System Of Simultaneous Linear Equations Research Articles

A COMPUTER PROGAM FOR COMPUTING SISMIC WAVE REFLECTION AND TRANSMISSION COEFFICIENTS OF A SINGLE INTERFACE

Reflection and transmission properties of a single interface form the basic step in understanding and solving many problems of seismic waves propagation in multilayered media. The computational procedure of these coefficients is an essential element for numerical modeling or simulation purposes. A computer program for computing seismic wave reflection and transmission coefficients of a single interface has been effectively performed using the Gauss-Jordan elimination technique which is operated on a 4x 4 system of simultaneous linear equations with complex elements. The simultaneous equations have been ar ranged in matribx from. Elements of the matrix are obtained from the application of the boundary conditions to the solution of the seismic wave equations. The reflection and transmission coefficients for the compressional and shear waves constitute the unknowns of the simultaneous linear equations. Experiments using the program demonstrate the efficiency and the precislon of this procedure.

Trigonometric tension B-spline collocation approximations for time fractional Burgers’ equation

This manuscript’s aim is to form and examine the numerical simulation of Caputo-time fractional nonlinear Burgers’ equation via collocation approach with trigonometric tension B-splines as base functions. First, L1 discretization formula is utilized for the time fractional derivative and after linearizing the nonlinear term, the trigonometric tension B-spline interpolants are utilized to get a system of simultaneous linear equations that are solved via Gauss elimination method. Thus, numerical approximation at the desired time level is obtained. It is demonstrated via von-Neumann approach that proposed scheme produces stable solutions. The results of six different test examples having their analytical solutions are compared with the results in the literature to validate the accuracy and efficiency of the scheme.

Towards distortion-free imaging of the eye.

The high power of the eye and optical components used to image it result in "static" distortion, remaining constant across acquired retinal images. In addition, raster-based systems sample points or lines of the image over time, suffering from "dynamic" distortion due to the constant motion of the eye. We recently described an algorithm which corrects for the latter problem but is entirely blind to the former. Here, we describe a new procedure termed "DIOS" (Dewarp Image by Oblique Shift) to remove static distortion of arbitrary type. Much like the dynamic correction method, it relies on locating the same tissue in multiple frames acquired as the eye moves through different gaze positions. Here, the resultant maps of pixel displacement are used to form a sparse system of simultaneous linear equations whose solution gives the common warp seen by all frames. We show that the method successfully handles torsional movement of the eye. We also show that the output of the previously described dynamic correction procedure may be used as input for this new procedure, recovering an image of the tissue that is, in principle, a faithful replica free of any type of distortion. The method could be extended beyond ocular imaging, to any kind of imaging system in which the image can move or be made to move across the detector.

Mixed boundary value problems in power-law functionally graded circular annulus

The present work proposes a semi-analytical technique for solution of mixed boundary value problem in functionally graded circular annulus wherein shear modulus varies radially in power-law form while Poisson’s ratio is constant. The technique relies on two main steps. In the first step, corresponding to terms in periodic Fourier series applied individually as traction along the annulus surface, stress and displacement field in the annulus is computed harnessing Airy stress functions approach. In the second step, leveraging the strain–displacement relations in polar co-ordinates, mixed boundary conditions are rendered in terms of displacement all along the annulus surface. Assuming the unknown traction along the annulus surface in terms of periodic Fourier series with finite terms, the modified displacement boundary condition, family of solution from the first step and orthogonality of sine and cosine functions is used to generate a system of simultaneous linear equations for the series coefficients. Knowing the coefficients of periodic Fourier series, stress and displacement field can be computed everywhere in the annulus. The first step is digitized in terms of MAPLE functions, exhaustively validated through traction distribution comprising of normal, shear traction on part of the boundary and pair of equal and diametrically opposite point load along the boundary. The second step is corroborated through a problem where the inner surface is subjected to a specified traction distribution and mixed boundary conditions exist along the outer surface in the form of complete constraint along a part and traction-free condition elsewhere.

Dynamic stress concentration of an elliptical cavity in a semi-elliptical hill under SH-waves

The method of wave-function expansion in elliptical coordinates, elliptical cosine half-range expansion and Mathieu function were applied to obtain an exact analytical solution of the dynamic stress concentration factor (DSCF) around an elliptical cavity in a shallow, semi-elliptical hill. An infinite system of simultaneous linear equations for solving this problem was established by substituting the wave expression obtained by the Mathieu function including the standing wave expression of elliptical lining given herein into the boundary condition obtained by the region-matching method. The finite equations system with unknown coefficients obtained by truncation were solved numerically, and the results in the case of an ellipse degenerating into a circle were compared with previous results to verify the accuracy of the method. The effects of different aspect ratios, incident wave angles and aperture ratios on the dynamic stress concentration around the elliptical cavity were described. Some numerical results, when the elliptical hill was changed into a circular one, were analyzed and compared in detail. In engineering, this model can be regarded as a semi-cylindrical hill with an elliptical cylindrical unlined tunnel under the action of SH waves, and the results are significant in aseismic design.

