System Of Simultaneous Linear Equations Research Articles

Hydroelastic interaction between obliquely incident waves and a semi-infinite elastic plate on a two-layer fluid

The hydroelastic response of a semi-infinite thin elastic plate floating on a two-layer fluid of finite depth due to obliquely incident waves is investigated. The upper and lower fluids with different densities separated by a sharp and stable interface are assumed to be inviscid and incompressible and the motion to be irrotational. Simply time-harmonic incident waves of the surface and interfacial wave modes with a given angular frequency are considered within the framework of linear potential flow theory. With the aid of the methods of matched eigenfunction expansion and the inner product of the two-layer fluid, a closed system of simultaneous linear equations is derived for the reflection and transmission coefficients of the series solutions. Based on the dispersion relations for the gravity waves and the flexural–gravity waves in a two-layer fluid and Snell’s law for refraction, we obtain a critical angle for the incident waves of the surface wave mode and three critical angles for the incident waves of the interfacial wave mode, which are related to the existence of the propagating waves. Graphical representations of the series solutions show the interaction between the water waves and the plate. The effects of several physical parameters, including the density and depth ratios of the fluid and the thickness of the plate, on the wave scattering and the hydroelastic response of the plate are studied. It is found that the variation of the thickness of the plate may change the wave numbers and the critical angles. The density ratio is the main factor to influence the wave numbers of the interfacial wave modes. Finally, the stress state is considered.

Scattering of anti-plane (SH) waves by a semi-elliptical hill: I—Shallow hill

The paper presents an exact, analytical solution to the boundary-valued problem of the two-dimensional scattering of anti-plane (SH) waves by a shallow, semi-elliptical hill on an elastic half-space, based on the method of wave-function expansion in elliptical coordinate and elliptical cosine half-range expansion. It is expressed in terms of an infinite system of simultaneous linear equations that is later truncated for numerical computation. Some numerical results when semi-elliptical tends to semi-circular are compared with the existing results presented by Lee et al. (2006) [12]. Complicated effects on ground motion due to the existence of an elliptical hill at various aspect ratios and angles of wave incidence are illustrated.

Quantification of Quercus petraea Liebl. forking based on a 23-year-long longitudinal survey

For Quercus petraea (sessile oak), the origin of forks is either traumatic or related to crown base formation (“metamorphosis”). The objectives of this study were to quantify forking, distinguish the two origins and test the effect of initial stocking density and tree attributes on the number of forks and height to the lowest one.The experiment took place in a 23-year-old sessile oak plantation in Normandy, France, where three initial planting densities were compared (1333, 2667 and 5333seedlings per ha). The forks were observed every 3years after the plantation. Fork number was modelled with a system of simultaneous linear equations that respectively predicted tree girth, height and the log-transformed fork frequency, with age and initial stocking density as explanatory variables. This model 1 was fitted as a seemingly unrelated regression with random effects to account for the tree and plot clustering. For trees having at least one fork, lowest fork height was quantified using a linear mixed model including predictors compatible with the response variables of model 1, age and stocking.The results showed that the number of forks can be predicted using the expected explanatory variables and it is significantly correlated with height and girth as well. Traumatic and metamorphosis-related origins were not clearly distinguished, but significant differences between the initial seedling densities were found, with the lowest density showing the highest number of forks. Lowest fork height depended on response variables of model 1, age and stocking. The stocking effect was significant but small.This study presents the first statistical quantification of tree forking ever made. From this quantification, it seemed obvious that the initial density of 2667 seedlings per ha could be a relevant initial density to avoid the emergence of too many forks and the insertion of the lowest forks too low down.

LU Decomposition Method for Solving Fully Fuzzy Linear System with Trapezoidal Fuzzy Numbers

System of simultaneous linear equations plays a vital role in mathematics, Operations Research, Statistics, Physics, Engineering and Social Sciences etc. In many applications at least some of the system's parameters and measurements are represented by fuzzy numbers rather than crisp numbers. Therefore it is imperative to develop mathematical models and numerical procedures to solve such a fuzzy linear system. The general model of a fuzzy linear system whose coefficient matrix is crisp and the right hand side column is an arbitrary fuzzy vector. In the fully fuzzy linear system all the parameters are considered to be fuzzy numbers. Since triangular fuzzy numbers is a special case of trapezoidal fuzzy numbers, hence in this paper we considered fully fuzzy linear system with trapezoidal fuzzy numbers. LU decomposition method for a crisp matrix is well known in solving linear system of equations. We discuss LU decomposition of the coefficient matrix of the fully fuzzy linear system, in which the coefficients are trapezoidal fuzzy numbers.

