Showing papers on "Mathematical model published in 1985"

Computer generation of structural models of amorphous Si and Ge.

Abstract: We have developed and applied a computer algorithm that generates realistic random-network models of $a$-Si with periodic boundary conditions. These are the first models to have correlation functions that show no serious discrepancy with experiment. The algorithm provides a much-needed systematic approach to model construction that can be used to generate models of a large class of amorphous materials.

Quantum mechanics of the scalar field in the new inflationary universe.

Abstract: An attempt is made to clarify the quantum theory of the ``slow-rollover'' phase transition which characterizes the new inflationary universe model. We discuss the theory of the upside-down harmonic oscillator as a toy model, with particular emphasis on the fact that the system can be described at late times by a classical probability distribution. An approximate but exactly soluble model for the scalar field is then constructed, based on three principal assumptions: (1) exact de Sit- ter expansion for all time; (2) a quadratic potential function which changes from stable to unstable as a function of time; and (3) an initial state which is thermal in the asymptotic past. It is proposed that this model would be the proper starting point for a perturbative calculation in more realistic models. The scalar field can also be described at late times by a classical probability distribution, and numerical calculations are carried out to illustrate how this distribution depends on the parameters of the model. For a suitable choice of these parameters, a sufficient period of inflation can be easily obtained. Density fluctuations can be calculated exactly in this model, and the results agree very well with those previously obtained using approximate methods.

A Multiphase Approach to the Modeling of Porous Media Contamination by Organic Compounds: 2. Numerical Simulation

Abstract: A system of equations, derived in part 1 of this paper, which describes the multiphase migration of an organic contaminant in the subsurface is presented. Although this system is not amenable to solution by analytical means, an approximate solution can be sought by a finite difference discretization of the governing equations. A one-dimensional, implicit numerical model is developed in this manner. To handle the solution of the resultant system of nonlinear algebraic equations, a Newton-Raphson iteration scheme is employed. In order to apply the finite difference model to a specific problem a number of parameters must be evaluated. These include three-phase relative permeabilities, saturation-pressure relations, partition coefficients, and mixture densities and viscosities. As a demonstration of the model's applicability, the migration of a two-component hydrocarbon mixture in a soil column is simulated. A mass balance is performed, and convergence of the iteration scheme as well as convergence of the difference scheme in space and time are examined heuristically.

Hysteresis models from the mathematical and control theory points of view

Abstract: A new approach to Preisach’s model which emphasizes the phenomenological nature of this model and its mathematical generality is discussed. This approach enables the formulation of necessary and sufficient conditions under which actual hysteresis nonlinearities can be represented by Preisach’s model.

Geomagnetic field analysis—III. Magnetic fields on the core—mantle boundary

Abstract: Summary. The method of stochastic inversion, previously applied to secular variation data, is applied to main field data. Adaptations to the method are required: non-linear, as well as linear, data are used; allowance is made for crustal components in the observatory data; and the prior information is specified differently. The requirement that the models should satisfy a finite lower bound on the Ohmic heating in the core provides strong prior information and gives finite error estimates at the core—mantle boundary. The new method is applied to data from the epochs 1969.5 and 1980.0. The resulting field models are very much more complex than other models, such as the IGRF models extrapolated to the core, and show considerable small-scale detail which, on the basis of the error analysis, can be believed. The flux integral over the northern hemisphere is computed at each epoch; the difference between the two epochs is approximately one standard deviation, suggesting that the question as to whether the decay of the dipole is consistent with the frozen-flux hypothesis has been resolved in favour of the hypothesis.

An Examination of Response-Surface Methodologies for Uncertainty Analysis in Assessment Models

Abstract: Two techniques of uncertainty analysis were applied to a mathematical model that estimates the dose-equivalent to man from the concentration of radioactivity in air, water, and food. The response-surface method involved screening of the model to determine the important parameters, development of the response-surface equation, calculating the moments using the response-surface model, and fitting a Pearson or Johnson distribution using the calculated moments. The second method sampled model inputs by Latin hypercube methods and iteratively simulated the model to obtain an empirical estimation of the cdf. Comparison of the two methods indicates that it is often difftcult to ascertain the adequacy or reliability of the response-surface method. The empirical method is simpler to implement and, because all model inputs are included in the analysis, it is also a more reliable estimator of the cumulative distribution function of the model output than the response-surface method.

