ groundwater remediation systems, to evaluate the surface–subsurface water interaction and to manage the water resources.

Keywords: Complex potential, conformal mapping, groundwater remediation, image well theory, Laplace’s equation, superposition principle.

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Authors: Masatora Daito, Noriyuki Tanida

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The underground shopping mall has the constructional problem of the fire evacuation. Also, the people sometimes lose their direction and information of current time in the mall. If the emergencies such as terrorist explosions or gas explosions are happened, they have to go out soon. Under such circumstances, inside of the mall has high risk for life. In this research, the authors propose a way that he/she can go out from the underground shopping mall quickly. If the narrow exits are discovered by using active RFID (Radio Frequency Identification) tags and using cellular phones, they can evacuate as soon as possible. To verify this hypothesis, the authors design the model and carry out the agent-based simulation. They treat, as a case study, the Tenjin mall in Fukuoka Prefecture in Japan. The result of the simulation is that the case of the pedestrian with using active RFID tags and cellular phones reduced the amount of time to spend on the evacuation. Even if the diffusion of RFID tags and cellular phones was not perfect, they could show the effectiveness of reducing the time of evacuation.

Keywords: Evacuation, active RFID tag and cellular phone, underground shopping mall, agent-based simulation.

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Authors: Arezoo Sadrinezhad, Kallol Sett, S. I. Hariharan

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In this paper, a probabilistic framework based on Fokker-Planck-Kolmogorov (FPK) approach has been applied to simulate triaxial cyclic constitutive behavior of uncertain soils. The framework builds upon previous work of the writers, and it has been extended for cyclic probabilistic simulation of triaxial undrained behavior of soils. von Mises elastic-perfectly plastic material model is considered. It is shown that by using probabilistic framework, some of the most important aspects of soil behavior under cyclic loading can be captured even with a simple elastic-perfectly plastic constitutive model.

Keywords: Elasto-plasticity, uncertainty, soils, Fokker-Planck equation, Fourier Spectral method, Finite Difference method.

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Authors: L. Dahmani

Abstract:

The first and basic cause of the failure of concrete is repeated freezing (thawing) of moisture contained in the pores, microcracks, and cavities of the concrete. On transition to ice, water existing in the free state in cracks increases in volume, expanding the recess in which freezing occurs. A reduction in strength below the initial value is to be expected and further cycle of freezing and thawing have a further marked effect. By using some experimental parameters like nuclear magnetic resonance variation (NMR), enthalpy-temperature (or heat capacity) variation, we can resolve between the various water states and their effect on concrete properties during cooling through the freezing transition temperature range. The main objective of this paper is to describe the principal type of water responsible for the reduction in strength and structural damage (frost damage) of concrete following repeated freeze –thaw cycles. Some experimental work was carried out at the institute of cryogenics to determine what happens to water in concrete during the freezing transition. 

Keywords: Concrete, frost proof, strength, water diffusion.

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Authors: Weihua Ruan, Kuan-Chou Chen

Abstract:

This paper is concerned with a system of Hamilton-Jacobi-Bellman equations coupled with an autonomous dynamical system. The mathematical system arises in the differential game formulation of political economy models as an infinite-horizon continuous-time differential game with discounted instantaneous payoff rates and continuously and discretely varying state variables. The existence of a weak solution of the PDE system is proven and a computational scheme of approximate solution is developed for a class of such systems. A model of democratization is mathematically analyzed as an illustration of application.

Keywords: Differential games, Hamilton-Jacobi-Bellman equations, infinite horizon, political-economy models.

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Authors: M. Petchsangsai, N. Wonglertnirant, T. Rojanarata, P. Opanasopit, T. Ngawhirunpat

Abstract:

The objective of the present study was to evaluate the potential of hollow microneedles for enhancing the transdermal delivery of Bovine Serum Albumin (MW~66,000 Da)-Fluorescein Isothiocyanate (BSA-FITC) conjugate, a hydrophilic large molecular compound. Moreover, the effect of different formulations was evaluated. The series of binary mixtures composed of propylene glycol (PG) and pH 7.4 phosphate buffer solution (PBS) was prepared and used as a medium for BSA-FITC. The results showed that there was no permeation of BSA-FITC solution across the neonatal porcine skin without using hollow microneedles, whereas the cumulative amount of BSA-FITC released at 8 h through the neonatal porcine skin was about 60-70% when using hollow microneedles. Furthermore, the results demonstrated that the higher volume of PG in binary mixtures injected, the lower cumulative amount of BSA-FITC released and release rate of BSA-FITC from skin. These release profiles of BSA-FITC in binary mixtures were expressed by Fick-s law of diffusion. These results suggest the utilization of hollow microneedle to enhance transdermal delivery of protein and provide useful information for designing an effective hollow microneedle system.