An ultra compact spectrometer based on the optical transmission through a micro interferometer array

We present a chip-level integrated optical spectrometer and demonstrate its high resolution performance by using finite-difference time-domain (FDTD) simulations. In this miniature interferometer-based spectrometer, a two-dimensional array of micron-scale phase objects made from polymethyl methacrylate (PMMA) is mounted on an imaging device. Our analysis shows that the two-dimensional intensity distribution obtained by the imaging device can be described by a system of simultaneous linear equations when the phase delay within each interferometer element is pre-designed appropriately. One can readily recover the spectral contents of the input radiation with picometer resolution. The proposed design takes advantage of the well-established imaging integrated circuits and precision molding technologies for producing miniature, low-cost and high performance optical spectrometers, which may find applications in a wide range of applications.

Effects of the transition rate uncertainty on the steady state probabilities of Markov models using interval arithmetic

Markov models are often used to assess the reliability or availability of a system. Such assessment generally involves calculations of the steady state probability for each possible state in the system by solving a linear system of simultaneous equations based on the transition rates between states or using ad hoc algorithms such as the state reduction algorithm. Transition rates are usually subject to uncertainty. This paper proposes the use of interval arithmetic as an alternative approach for dealing with uncertainties in the transition rates that are modelled through lower and upper bounds. Interval arithmetic takes into account the uncertainty of all the parameters and is able to provide strict bounds with only one evaluation. The interval version of the state reduction algorithm, with additional procedures to enhance its precision, is proposed as an alternative method to evaluate uncertainty propagation. Several examples illustrate the proposed approach. Results are compared with other classical approaches, such as Monte Carlo simulation or solving a linear system of simultaneous equations.

An Application of Sylvester's Rank Inequality

SummaryUsing two well known criteria for the diagonalizability of a square matrix plus an extended form of Sylvester's Rank Inequality, the author presents a new condition for the diagonalization of a real matrix from which one can obtain the eigenvectors by simply multiplying some associated matrices without solving a linear system of simultaneous equations.

The Effectiveness of Genetic Algorithm in Solving Simultaneous Equations

There are several known conventional algorithms for solving system of linear equations, which are based on some theoretical principles. Finding solution to these set of equations through the evolutionary process of genetic algorithm is a new and developing research area of interest. The Genetic Algorithm approach follows the concept of solution evolution by stochastically developing generations of solutions population using a definite fitness function to determine the best fit solution to the problem. In this study, we experimented with a new non-conventional approach, based on biological evolution, to solving system of simultaneous linear equations. We discussed the origin of Genetic Algorithm and explore its applicability in solving system of simultaneous equations. We used Genetic Algorithm, on one hand, and Gaussian elimination method, on the other hand, to solve seven different systems of simultaneous linear equations. We then compared the results obtained from the two methods. It was observed that the Genetic Algorithm was very effective in discovering all possible sets of solutions that are applicable to any given system of simultaneous linear equations. Conventional numerical methods, such as Gaussian elimination method, produced a single set of solutions for a particular system of simultaneous linear equations, but GA was able to produce more than one set of solutions for certain systems of equations. For example, during our experiments with GA equation solver, one particular set of equations produced three different sets of perfect solutions, which perfectly fit into the equations. General Terms Genetic Algorithms

DVCC-Based Non-linear Feedback Neural Circuit for Solving System of Linear Equations

A neural circuit to solve a system of simultaneous linear equations is presented. The circuit employs non-linear feedback to achieve a transcendental energy function that ensures fast convergence to the exact solution while enjoying reduction in hardware complexity over existing schemes. A new building block for analog signal processing, the digitally controlled differential voltage current conveyor (DC-DVCC) is introduced and is utilized for the non-linear synaptic interconnections between neurons. The proof of the energy function has been given and it is shown that the gradient network converges exactly to the solution of the system of equations. PSPICE simulation results are presented for linear systems of equations of various sizes and are found to be in close agreement with the algebraic solution. The use of CMOS DC-DVCCs and operational amplifiers facilitates monolithic integration.

A neural circuit with transcendental energy function for solving system of linear equations

A feedback neural network based circuit to solve a system of simultaneous linear equations is presented. The circuit has an associated transcendental energy function that ensures fast convergence to the exact solution while enjoying reduction in hardware complexity over existing schemes. The proof of the energy function has been given and it is shown that the gradient network converges exactly to the solution of the system of equations. PSPICE simulation results are presented for linear systems of equations of various sizes and are found to agree exactly with the algebraic solution. Hardware implementation for small sized problems further confirm the circuit operation.