Dynamic Vector Mode Regression

We study the semiparametric estimation of the conditional mode of a random vector that has a continuous conditional joint density with a well-defined global mode. A novel full-system estimator is proposed and its asymptotic properties are studied. We specifically consider the estimation of vector autoregressive conditional mode models and of systems of linear simultaneous equations defined by mode restrictions. The proposed estimator is easy to implement and simulations suggest that it is reasonably behaved in finite samples. An empirical example illustrates the application of the proposed methods, including its use to obtain multistep forecasts and to construct impulse response functions.

Waves propagating over a two-layer porous barrier on a seabed

A research of wave propagation over a two-layer porous barrier, each layer of which is with different values of porosity and friction, is conducted with a theoretical model in the frame of linear potential flow theory. The model is more appropriate when the seabed consists of two different properties, such as rocks and breakwaters. It is assumed that the fluid is inviscid and incompressible and the motion is irrotational. The wave numbers in the porous region are complex ones, which are related to the decaying and propagating behaviors of wave modes. With the aid of the eigenfunction expansions, a new inner product of the eigenfunctions in the two-layer porous region is proposed to simplify the calculation. The eigenfunctions, under this new definition, possess the orthogonality from which the expansion coefficients can be easily deduced. Selecting the optimum truncation of the series, we derive a closed system of simultaneous linear equations for the same number of the unknown reflection and transmission coefficients. The effects of several physical parameters, including the porosity, friction, width, and depth of the porous barrier, on the dispersion relation, reflection and transmission coefficients are discussed in detail through the graphical representations of the solutions. It is concluded that these parameters have certain impacts on the reflection and transmission energy.

Numerical computation of some iterative techniques for solving system of linear equations of multivariable

In this paper we introduce, numerical computation of some iterative techniques for solving system of linear simultaneous equations of 4 or more variables. Many iterative techniques is presented by the different formulae. Using Jacobi method, Seidel method and SOR method and their results are compared. The software, Matlab 2009a was used to find the solution of the linear simultaneous equations having diagonally dominant in coefficient matrix. Numerical rate of convergence of solution has been found in each calculation. It was observed that the Seidel method converges at the 12iteration while Jacobi and SOR methods converge to the exact value of X(x, y, z, t) with error level of accuracy 〖10〗^(-15) at the 22th iteration respectively. However, when we compare performance, we must compare both cost, speed of convergence. Some numerical examples are given to illustrate the efficiency and robustness of the techniques. It was then concluded that Seidel is the most effective technique.

Econometric model of non-life technical provisions: the Czech insurance market case study

The paper introduces and discusses a complex econometric model of non-life technical provisions based on the Czech non-life insurance market data. Selected economic-actuarial relations among given insurance variables are described by means of the dynamic linear system of simultaneous equations used in econometrics. In particular, the provision for outstanding claims, the provision for unearned premium, the other (marginal) technical provisions, the acquisition and administrative expenses, the benefit expenses, and their mutual interactions are studied in detail. The suggested simultaneous equations model is estimated, statistically verified, and interpreted with special regard to the actuarial point of view. The proposed modelling scheme can be further employed for prognosing the considered non-life technical provisions. Particularly, such forecasts can be taken into account by non-life insurance companies in their internal calculations (e.g. for financial planning purposes, for testing the sufficiency of non-life technical provisions, or for liability adequacy tests LAT) or by an insurance regulator or supervisory authority (e.g. for performing stress tests). Alternatively, this approach might motivate development of internal models applicable in the Solvency II framework. Both deterministic and randomly generated scenarios are analysed which can deliver relevant outputs for formulating crucial recommendations and conclusions.

Quantitative analysis of the clay minerals in the Shurijeh Reservoir Formation using combined X-ray analytical techniques