Keywords: Hydrophilic macromolecules, Microneedles, Propylene glycol, Transdermal drug delivery

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Authors: Choon Ki Ahn, Jung Hun Park, Wook Hyun Kwon

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CEMTool is a command style design and analyzing package for scientific and technological algorithm and a matrix based computation language. In this paper, we present new 2D & 3D finite element method (FEM) packages for CEMTool. We discuss the detailed structures and the important features of pre-processor, solver, and post-processor of CEMTool 2D & 3D FEM packages. In contrast to the existing MATLAB PDE Toolbox, our proposed FEM packages can deal with the combination of the reserved words. Also, we can control the mesh in a very effective way. With the introduction of new mesh generation algorithm and fast solving technique, our FEM packages can guarantee the shorter computational time than MATLAB PDE Toolbox. Consequently, with our new FEM packages, we can overcome some disadvantages or limitations of the existing MATLAB PDE Toolbox.

Keywords: CEMTool, Finite element method (FEM), Numericalanalysis, Partial differential equation (PDE)

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Authors: António L. Topa

Abstract:

A complete spectral representation for the electromagnetic field of planar multilayered waveguides inhomogeneously filled with omega media is presented. The problem of guided electromagnetic propagation is reduced to an eigenvalue equation related to a 2 ´ 2 matrix differential operator. Using the concept of adjoint waveguide, general bi-orthogonality relations for the hybrid modes (either from the discrete or from the continuous spectrum) are derived. For the special case of homogeneous layers the linear operator formalism is reduced to a simple 2 ´ 2 coupling matrix eigenvalue problem. Finally, as an example of application, the surface and the radiation modes of a grounded omega slab waveguide are analyzed.

Keywords: Metamaterials, linear operators, omega media, layered waveguide, orthogonality relations

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Authors: Shan Gao

Abstract:

In this paper, we consider a risk model involving two independent classes of insurance risks and random premium income. We assume that the premium income process is a Poisson Process, and the claim number processes are independent Poisson and generalized Erlang(n) processes, respectively. Both of the Gerber- Shiu functions with zero initial surplus and the probability generating functions (p.g.f.) of the Gerber-Shiu functions are obtained.

Keywords: Poisson process, generalized Erlang risk process, Gerber-Shiu function, generating function, generalized Lundberg equation.

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Authors: R. K. Saxena, Ravi Saxena

Abstract:

In view of their importance and usefulness in reliability theory and probability distributions, several generalizations of the inverse Gaussian distribution and the Krtzel function are investigated in recent years. This has motivated the authors to introduce and study a new generalization of the inverse Gaussian distribution and the Krtzel function associated with a product of a Bessel function of the third kind )(zKQ and a Z - Fox-Wright generalized hyper geometric function introduced in this paper. The introduced function turns out to be a unified gamma-type function. Its incomplete forms are also discussed. Several properties of this gamma-type function are obtained. By means of this generalized function, we introduce a generalization of inverse Gaussian distribution, which is useful in reliability analysis, diffusion processes, and radio techniques etc. The inverse Gaussian distribution thus introduced also provides a generalization of the Krtzel function. Some basic statistical functions associated with this probability density function, such as moments, the Mellin transform, the moment generating function, the hazard rate function, and the mean residue life function are also obtained.KeywordsFox-Wright function, Inverse Gaussian distribution, Krtzel function & Bessel function of the third kind.

Keywords: Fox-Wright function, Inverse Gaussian distribution, Krtzel function & Bessel function of the third kind.

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Authors: Abdelrahman Elsehsah, Hany Madkour, Khalid Farah

Abstract:

This article presents a 3-D modified non-linear elastic model in the strain space. The Helmholtz free energy function is introduced with the existence of a dissipation potential surface in the space of thermodynamic conjugate forces. The constitutive equation and the damage evolution were derived as well. The modified damage has been examined to model the nonlinear behavior of reinforced concrete (RC) slabs with an opening. A parametric study with RC was carried out to investigate the impact of different factors on the behavior of RC slabs. These factors are the opening area, the opening shape, the place of opening, and the thickness of the slabs. And the numerical results have been compared with the experimental data from literature. Finally, the model showed its ability to be applied to the structural analysis of RC slabs.

Keywords: 3-D numerical analysis, damage mechanics, RC slab with opening.