Combined diffraction and radiation of ocean waves around an OWC device

In a previous paper we provided an analytical solution for the problem of diffraction of plane surface waves by a suspended hollow cylindrical shell structure. Following on from that work, in this paper we formulate and solve the related problem of radiation of ocean waves by an oscillating surface pressure within the cylinder. We find that the same technique of reducing the problem to an infinite system of linear simultaneous equations can still be employed, and use this technique to find a rapidly converging solution. We present an analysis of convergence and accuracy for the radiation problem, as well as results for the combined diffraction-radiation problem, which we obtain by a linear superposition of the two wave fields.

Why we should give up the sin2ψ method

The sin2ψ method can be formulated as a single system of simultaneous linear equations. Using this it is easy to show that the sin2ψ method is not a least-squares method. It further helps to compare the accuracies of the stress tensors obtained by the sin2ψ method and the method of least squares. Quantitative comparisons have been made for different fictitious measurements. It is shown that the unnecessary loss in accuracy by using the sin2ψ method is quite significant and by no means negligible. The same course of action has been applied to compare the so-called Dölle-Hauk method with a least-squares method; the result is similar. Some other methods for X-ray stress determination, most often similar to the sin2ψ method, and their shortcomings are also discussed briefly, together with the corresponding, more effective and more versatile least-squares method.

Asymptotic expansions and higher order properties of semi-parametric estimators in a system of simultaneous equations

Asymptotic expansions are made for the distributions of the Maximum Empirical Likelihood (MEL) estimator and the Estimating Equation (EE) estimator (or the Generalized Method of Moments (GMM) in econometrics) for the coefficients of a single structural equation in a system of linear simultaneous equations, which corresponds to a reduced rank regression model. The expansions in terms of the sample size, when the non-centrality parameters increase proportionally, are carried out to O ( n − 1 ) . Comparisons of the distributions of the MEL and GMM estimators are made. Also, we relate the asymptotic expansions of the distributions of the MEL and GMM estimators to the corresponding expansions for the Limited Information Maximum Likelihood (LIML) and the Two-Stage Least Squares (TSLS) estimators. We give useful information on the higher order properties of alternative estimators including the semi-parametric inefficiency factor under the homoscedasticity assumption.

Method-of-moment view of linear simultaneous equation systems

In this paper, we review the modern method-of-moment-based approaches to identification and estimation of linear simultaneous equation systems. First, we present the rank condition for the structural form (SF) parameter identification. The rank condition comes naturally and is much easier to understand than that in the conventional reduced-form-based indirect approach. Then, we show how to estimate all SF parameters jointly (in a single step) with method-of-moment estimators. As it turns out, using only unconditional moments, but not any conditional moments, greatly simplifies the identification and estimation issues, and makes light work of conveying the essential ideas involved.

Ranking Economics Departments Worldwide on the Basis of PhD Placement

An objective ranking of economics departments worldwide in terms of graduate education is derived. The central idea is that the value of a department is the sum of the values of its PhD graduates, as reflected in the values of their current employing departments. The scores are thus derived as solutions to a linear system of simultaneous equations in the values. The sample includes the top fifty-four departments, the composition of which is determined endogenously using a criterion requiring a minimum of four placements in the departments comprising the sample. Two other related rankings are proposed, which place more emphasis on more recent faculty recruitments. The results point to a very high concentration in the economics PhD education market worldwide, confirming the dominance of the top U.S. departments, in particular of Harvard and M.I.T. Nevertheless, a modest de-concentration trend is under way. The rankings are in close agreement with the 1994 National Research Council survey ranking based on the perceived quality of PhD programs.

Partial identification of probability distributions with misclassified data

This paper addresses the problem of data errors in discrete variables. When data errors occur, the observed variable is a misclassified version of the variable of interest, whose distribution is not identified. Inferential problems caused by data errors have been conceptualized through convolution and mixture models. This paper introduces the direct misclassification approach. The approach is based on the observation that in the presence of classification errors, the relation between the distribution of the ‘true’ but unobservable variable and its misclassified representation is given by a linear system of simultaneous equations, in which the coefficient matrix is the matrix of misclassification probabilities. Formalizing the problem in these terms allows one to incorporate any prior information into the analysis through sets of restrictions on the matrix of misclassification probabilities. Such information can have strong identifying power. The direct misclassification approach fully exploits it to derive identification regions for any real functional of the distribution of interest. A method for estimating the identification regions and construct their confidence sets is given, and illustrated with an empirical analysis of the distribution of pension plan types using data from the Health and Retirement Study.