Simple model for transient soil loading in earthquake analysis. I. Basic model and its application

Abstract: Several models describing soil response under cyclic loading and the 'liquefaction' potential have been introduced in recent years with limited success. Most of these are over-complex for realistic parameter identification and have not been widely adopted for practical use. This paper introduces a relatively simple modification of the well-known critical state model which accounts reasonably well for the phenomena observed under cyclic tests and indeed improves the performance of critical state models in monotonic loading. This model is compared with experimental results and with the 'densification model' introduced earlier by the authors and shows good predictive capacity. The model is of a generalized plasticity-bounding surface type. In its simplest form, suitable for clay-like materials, it requires the identifications of a single parameter additional to those required for a standard, critical state model. See also IRRD 288367. (Author/TRRL)

Earth Surface Systems

Abstract: I Introduction.- 1 Systems and Models.- 1.1 Defining Systems.- 1.1.1 Systems as Form and Process Structures.- 1.1.2 Systems as Simple and Complex Structures.- 1.1.3 Systems and Their Surroundings.- 1.1.4 A Problem of Scale.- 1.2 Models of Systems.- 1.2.1 Conceptual Models.- 1.2.2 Scale Models.- 1.2.3 Mathematical Models.- II Conceptual Models.- 2 Simple and Complex Systems.- 2.1 Simple Systems.- 2.2 Systems of Complex Disorder.- 2.2.1 Irreversible Processes.- 2.2.2 Accounting Models: the Laws of Thermodynamics.- 2.3 Systems of Complex Order.- 2.3.1 Nonequilibrium Systems.- 2.3.2 Systems Far from Equilibrium.- 2.3.3 Open Systems at the Earth's Surface.- 3 Form and Process Systems.- 3.1 Models of System Form.- 3.1.1 Models of System Constitution.- 3.1.2 Models of System Geometry.- 3.2 Models of System Process.- 3.2.1 Land-Surface Cascades.- 3.2.2 Solid-Phase and Liquid-Phase Cascades.- 3.3 Models of System Form and Process.- 3.3.1 Concepts of Landscape Development.- 3.3.2 Concepts of Soil Development.- 3.3.3 Concepts of Soil-Landscape Development.- III Mathematical Models.- 4 Deductive Stochastic Models.- 4.1 Introduction to Probability.- 4.1.1 The Classical View of Probability.- 4.1.2 The Relative Frequency View of Probability.- 4.1.3 Axioms of Probability Theory.- 4.2 Independent Events in Time.- 4.2.1 Binomial Processes.- 4.2.2 Poisson Processes.- 4.3 Independent Events in Space.- 4.3.1 Point Patterns.- 4.3.2 Line Patterns.- 4.3.3 Area Patterns.- 4.4 Random-Walk Models.- 4.4.1 Stream Networks.- 4.4.2 Alluvial Fans.- 4.5 Markov Chains.- 4.5.1 Transition Probabilities.- 4.5.2 Sedimentary Sequences.- 4.5.3 Volcanic Activity.- 4.6 Entropy Models.- 4.6.1 Entropy Maximization.- 4.6.2 Entropy Minimization.- 4.6.3 Developments of the Thermodynamic Approach.- 5 Inductive Stochastic Models.- 5.1 Box and Jenkins's Models: an Introduction.- 5.1.1 System Definition.- 5.1.2 Stages in Systems Analysis.- 5.2 Autoregressive Moving-Average Models of Time Series.- 5.2.1 Model Formulation.- 5.2.2 Modelling Procedures.- 5.2.3 The Lagan Rainfall Series.- 5.3 Autoregressive Moving-Average Models of Distance Series.- 5.3.1 Model Formulation.- 5.3.2 River Meanders.- 5.3.3 Landforms.- 5.4 Transfer Function Models.- 5.4.1 Model Formulation.- 5.4.2 Rainfall and Runoff in the Lagan Drainage Basin.- 5.4.3 Channel Form in the Afon Elan, Wales.- 5.5 Problems of Inductive Stochastic Modelling.- 6 Statistical Models.- 6.1 Simple Regression and Correlation.- 6.1.1 The Regression Line.- 6.1.2 The Correlation Coefficient.- 6.1.3 Problems of Correlation.- 6.1.4 Linear Relations.- 6.1.5 Linear Versus Nonlinear Relations.- 6.2 Multiple Regression.- 6.2.1 "Simple" Multiple Regression.- 6.2.2 Trend Surface Analysis.- 6.2.3 Stepwise Regression.- 6.2.4 Problems of Multiple Regression.- 6.3 Correlation Systems.- 6.3.1 Principal Component Analysis.- 6.3.2 Principal Coordinate Analysis.- 6.3.3 Factor Analysis.- 6.3.4 Canonical Correlation.- 6.3.5 Problems with Correlation Systems.- 7 Deterministic Models of Water and Solutes.- 7.1 Ice.- 7.1.1 Glaciers.- 7.1.2 Ice Sheets.- 7.2 Water.- 7.2.1 Overland Flow.- 7.2.2 Open Channel Flow.- 7.2.3 Flow in Porous Media.- 7.2.4 Unsaturated Flow.- 7.3 Solutes.- 7.3.1 Seas and Lakes.- 7.3.2 Solutes in Groundwater.- 7.3.3 Solutes in Soils.- 8 Deterministic Models of Slopes and Sediments.- 8.1 Discrete Component Models.- 8.2 Analytical Models.- 8.2.1 Heuristic Models.- 8.2.2 Models Based on the Continuity Equation.- 8.3 Simulation Models.- 8.3.1 Landscape Simulation.- 8.3.2 Drainage Basin Simulation.- 8.3.3 Nearshore Bar Formation.- 8.3.4 Sand Dune Formation.- 9 Dynamical Systems Models.- 9.1 Model Building.- 9.1.1 State and State Change.- 9.1.2 Transfer Equations.- 9.2 System Stability.- 9.2.1 State Space.- 9.2.2 Sensitivity Analysis.- 9.3 Biogeochemical Cycles.- 9.3.1 The Global Cycle of Phosphorus.- 9.3.2 The Global Cycle of Carbon Dioxide and Oxygen.- 9.3.3 Strontium and Manganese in a Tropical Rain Forest.- 9.3.4 Water in Soils.- 9.3.5 Nutrients in Lake Erie.- 9.4 Dissipative Structures.- 9.4.1 Bifurcations and Catastrophes.- 9.4.2 Thresholds.- 9.4.3 Dominance Domains.- 10 Conclusion and Prospect.- 10.1 Models as a Complement to Field Studies.- 10.2 Models as a Testing Ground for Long-Term Change.- 10.3 Models as Good Predictors of Complex Situations.- References.

