Search results for: Linear diffusion layer

Authors: Chaitanya Chawla, Divya Panwar, Gurneesh Singh Anand, M. P. S Bhatia

Abstract:

This paper presents a deep-learning mechanism for classifying computer generated images and photographic images. The proposed method accounts for a convolutional layer capable of automatically learning correlation between neighbouring pixels. In the current form, Convolutional Neural Network (CNN) will learn features based on an image's content instead of the structural features of the image. The layer is particularly designed to subdue an image's content and robustly learn the sensor pattern noise features (usually inherited from image processing in a camera) as well as the statistical properties of images. The paper was assessed on latest natural and computer generated images, and it was concluded that it performs better than the current state of the art methods.

Keywords: Image forensics, computer graphics, classification, deep learning, convolutional neural networks.

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Authors: Brigitte Rafael, Stefan M. Oertl

Abstract:

Music segmentation is a key issue in music information retrieval (MIR) as it provides an insight into the internal structure of a composition. Structural information about a composition can improve several tasks related to MIR such as searching and browsing large music collections, visualizing musical structure, lyric alignment, and music summarization. The authors of this paper present the MTSSM framework, a twolayer framework for the multi-track segmentation of symbolic music. The strength of this framework lies in the combination of existing methods for local track segmentation and the application of global structure information spanning via multiple tracks. The first layer of the MTSSM uses various string matching techniques to detect the best candidate segmentations for each track of a multi-track composition independently. The second layer combines all single track results and determines the best segmentation for each track in respect to the global structure of the composition.

Keywords: Pattern Recognition, Music Information Retrieval, Machine Learning.

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Authors: Zia R. Tahir, P. Mandal

Abstract:

A numerical study is presented on buckling and post buckling behaviour of laminated carbon fiber reinforced plastic (CFRP) thin-walled cylindrical shells under axial compression using asymmetric meshing technique (AMT). Asymmetric meshing technique is a perturbation technique to introduce disturbance without changing geometry, boundary conditions or loading conditions. Asymmetric meshing affects predicted buckling load, buckling mode shape and post-buckling behaviour. Linear (eigenvalue) and nonlinear (Riks) analyses have been performed to study the effect of asymmetric meshing in the form of a patch on buckling behaviour. The reduction in the buckling load using Asymmetric meshing technique was observed to be about 15%. An isolated dimple formed near the bifurcation point and the size of which increased to reach a stable state in the post-buckling region. The load-displacement curve behaviour applying asymmetric meshing is quite similar to the curve obtained using initial geometric imperfection in the shell model.

Keywords: CFRP Composite Cylindrical Shell, Finite Element Analysis, Perturbation Technique, Asymmetric Meshing Technique, Linear Eigenvalue analysis, Non-linear Riks Analysis

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Authors: Tebogo Emma Makaba, Barnabas Ndlovu Gatsheni

Abstract:

The cities of Johannesburg and Pretoria both located in the Gauteng province are separated by a distance of 58 km. The traffic queues on the Ben Schoeman freeway which connects these two cities can stretch for almost 1.5 km. Vehicle traffic congestion impacts negatively on the business and the commuter’s quality of life. The goal of this paper is to identify variables that influence the flow of traffic and to design a vehicle traffic prediction model, which will predict the traffic flow pattern in advance. The model will unable motorist to be able to make appropriate travel decisions ahead of time. The data used was collected by Mikro’s Traffic Monitoring (MTM). Multi-Layer perceptron (MLP) was used individually to construct the model and the MLP was also combined with Bagging ensemble method to training the data. The cross—validation method was used for evaluating the models. The results obtained from the techniques were compared using predictive and prediction costs. The cost was computed using combination of the loss matrix and the confusion matrix. The predicted models designed shows that the status of the traffic flow on the freeway can be predicted using the following parameters travel time, average speed, traffic volume and day of month. The implications of this work is that commuters will be able to spend less time travelling on the route and spend time with their families. The logistics industry will save more than twice what they are currently spending.

Keywords: Bagging ensemble methods, confusion matrix, multi-layer perceptron, vehicle traffic flow.