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Authors: Rui Sun, Tamer M. Ismail, Xiaohan Ren, M. Abd El-Salam

Abstract:

Incineration of municipal solid waste (MSW) is one of the key scopes in the global clean energy strategy. A computational fluid dynamics (CFD) model was established in order to reveal these features of the combustion process in a fixed porous bed of MSW. Transporting equations and process rate equations of the waste bed were modeled and set up to describe the incineration process, according to the local thermal conditions and waste property characters. Gas phase turbulence was modeled using k-ε turbulent model and the particle phase was modeled using the kinetic theory of granular flow. The heterogeneous reaction rates were determined using Arrhenius eddy dissipation and the Arrhenius-diffusion reaction rates. The effects of primary air flow rate and temperature in the burning process of simulated MSW are investigated experimentally and numerically. The simulation results in bed are accordant with experimental data well. The model provides detailed information on burning processes in the fixed bed, which is otherwise very difficult to obtain by conventional experimental techniques.

Keywords: Computational Fluid Dynamics (CFD) model, Waste Incineration, Municipal Solid Waste (MSW), Fixed Bed, Primary air.

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Authors: Vu Thu Trang, Lam Xuan Thanh, Samira Sarter, Tomoko Shimamura, Hiroaki Takeuchi　

Abstract:

Antimicrobial activities of aminoreductone (AR), a product formed in the initial stage of Maillard reaction, were screened against pathogenic bacteria. A significant growth inhibition of AR against all 7 isolates (Staphylococcus aureus ATCC® 25923™, Salmonella typhimurium ATCC® 14028™, Bacillus cereus ATCC® 13061™, Bacillus subtilis ATCC® 11774™, Escherichia coli ATCC® 25922™, Enterococcus faecalis ATCC® 29212™, Listeria innocua ATCC® 33090™) were observed by the standard disc diffusion methods. The inhibition zone for each isolate by AR (2.5 mg) ranged from 15±0mm to 28.3±0.4mm in diameter. The minimum inhibitory concentration (MIC) of AR ranging from 20mM to 26mM was proven in the 7 isolates tested. AR also showed the similar effect of growth inhibition in comparison with antibiotics frequently used for the treatment of infections bacteria, such as amikacin, ciprofloxacin, meropennem and levofloxacin. The results indicated that foods containing AR are valuable sources of bioactive compounds towards pathogenic bacteria.

Keywords: Pathogenic bacteria, aminoreductone, Maillard reaction, antimicrobial activity.

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Authors: Ibrahim Abu-Alshaikh

Abstract:

In this study, a new root-finding method for solving nonlinear equations is proposed. This method requires two starting values that do not necessarily bracketing a root. However, when the starting values are selected to be close to a root, the proposed method converges to the root quicker than the secant method. Another advantage over all iterative methods is that; the proposed method usually converges to two distinct roots when the given function has more than one root, that is, the odd iterations of this new technique converge to a root and the even iterations converge to another root. Some numerical examples, including a sine-polynomial equation, are solved by using the proposed method and compared with results obtained by the secant method; perfect agreements are found.

Keywords: Iterative method, root-finding method, sine-polynomial equations, nonlinear equations.

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Authors: Ezzah Liana Ahmad Fauzi, Syakila Ahmad, Ioan Pop

Abstract:

The steady mixed convection boundary layer flow from a vertical cone in a porous medium filled with a nanofluid is numerically investigated using different types of nanoparticles as Cu (copper), Al2O3 (alumina) and TiO2 (titania). The boundary value problem is solved by using the shooting technique by reducing it into an ordinary differential equation. Results of interest for the local Nusselt number with various values of the constant mixed convection parameter and nanoparticle volume fraction parameter are evaluated. It is found that dual solutions exist for a certain range of mixed convection parameter.

Keywords: boundary layer, mixed convection, nanofluid, porous medium, vertical cone.

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Authors: Chinwendu. B. Eleje, Udechukwu P. Egbuhuzor

Abstract:

One of the difficulties encountered in solving nonlinear Boundary Value Problems (BVP) by many researchers is finding approximated solutions with minimum deviations from the exact solutions without so much rigor and complications. In this paper, we propose an approach to solve a two point BVP which involves a combination of Taylor series expansion method and Newton Raphson method. Furthermore, the fourth and sixth order approximated solutions are obtained and we compare their relative error and rate of convergence to the exact solution. Finally, some numerical simulations are presented to show the behavior of the solution and its derivatives.

Keywords: Newton Raphson method, non-linear boundary value problem, Taylor series approximation, Michaelis-Menten equation.