Maximum-Likelihood Reconstruction for Single-Photon Emission Computed-Tomography

Abstract: A mathematical model is formulated for a gamma camera used to observe single-photon emissions from multiple view angles. The model accounts for the statistics of radioactive decays, nonuniform attenuation, and a depth-dependent point-spread function. The maximum-likelihood method of statistics is used with the model to derive an algorithm for estimating the distribution of radioactivity.

Analysis of River Wave Types

Abstract: In this paper we consider long-period, shallow-water waves in rivers that are a consequence of unsteady flow. River waves result from hydroelectric power generation or flow control at a dam, the breach of a dam, the formation or release of an ice jam, and rainfall-runoff processes. The Saint-Venant equations are generally used to describe river waves. Dynamic, gravity, diffusion, and kinematic river waves have been defined, each corresponding to different forms of the momentum equation and each applying to some subset of the overall range of river hydraulic properties and time scales of wave motion. However, the parameter ranges corresponding to each wave description are not well defined, and the transitions between wave types have not been explored. This paper is an investigation into these areas, which are fundamental to river wave modeling. The analysis is based on the concept that river wave behavior is determined by the balance between friction and inertia. The Saint-Venant equations are combined to form a system equation that is written in dimensionless form. The dominant terms of the system equation change with the relative magnitudes of a group of dimensionless scaling parameters that quantify the friction-inertia balance. These scaling parameters are continuous, indicating that the various river wave types and the transitions between them form a spectrum. Additional data describing the physical variability of a river and wave are incorporated into the analysis by interpreting the scaling parameters as random variables. This probabilistic interpretation provides an improved estimate of the friction-inertia balance, further insight into the continuous nature of wave transitions, and a measure of the reliability of wave type assessments near a transition. Case studies are used to define the scaling parameter ranges representing each wave type and transition and to provide data with which to evaluate the usefulness of the analysis for general application.