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Authors: Kwok Tak Kit

Abstract:

This research focuses mainly on the transformation of circular economy and specifically on the building and construction industry in Hong Kong. This research is going to investigate the current status of development and transformation of linear economy of Hong Kong into circular economy. The research is based on literature review and different published reports to outline the possibilities for further development and transformation of circular economy in Hong Kong. Different key factors affecting the transformation particularly in building and construction industry are discussed.

Keywords: Circular economy, LEED net zero, carbon emission, waste reduction, waste management.

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Authors: Ayesha Aslam, F A Bhatti

Abstract:

A simple impedance matching technique for inset feed grooved microstrip patch antenna based on the concept of coplanar waveguide feed line has been developed and investigated for a printed antenna at X-Band frequency of 10GHz. The proposed technique has been used in the design of Linear Grooved Microstrip patch antenna array. The characteristics of the antenna are determined in terms of Return loss, VSWR, gain, radiation pattern etc. The measured and simulated results presented are found to be in good agreement.

Keywords: Gain, Microstrip patch, return loss, VSWR, Radiation pattern, CPW Feed, Inset feed.

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Authors: Abhishek Soni, A. Kumaraswamy, M. S. Mahesh

Abstract:

Bullet penetration in steel plate is investigated with the help of three-dimensional, non-linear, transient, dynamic, finite elements analysis using explicit time integration code LSDYNA. The effect of large strain, strain-rate and temperature at very high velocity regime was studied from number of simulations of semi-spherical nose shape bullet penetration through single layered circular plate with 2 mm thickness at impact velocities of 500, 1000, and 1500 m/s with the help of Johnson Cook material model. Mie-Gruneisen equation of state is used in conjunction with Johnson Cook material model to determine pressure-volume relationship at various points of interests. Two material models viz. Plastic-Kinematic and Johnson- Cook resulted in different deformation patterns in steel plate. It is observed from the simulation results that the velocity drop and loss of kinetic energy occurred very quickly up to perforation of plate, after that the change in velocity and changes in kinetic energy are negligibly small. The physics behind this kind of behaviour is presented in the paper.

Keywords: AISI 4340 steel, ballistic impact simulation, bullet penetration, non-linear FEM.

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Authors: Nur Nadiah Abd Hamid , Ahmad Abd. Majid, Ahmad Izani Md. Ismail

Abstract:

Linear two-point boundary value problems of order two are solved using cubic trigonometric B-spline interpolation method (CTBIM). Cubic trigonometric B-spline is a piecewise function consisting of trigonometric equations. This method is tested on some problems and the results are compared with cubic B-spline interpolation method (CBIM) from the literature. CTBIM is found to approximate the solution slightly more accurately than CBIM if the problems are trigonometric.

Keywords: trigonometric B-spline, two-point boundary valueproblem, spline interpolation, cubic spline

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Authors: George Yungchih Wang

Abstract:

Real options theory suggests that managerial flexibility embedded within irreversible investments can account for a significant value in project valuation. Although the argument has become the dominant focus of capital investment theory over decades, yet recent survey literature in capital budgeting indicates that corporate practitioners still do not explicitly apply real options in investment decisions. In this paper, we explore how real options decision criteria can be transformed into equivalent capital budgeting criteria under the consideration of uncertainty, assuming that underlying stochastic process follows a geometric Brownian motion (GBM), a mixed diffusion-jump (MX), or a mean-reverting process (MR). These equivalent valuation techniques can be readily decomposed into conventional investment rules and “option impacts", the latter of which describe the impacts on optimal investment rules with the option value considered. Based on numerical analysis and Monte Carlo simulation, three major findings are derived. First, it is shown that real options could be successfully integrated into the mindset of conventional capital budgeting. Second, the inclusion of option impacts tends to delay investment. It is indicated that the delay effect is the most significant under a GBM process and the least significant under a MR process. Third, it is optimal to adopt the new capital budgeting criteria in investment decision-making and adopting a suboptimal investment rule without considering real options could lead to a substantial loss in value.