The Shurijeh Reservoir Formation of Neocomian age is represented by a sandstone sequence, occasionally interbedded with shale, in the Gonbadli gas field, Kopet-Dagh Basin, Northeastern Iran. In this study X-ray diffraction (XRD) and X-ray fluorescence (XRF) techniques were used together to characterize the Shuirjeh clay minerals in 76 core samples collected from two deep Gonbadli wells. The results of XRF analysis showed high percentages of silicon and moderate to low percentages of aluminum, sulfur, calcium, potassium, sodium, magnesium, and iron in both wells. The XRD analysis indicated that the above elements were concentrated in the form of quartz, anhydrite, dolomite, calcite, plagioclase, K-feldspar, hematite, and clay minerals. Further XRD examination of the clay fraction revealed that illite, chlorite, and kaolinite were the major types of clay minerals. Unlike, glauconite, smectite, and a mixed layer clays of both the illite–smectite and chlorite–smectite types were observed only in very few samples. The percentages of individual clay minerals were determined using external standard calibration curves and successfully validated by a system of simultaneous linear equations acquired from detailed elemental information based on the XRF analysis. The error reached ±5% for the main mineral constituent and ±15% for minor minerals. A local regression relationship was also derived, based on the XRF elemental information, which can be used to estimate the clay contents of other Shurijeh drilled wells with data of pulsed-neutron spectroscopy tools. According to the proposed quantitative approach, the amount of illite varied considerably, reaching 18.3%. In contrast, the amounts of kaolinite and chlorite were generally small, i.e., less than 8.4%. The amount of total clay minerals changed greatly from a minimum of 5% to a maximum of 32.5%. An increase in illite with increasing burial depth and temperature was an obvious indication of deep burial diagenesis in this formation.

Efficient Generalized Inverse for Solving Simultaneous Linear Equations

Solving large scale system of Simultaneous Linear Equations (SLE) has been (and continue to be) a major challenging problem for many real-world engineering and science applications. Solving SLE with singular coefficient matrices arises from various engineering and sciences applications [1]-[6]. In this paper, efficient numerical procedures for finding the generalized (or pseudo) inverse of a general (square/rectangle, symmetrical/unsymmetrical, non-singular/singular) matrix and solving systems of Simultaneous Linear Equations (SLE) are formulated and explained. The developed procedures and its associated computer software (under MATLAB [7] computer environment) have been based on “special Cholesky factorization schemes” (for a singular matrix). Test matrices from different fields of applications have been chosen, tested and compared with other existing algorithms. The results of the numerical tests have indicated that the developed procedures are far more efficient than the existing algorithms.

Miniature spectrometer based on diffraction in a dispersive hole array.

We present an ultra-compact spectrometer that uses a 10×10 hole array as the dispersive component. Our analysis shows that the two-dimensional intensity distribution can be modeled by a system of simultaneous linear equations when the size of each hole in the dispersive component has been pre-designed appropriately. One can readily recover the spectral contents of the input radiation by solving the linear equation system with regularized procedure. Experimental results show that the reconstruction range is at least within the entire visible band, which can be further extended if a near-infrared CCD is used. One therefore envisions strong potential for many wavelength analysis applications.

Assessment of the transition-rates importance of Markovian systems at steady state using the unscented transformation

The Unscented Transformation (UT) is a technique to understand and compute how the uncertainty of a set of random variables, with known mean and variance is propagated on the outputs of a model, through a reduced set of model evaluations as compared with other approaches (e.g., Monte Carlo). This computational effort reduction along with the definition of a proper UT model allows proposing an alternative approach to quantify the transition rates (TR) having the highest contribution to the variance of the steady-state probability, for each possible state of a system represented by a Markov model. The so called “main effects” of each transition rate, as well as high order component interactions are efficiently derived from the solution of only (2n+1) linear system of simultaneous equations, being n the number of transition rates in the model.

Dual-energy X-ray analysis using synchrotron computed tomography at 35 and 60 keV for the estimation of photon interaction coefficients describing attenuation and energy absorption.

A novel method for dual-energy X-ray analysis (DEXA) is tested using measurements of the X-ray linear attenuation coefficient μ. The key is a mathematical model that describes elemental cross sections using a polynomial in atomic number. The model is combined with the mixture rule to describe μ for materials, using the same polynomial coefficients. Materials are characterized by their electron density Ne and statistical moments Rk describing their distribution of elements, analogous to the concept of effective atomic number. In an experiment with materials of known density and composition, measurements of μ are written as a system of linear simultaneous equations, which is solved for the polynomial coefficients. DEXA itself involves computed tomography (CT) scans at two energies to provide a system of non-linear simultaneous equations that are solved for Ne and the fourth statistical moment R4. Results are presented for phantoms containing dilute salt solutions and for a biological specimen. The experiment identifies 1% systematic errors in the CT measurements, arising from third-harmonic radiation, and 20-30% noise, which is reduced to 3-5% by pre-processing with the median filter and careful choice of reconstruction parameters. DEXA accuracy is quantified for the phantom as the mean absolute differences for Ne and R4: 0.8% and 1.0% for soft tissue and 1.2% and 0.8% for bone-like samples, respectively. The DEXA results for the biological specimen are combined with model coefficients obtained from the tabulations to predict μ and the mass energy absorption coefficient at energies of 10 keV to 20 MeV.