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Authors: M.A. Kamely, S.M. Kamil, C.W. Chong

Abstract:

A mathematical model of the surface roughness has been developed by using response surface methodology (RSM) in grinding of AISI D2 cold work tool steels. Analysis of variance (ANOVA) was used to check the validity of the model. Low and high value for work speed and feed rate are decided from design of experiment. The influences of all machining parameters on surface roughness have been analyzed based on the developed mathematical model. The developed prediction equation shows that both the feed rate and work speed are the most important factor that influences the surface roughness. The surface roughness was found to be the lowers with the used of low feed rate and low work speed. Accuracy of the best model was proved with the testing data.

Keywords: Mathematical Modeling, Response surfacemethodology, Surface roughness, Cylindrical Grinding.

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Authors: N. Behidj, K. Benyounes, T. Dahmane, A. Allem

Abstract:

The aim of this study was to investigate the antimicrobial activity of flavonoids isolated from the aerial part of a medicinal plant which is Thymus inodorusby the middle agar diffusion method on following microorganisms. We have Staphylococcus aureus, Escherichia coli, Pseudomonas fluorescens, AspergillusNiger, Aspergillus fumigatus and Candida albicans. During this study, flavonoids extracted by stripping with steam are performed. The yields of flavonoids is 7.242% for the aqueous extract and 28.86% for butanol extract, 29.875% for the extract of ethyl acetate and 22.9% for the extract of di - ethyl. The evaluation of the antibacterial effect shows that the diameter of the zone of inhibition varies from one microorganism to another. The operation values obtained show that the bacterial strain P fluoresces, and 3 yeasts and molds; A. Niger, A. fumigatus and C. albicansare the most resistant. But it is noted that, S. aureus is shown more sensitive to crude extracts, the stock solution and the various dilutions. Finally for the minimum inhibitory concentration is estimated only with the crude extract of Thymus inodorus flavonoid.Indeed, these extracts inhibit the growth of Gram + bacteria at a concentration varying between 0.5% and 1%. While for bacteria to Gram -, it is limited to a concentration of 0.5%.

Keywords: Antimicrobial activity, flavonoids, strains, Thymus inodorus.

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Authors: M. K. Moazzam, A. Salmanpour, M. Nirouei

Abstract:

In this paper we have numerically analyzed terahertzrange wavelength conversion using nondegenerate four wave mixing (NDFWM) in a SOA integrated DFB laser (experiments reported both in MIT electronics and Fujitsu research laboratories). For analyzing semiconductor optical amplifier (SOA), we use finitedifference beam propagation method (FDBPM) based on modified nonlinear SchrÖdinger equation and for distributed feedback (DFB) laser we use coupled wave approach. We investigated wavelength conversion up to 4THz probe-pump detuning with conversion efficiency -5dB in 1THz probe-pump detuning for a SOA integrated quantum-well

Keywords: distributed feedback laser, nondegenerate fourwave mixing, semiconductor optical amplifier, wavelengthconversion

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Authors: Xian Ming Gu, Ting Zhu Huang, Hou Biao Li

Abstract:

In this paper, getting an high-efficiency parallel algorithm to solve sparse block pentadiagonal linear systems suitable for vectors and parallel processors, stair matrices are used to construct some parallel polynomial approximate inverse preconditioners. These preconditioners are appropriate when the desired target is to maximize parallelism. Moreover, some theoretical results about these preconditioners are presented and how to construct preconditioners effectively for any nonsingular block pentadiagonal H-matrices is also described. In addition, the availability of these preconditioners is illustrated with some numerical experiments arising from two dimensional biharmonic equation.

Keywords: Parallel algorithm, Pentadiagonal matrix, Polynomial approximate inverse, Preconditioners, Stair matrix.

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Authors: Gyo Woo Lee

Abstract:

In this study, the phase transition characteristics of flame-synthesized γ-Al₂O₃ nanoparticles to α-Al₂O₃ have been investigated. The nanoparticles were synthesized by using a coflow hydrogen diffusion flame. The phase transition and particle characteristics of the Al₂O₃ nanoparticles were determined by examining the crystalline structure and the shape of the collected nanoparticles before and after the heat treatment. The morphology and crystal structure of the Al₂O₃ nanoparticles were determined from SEM images and XRD analyses, respectively. The measured specific surface area and averaged particle size were 63.44m²/g and 23.94nm, respectively. Based on the scanning electron microscope images and x-ray diffraction patterns, it is believed that the onset temperature of the phase transition to α-Al₂O₃ was existed near 1200^oC. The averaged diameters of the sintered particles heat treated at 1,260^oC were approximately 80nm.

Keywords: BET Specific Surface Area, Gamma-Al2O3 Nanoparticles, Flame Synthesis, Phase Transition, X-ray Diffraction.

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