Keywords: real options, capital budgeting, geometric Brownianmotion, mixed diffusion-jump, mean-reverting process

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Authors: M. Kumpugdee-Vollrath, T. G. D. Phu, M. Helmis

Abstract:

A suitable model membrane to study the pharmacological effect of pharmaceutical products is human stratum corneum because this layer of human skin is the outermost layer and it is an important barrier to be passed through. Other model membranes which were also used are for example skins from pig, mouse, reptile or fish. We are interested in fish skins in this project. The advantages of the fish skins are, that they can be obtained from the supermarket or fish shop. However, the fish skins should be freshly prepared and used directly without storage. In order to understand the effect of different model drugs e.g. lidocaine HCl, resveratrol, paracetamol, ibuprofen, acetyl salicylic acid on the properties of the model membrane from various types of fishes e.g. trout, salmon, cod, plaice permeation tests were performed and differential scanning calorimetry was applied.

Keywords: Fish skin, model membrane, permeation, DSC, lidocaine HCl, resveratrol, paracetamol, ibuprofen, acetyl salicylic acid.

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Authors: Aylin Alin

Abstract:

In this paper, penalized power-divergence test statistics have been defined and their exact size properties to test a nested sequence of log-linear models have been compared with ordinary power-divergence test statistics for various penalization, λ and main effect values. Since the ordinary and penalized power-divergence test statistics have the same asymptotic distribution, comparisons have been only made for small and moderate samples. Three-way contingency tables distributed according to a multinomial distribution have been considered. Simulation results reveal that penalized power-divergence test statistics perform much better than their ordinary counterparts.

Keywords: Contingency table, Log-linear models, Penalization, Power-divergence measure, Penalized power-divergence measure.

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Authors: Anjali Verma, Ram Jiwari, Jitender Kumar

Abstract:

This paper presents a new function expansion method for finding traveling wave solution of a non-linear equation and calls it the (G'/G)-expansion method. The shallow water wave equation is reduced to a non linear ordinary differential equation by using a simple transformation. As a result the traveling wave solutions of shallow water wave equation are expressed in three forms: hyperbolic solutions, trigonometric solutions and rational solutions.

Keywords: Shallow water wave equation, Exact solutions, (G'/G) expansion method.

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Authors: Guiding Gu, Guo Liu

Abstract:

We discuss the convergence property of the minimum residual (MINRES) method for the solution of complex shifted Hermitian system (αI + H)x = f. Our convergence analysis shows that the method has a faster convergence than that for real shifted Hermitian system (Re(α)I + H)x = f under the condition Re(α) + λmin(H) > 0, and a larger imaginary part of the shift α has a better convergence property. Numerical experiments show such convergence properties.

Keywords: complex shifted linear system, Hermitian matrix, MINRES method.

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Authors: Biao Qin, Jin Huang, Jiaojiao Ren, Wei Kang

Abstract:

This paper deals with the problem of passivity analysis for stochastic neural networks with leakage, discrete and distributed delays. By using delay partitioning technique, free weighting matrix method and stochastic analysis technique, several sufficient conditions for the passivity of the addressed neural networks are established in terms of linear matrix inequalities (LMIs), in which both the time-delay and its time derivative can be fully considered. A numerical example is given to show the usefulness and effectiveness of the obtained results.

Keywords: Passivity, Stochastic neural networks, Multiple time delays, Linear matrix inequalities (LMIs).

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Authors: Wei-hua Ai, Wei Yan, Xiang Li

Abstract:

Data compression is used operationally to reduce bandwidth and storage requirements. An efficient method for achieving lossless weather radar data compression is presented. The characteristics of the data are taken into account and the optical linear prediction is used for the PPI images in the weather radar data in the proposed method. The next PPI image is identical to the current one and a dramatic reduction in source entropy is achieved by using the prediction algorithm. Some lossless compression methods are used to compress the predicted data. Experimental results show that for the weather radar data, the method proposed in this paper outperforms the other methods.

Keywords: Lossless compression, weather radar data, optical linear prediction, PPI image

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Authors: Yiqin Lin, Liang Bao

Abstract:

This paper is devoted to the numerical solution of large-scale linear ill-posed systems. A multilevel regularization method is proposed. This method is based on a synthesis of the Arnoldi-Tikhonov regularization technique and the multilevel technique. We show that if the Arnoldi-Tikhonov method is a regularization method, then the multilevel method is also a regularization one. Numerical experiments presented in this paper illustrate the effectiveness of the proposed method.