Ranking Economics Departments Worldwide On the Basis of PhD Placement

An objective ranking of economics departments worldwide in terms of graduate education is derived. The central idea is that the value of a department is the sum of the values of its PhD graduates, as reflected in the values of their current employing departments. The scores are thus derived as solutions to a linear system of simultaneous equations in the values. The sample includes the top fifty-four departments, the composition of which is determined endogenously using a criterion requiring a minimum of four placements in the departments comprising the sample. Two other related rankings are proposed, which place more emphasis on more recent faculty recruitments. The results point to a very high concentration in the economics PhD education market worldwide, confirming the dominance of the top U.S. departments, in particular of Harvard and M.I.T. Nevertheless, a modest de-concentration trend is under way. The rankings are in close agreement with the 1994 National Research Council survey ranking based on the perceived quality of PhD programs.

Correlation, price discovery and co-movement of asset-backed securities and equity

Asset-backed securitisation has become a viable and increasingly attractive risk management and refinancing method either as a stand-alone form of structured finance or as securitised debt in collateralised debt obligations (CDO). The absence of industry standardisation, however, has prevented rising investment demand from translating into market liquidity comparable with traditional fixed income instruments, in all but a few selected market segments. In particular, low financial transparency and complex security designs inhibit profound analysis of secondary market pricing and how it relates to established forms of external finance. This paper represents the first attempt to measure the intertemporal, bivariate causal relationship between matched price series of equity and ABS issued by the same entity. In a two-dimensional linear system of simultaneous equations, it investigates the short-term dynamics and long-term consistency of daily secondary market data from the UK sterling asset-backed securities (ABS)/mortgage backed securities (MBS) market and exchange traded shares between 1998 and 2004 with and without the presence of cointegration. The causality framework delivers compelling empirical support for a strong co-movement between matched price series of ABS-equity pairs, where ABS markets seem to contribute more to price discovery over the long run. Controlling for cointegration, risk-free interest and average market risk of corporate debt hardly alters the results. Once the magnitude and direction of price discovery on various security characteristics, such as the ABS asset class, are qualified, however, it is found that ABS-equity pairs with large-scale commercial MBS/residential MBS and credit card/student loan ABS reveal stronger lead-lag relationships and joint price dynamics than whole business ABS.

TWO-DIMENSIONAL NUMERICAL MODEL FOR THERMAL POLLUTION OF SINGEL SOURCE IN RIVER

The aim of this research was to apply a numerical model capable of describing the thermal pollution in rivers. For this purpose a two-dimensional numerical model was applied for estimating temperature distribution in a river Momentum Conservation Equation and Thermal Energy Equation were used to describe the fdistribution and diffusion of temperature along the river subjected to thermal pollution point source. Furthermore the model incorporates the (k-e) turbulence model to calculate the distribution of turbulent viscosity The pressure distribution was determined using hydrostatic pressure equation. The partial differential equations were formalized and simplified to be solved using Alternative direction implicit- explicit method (ADI) with upwinding technique. The resulting system of linear simultaneous equations were then solved using Gauss-Elimination method. A laboratory physical model was built to find experimental data. These data were used for model verification with data obtained from Al-Daura power station and Tigris river. The model was found to be sensitive to the variation of river velocity and density difference and the model was found to be insensitive to the wind speed. The comparison of observed results from Al- Daura power station and laboratory physical model with those computed by the numerical model show a good agreement. The maximum absolute difference percentage are (16.2%, 8.6%) respectively

Ecological pricing and transformity: A solution method for systems rarely at general equilibrium

This paper outlines a new method for determining ecological prices/transformities in complicated ecological–economic systems where non-equilibrium prices are likely to be prevalent. That is, an eigenvalue–eigenvector mathematical solution method for solving an overdetermined, homogeneous system of simultaneous linear equations is proposed and tested. Previous mathematical approaches to the problem of ecological pricing are reviewed before proceeding with an explanation of the new solution method for the determination of ecological prices. The proposed method seeks to avoid the need to make a number of untenable assumptions used in previous methods—e.g., the assumption that there necessarily needs to be an equal number of processes and quantities in the ecological–economic system. The paper also provides practical advice and an algorithm for solving the ecological pricing problem. For example, advice on how to overcome the problem of ill-conditioned matrices which are sometimes encountered.