Econometric Model of the Czech Life Insurance Market

The aim of the article is to introduce a complex econometric model of cash-lows for the Czech life insurance market. Namely, technical-actuarial links among insurance variables observed in annually published summary balance sheets of life insurers are described by means of an econometric system of linear simultaneous equations. The suggested model is statistically veri ed and thus it can provide useful economic interpretations. Further, adjusted residual bootstrapping is introduced in this context as a straightforward alternative which can solve possible problems with questionable asymptotic distribution properties of residuals. This technique can be applied e.g. for signi cance testing purposes. Finally, an important practical illustration of scenario analysis is considered. Such an analysis might be really useful, e.g. for internal calculations of the Czech life insurers, nancial planning or stress testing in the framework of Solvency II. Two general approaches are presented: deterministic and stochastic. The second one is capable of delivering various empirical probabilities concerning possible future developments.

Control functions in nonseparable simultaneous equations models

The control function approach (Heckman and Robb (1985)) in a system of linear simultaneous equations provides a convenient procedure to estimate one of the functions in the system using reduced form residuals from the other functions as additional regressors. The conditions on the structural system under which this procedure can be used in nonlinear and nonparametric simultaneous equations has thus far been unknown. In this paper, we define a new property of functions called control function separability and show it provides a complete characterization of the structural systems of simultaneous equations in which the control function procedure is valid.

Electromagnetic Wave Scattering by Parallel Dielectric/Metamaterial Coated Elliptic Cylinders

An exact solution to the boundary value problem of multiple scattering by M parallel two-layer elliptic cylinders, based on the separation of variable technique and the addition theorem for Mathieu function is presented. It is expressed in terms of a system of simultaneous linear equations of infinite order. Numerical results for the special case of two parallel two-layer cylinders are presented and discussed for several geometrical cases and material parameters

Field programmable gate array based predictive control system for spacecraft rendezvous in elliptical orbits

SummaryA field programmable gate array (FPGA) based model predictive controller for two phases of spacecraft rendezvous is presented. Linear time‐varying prediction models are used to accommodate elliptical orbits, and a variable prediction horizon is used to facilitate finite time completion of the longer range manoeuvres, whilst a fixed and receding prediction horizon is used for fine‐grained tracking at close range. The resulting constrained optimisation problems are solved using a primal–dual interior point algorithm. The majority of the computational demand is in solving a system of simultaneous linear equations at each iteration of this algorithm. To accelerate these operations, a custom circuit is implemented, using a combination of Mathworks HDL Coder and Xilinx System Generator for DSP, and used as a peripheral to a MicroBlaze soft‐core processor on the FPGA, on which the remainder of the system is implemented. Certain logic that can be hard‐coded for fixed sized problems is implemented to be configurable online, in order to accommodate the varying problem sizes associated with the variable prediction horizon. The system is demonstrated in closed‐loop by linking the FPGA with a simulation of the spacecraft dynamics running in Simulink on a PC, using Ethernet. Timing comparisons indicate that the custom implementation is substantially faster than pure embedded software‐based interior point methods running on the same MicroBlaze and could be competitive with a pure custom hardware implementation.Copyright © 2014 John Wiley & Sons, Ltd.

Determination Effects of Process Parameters on CO2 Reactive absorption System by Mathematical Modeling

A steady state model for CO2- reactive absorption system was developed based on principle of mass transfer and chemical reaction. The pseudo-first order model is assumed and reversibility of chemical reaction was also neglected. The continuity model equation in term of material balance with chemical reaction across an elemental stage K on CO2 was developed. The model consists of system of linear simultaneous equations, and the equations representing the composition of CO2 in both liquid and gas phases were solved. The simulation studies were performed to investigate the effect of changing various process variables such as number of plate, gas flow rate, and CO2 composition in the feed vapor.

Water wave diffraction by a bottom-mounted circular cylinder clamped to an elastic plate floating on a two-layer fluid

The wave diffraction by a bottom-mounted circular cylinder, which is clamped to the center of a floating circular thin elastic plate, in the two-layer fluid of finite depth is investigated for the time-harmonic incident waves of the surface and interfacial wave modes. Each fluid layer is inviscid, incompressible and of constant density. The flexural–gravity waves are composed of the propagating, decaying propagating and evanescent wave modes. Within the framework of the linear potential flow theory, a closed system of simultaneous linear equations is derived to solve the undetermined expansion coefficients with the methods of the angular eigenfunction expansion and the inner product. Explicit numerical computations are employed to test the convergence of the two series for the angular expansions and the evanescent wave modes. The horizontal forces and the moments exerted on the circular cylinder due to different wave modes are discussed in the case of the incident waves of either the surface or interfacial wave mode. It is obtained that the evanescent wave modes are appreciable parts for a high frequency.