Keywords: Discrete ill-posed problem, Tikhonov regularization, discrepancy principle, Arnoldi process, multilevel method.

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Authors: K. Sandjak, B. Tiliouine

Abstract:

Structural analysis of flexible pavements has been and still is currently performed using multi-layer elastic theory. However, for thinly surfaced pavements subjected to low to medium volumes of traffics, the importance of non-linear stress-strain behavior of unbound granular materials (UGM) requires the use of more sophisticated numerical models for structural design and performance of such pavements. In the present work, nonlinear unbound aggregates constitutive model is implemented within an axisymmetric finite element code developed to simulate the nonlinear behavior of pavement structures including two local aggregates of different mineralogical nature, typically used in Algerian pavements. The performance of the mechanical model is examined about its capability of representing adequately, under various conditions, the granular material non-linearity in pavement analysis. In addition, deflection data collected by Falling Weight Deflectometer (FWD) are incorporated into the analysis in order to assess the sensitivity of critical pavement design criteria and pavement design life to the constitutive model. Finally, conclusions of engineering significance are formulated. 

Keywords: Nonlinear resilient behavior, unbound granular materials, RLT test results, FWD backcalculations, finite element simulations, pavement response and performance.

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Authors: H. Abaali

Abstract:

The power buck converter is the most widely used DC/DC converter topology. They have a very large application area such as DC motor drives, photovoltaic power system which require fast transient responses and high efficiency over a wide range of load current. This work proposes, the modelling of DC/DC power buck converter using state-space averaging method and the current-mode control using a proportional-integral controller. The efficiency of the proposed model and control loop are evaluated with operating point changes. The simulation results proved the effectiveness of the linear model of DC/DC power buck converter.

Keywords: DC/DC power buck converter, Linear current control, State-space averaging method.

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Authors: Qiang Niu, Linzhang Lu

Abstract:

Restarted GMRES methods augmented with approximate eigenvectors are widely used for solving large sparse linear systems. Recently a new scheme of augmenting with error approximations is proposed. The main aim of this paper is to develop a restarted GMRES method augmented with the combination of harmonic Ritz vectors and error approximations. We demonstrate that the resulted combination method can gain the advantages of two approaches: (i) effectively deflate the small eigenvalues in magnitude that may hamper the convergence of the method and (ii) partially recover the global optimality lost due to restarting. The effectiveness and efficiency of the new method are demonstrated through various numerical examples.

Keywords: Arnoldi process, GMRES, Krylov subspace, systems of linear equations.

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Authors: M. Kheirandish, S. Borhani

Abstract:

In this experiment Polystyrene/Zinc-oxide (PS/ZnO) nanocomposite fibers were produced by electrospinning technique using limonene as a green solvent. First, the morphology of electrospun pure polystyrene (PS) and PS/ZnO nanocomposite fibers investigated by SEM. Results showed the PS fiber diameter decreased by increasing concentration of Zinc Oxide nanoparticles (ZnO NPs). Thermo Gravimetric Analysis (TGA) results showed thermal stability of nanocomposites increased by increasing ZnO NPs in PS electrospun fibers. Considering Differential Scanning Calorimeter (DSC) thermograms for electrospun PS fibers indicated that introduction of ZnO NPs into fibers affects the glass transition temperature (Tg) by reducing it. Also, UV protection properties of nanocomposite fibers were increased by increasing ZnO concentration. Evaluating the effect of metal oxide NPs amount on mechanical properties of electrospun layer showed that tensile strength and elasticity modulus of the electrospun layer of PS increased by addition of ZnO NPs. X-ray diffraction (XRD) pattern of nanopcomposite fibers confirmed the presence of NPs in the samples.

Keywords: Electrospininng, nanoparticle, polystyrene, ZnO.

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Authors: Shao-Tzu Huang, Chen-Chien Hsu, Wei-Yen Wang

Abstract:

Matching high dimensional features between images is computationally expensive for exhaustive search approaches in computer vision. Although the dimension of the feature can be degraded by simplifying the prior knowledge of homography, matching accuracy may degrade as a tradeoff. In this paper, we present a feature matching method based on k-means algorithm that reduces the matching cost and matches the features between images instead of using a simplified geometric assumption. Experimental results show that the proposed method outperforms the previous linear exhaustive search approaches in terms of the inlier ratio of matched pairs.

Keywords: Feature matching, k-means clustering, scale invariant feature transform, linear exhaustive search.

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Authors: S. Angalaparameswari, T.S. Mohamed Saleem, M. Alagusundaram, S. Ramkanth, V.S. Thiruvengadarajan, K. Gnanaprakash, C. Madhusudhana Chetty, G. Pratheesh

Abstract:

The present research was designed to investigate the anti-microbial activity of aristolochic acid from the root of Aristolochia bracteata. From the methanolic & ethyl extract extracts of Aristolochia bracteata aristolochic acid I was isolated and conformed through IR, NMR & MS. The percentage purity of aristolochic acid I was determined by UV & HPLC method. Antibacterial activity of extracts of Aristolochia bracteata and the isolated compound was determined by disc diffusion method. The results reveled that the isolated aristolochic acid from methanolic extract was more pure than the compound from ethyl acetate extract. The various extracts (500μg/disc) of Aristolochia bracteata showed moderate antibacterial activity with the average zone of inhibition of 7-18 mm by disc diffusion method. Among the extracts, ethyl acetate & methanol extracts were shown good anti-microbial activity and the growth of E.coli (18 mm) was strongly inhibited. Microbial assay of isolated compound (Aristolochic acid I) from ethyl acetate & methanol extracts were shown good antimicrobial activity and the zone of inhibition of both at higher concentration 50 μg/ml was similar with the standard aristolochic acid. It may be concluded that the isolated compound of aristolochic acid I has good anti-bacterial activity.

Keywords: Aristolochic acid I, Anti-microbial activity, Aristolochia bracteata, Bacillus subtilis, E.coli

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Authors: Yu-Wei Chen, Li-Ren Han

Abstract:

In this paper, we employ the approach of linear programming to propose a new interactive broadcast method. In our method, a film S is divided into n equal parts and broadcast via k channels. The user simultaneously downloads these segments from k channels into the user-s set-top-box (STB) and plays them in order. Our method assumes that the initial p segments will not have fast-forwarding capabilities. Every time the user wants to initiate d times fast-forwarding, according to our broadcasting strategy, the necessary segments already saved in the user-s STB or are just download on time for playing. The proposed broadcasting strategy not only allows the user to pause and rewind, but also to fast-forward.

Keywords: Broadcasting, Near Video-on-Demand (VOD), Linear Programming, Video-Cassette-Recorder (VCR) Functions, Waiting Time.

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Authors: A. D. Parekh, P. R. Tailor

Abstract:

The Montreal protocol and Kyoto protocol underlined the need of substitution of CFC’s and HCFC’s due to their adverse impact on atmospheric ozone layer which protects earth from U.V rays. The CFCs have been entirely ruled out since 1995 and a long-term basis HCFCs must be replaced by 2020. All this events motivated HFC refrigerants which are harmless to ozone layer. In this paper thermodynamic analysis of cascade refrigeration system has been done using three different refrigerant pairs R13-R12, R290-R23, and R404A-R23. Effect of various operating parameters i.e. evaporator temperature, condenser temperature, temperature difference in cascade condenser and low temperature cycle condenser temperature on performance parameters viz. COP, exergetic efficiency and refrigerant mass flow ratio have been studied. Thermodynamic analysis shows that out of three refrigerant pairs R12-R13, R290-R23 and R404A-R23 the COP of R290-R23 refrigerant pair is highest.

Keywords: Thermodynamic analysis, cascade refrigeration system, COP, exergetic efficiency.

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Authors: Ana Clara Santos, Maria Manuela Portela, Bettina Schaefli

Abstract:

This work assesses the performance of an analytical model framework to generate daily flow duration curves, FDCs, based on climatic characteristics of the catchments and on their streamflow recession coefficients. According to the analytical model framework, precipitation is considered to be a stochastic process, modeled as a marked Poisson process, and recession is considered to be deterministic, with parameters that can be computed based on different models. The analytical model framework was tested for three case studies with different hydrological regimes located in Switzerland: pluvial, snow-dominated and glacier. For that purpose, five time intervals were analyzed (the four meteorological seasons and the civil year) and two developments of the model were tested: one considering a linear recession model and the other adopting a nonlinear recession model. Those developments were combined with recession coefficients obtained from two different approaches: forward and inverse estimation. The performance of the analytical framework when considering forward parameter estimation is poor in comparison with the inverse estimation for both, linear and nonlinear models. For the pluvial catchment, the inverse estimation shows exceptional good results, especially for the nonlinear model, clearing suggesting that the model has the ability to describe FDCs. For the snow-dominated and glacier catchments the seasonal results are better than the annual ones suggesting that the model can describe streamflows in those conditions and that future efforts should focus on improving and combining seasonal curves instead of considering single annual ones.

Keywords: Analytical streamflow distribution, stochastic process, linear and non-linear recession, hydrological modelling, daily discharges.

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Authors: Hasan.O. Farahneh, Ayman A. Issa

Abstract:

Software development is moving towards agility with use cases and scenarios being used for requirements stories. Estimates of software costs are becoming even more important than before as effects of delays is much larger in successive short releases context of agile development. Thus, this paper reports on the development of new linear use case based software cost estimation model applicable in the very early stages of software development being based on simple metric. Evaluation showed that accuracy of estimates varies between 43% and 55% of actual effort of historical test projects. These results outperformed those of wellknown models when applied in the same context. Further work is being carried out to improve the performance of the proposed model when considering the effect of non-functional requirements.

Keywords: Metrics, Software Cost Estimation, Use Cases

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Authors: Praloy Kumar Biswas, Prof. Dipanwita Roy Chowdhury

Abstract:

Grobner basis calculation forms a key part of computational commutative algebra and many other areas. One important ramification of the theory of Grobner basis provides a means to solve a system of non-linear equations. This is why it has become very important in the areas where the solution of non-linear equations is needed, for instance in algebraic cryptanalysis and coding theory. This paper explores on a parallel-distributed implementation for Grobner basis calculation over GF(2). For doing so Buchberger algorithm is used. OpenMP and MPI-C language constructs have been used to implement the scheme. Some relevant results have been furnished to compare the performances between the standalone and hybrid (parallel-distributed) implementation.

Keywords: Grobner basis, Buchberger Algorithm, Distributed- Parallel Computation, OpenMP, MPI.

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Authors: Prasannakumar Palaniappan, Dong Ho Shin, Chul Gyu Song

Abstract:

The photoacoustic images are obtained from a custom developed linear array photoacoustic tomography system. The biological specimens are imitated by conducting phantom tests in order to retrieve a fully functional photoacoustic image. The acquired image undergoes the active region based contour filtering to remove the noise and accurately segment the object area for further processing. The universal back projection method is used as the image reconstruction algorithm. The active contour filtering is analyzed by evaluating the signal to noise ratio and comparing it with the other filtering methods.

Keywords: Contour filtering, linear array, photoacoustic tomography, universal back projection.

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Authors: Aneta Oblouková, Eva Vítková

Abstract:

The article deals with the issue of water and sewerage charge rate prices in the Czech Republic. The research is specifically focused on the analysis of the development of the average prices of water and sewerage charge rate in the Czech Republic in 1994-2021 and on the validation of the chosen methodology relevant for the prediction of the development of the average prices of water and sewerage charge rate in the Czech Republic. The research is based on data collection. The data for this research were obtained from the Czech Statistical Office. The aim of the paper is to validate the relevance of the mathematical linear trend estimate technique for the calculation of the predicted average prices of water and sewerage charge rates. The real values of the average prices of water and sewerage charge rates in the Czech Republic in 1994-2018 were obtained from the Czech Statistical Office and were converted into a mathematical equation. The same type of real data was obtained from the Czech Statistical Office for 2019-2021. Prediction of the average prices of water and sewerage charge rates in the Czech Republic in 2019-2021 was also calculated using a chosen method – a linear trend estimation technique. The values obtained from the Czech Statistical Office and the values calculated using the chosen methodology were subsequently compared. The research result is a validation of the chosen mathematical technique to be a suitable technique for this research.

Keywords: Czech Republic, linear trend estimation, price prediction, water and sewerage charge rate.

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

Abstract:

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

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

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