Mathematical model to predict 3-D wind loading on buildings

Abstract: A mathematical model to predict 3‐D wind loading on buildings with rectangular geometry and wind acting normally to a face is formulated. This methodology allows the evaluation of force distribution in the frequency domain including mean wind loads and fluctuating loads due to alongwind and acrosswind atmospheric turbulence and wake excitation; self‐excited forces are not taken into consideration. The reliability of the proposed technique is verified by comparing predicted results with experimental data available in the literature. The satisfactory agreement between theoretical and experimental results demonstrates the good applicability of this methodology, but also emphasizes the necessity of carrying out extensive wind tunnel and full‐scale experiments in order to improve the knowledge of the most relevant aerodynamic parameters on which the correctness of the predicted results strongly depends.

Statistics and the Scientific Method

Abstract: Regression models have not been so useful in the social sciences. In an attempt to see why, such models are contrasted with successful mathematical models in the natural sciences, including Kepler’s three laws of motion for the planets.

Computational model for interfaces

Abstract: Decompositions of the plane into disjoint components separated by curves occur frequently. We describe a package of subroutines which provides facilities for defining, building, and modifying such decompositions and for efficiently solving various point and area location problems. Beyond the point that the specification of this package may be useful to others, we reach the broader conclusion that well-designed data structures and support routines allow the use of more conceptual or non-numerical portions of mathematics in the computational process, thereby extending greatly the potential scope of the use of computers in scientific problem solving. Ideas from conceptual mathematics, symbolic computation, and computer science can be utilized within the framework of scientific computing and have an important role to play in that area.

Solving Equations with Physical Understanding

Abstract: In creating mathematical models of real processes, scientists, engineers and students frequently encounter differential equations whose exact solutions are necessarily complicated and are normally solvable only by computer or through complex formal mathematics. This practical book demonstrates how approximate methods may be used to minimise these mathematical difficulties, giving the reader physical understanding both of the solution process and the final result. Intended for undergraduates and graduate students, teachers of physics, engineering and other applied sciences, professional and applied scientists and engineers.

A Stochastic Approach for Estimating the Effectiveness of a Route Guidance System and Its Related Parameters

Abstract: Mathematical models are developed for estimating the effectiveness of a route guidance system (RGS) in which the control strategy is to guide vehicles toward the shortest travel time routes. These models are formulated by considering the stochastic nature of travel time. The parameters which characterize the models are defined and a quantitative analysis of the models is given for various values of the parameters. It is shown that the effectiveness of RGS can be estimated by the number of nodes along the route. The results derived from the models are shown to agree with experimental data collected from a route guidance system constructed in Tokyo. Some practical applications of the results are discussed.

Mathematical models for nonpoint water pollution control

Abstract: “MATHEMATICAL models” have been used extensively since the late 1960s. During this same period, computers have become commonplace. And because mathematical models are expressed in a computer code, we can now understand what mathematical models are. While the term “mathematical model” may be new, all of us have used mathematical models since we took algebra in school. Such models are mathematical representaions of physical, chemical, biological, social, economic, and related processes. Many formulae that we use in our day-today activities are mathematical models. The expressions needed to provide a good assessment of nonpoint-source water pollution problems can be complex. But in most cases they are collections of relatively simple concepts put together in a way that can be used to study complex problems. One familiar model is the universal soil loss equation (USLE), a regression equation used to calculate long-term average annual soil loss from small areas ( 41 ). The USLE has been coded for solution by computer and is used in most Soil Conservation Service (SCS) field offices. The need for models Planning and control. Perspectives on water resource problems have changed in recent decades. The issues themselves have not changed, but knowledge …

An extended traffic model for freeway control

Abstract: This report presents a basic nonlinear invariant model for simulating motorway traffic flow, and extends it to show more realistic behaviour at bottlenecks. Section i lists seven different reasons for developing mathematical models of road traffic flow, and briefly discusses some earlier models. Section II formulates the basic model, presenting and interpreting its equations in terms of their physical meaning; the model is given in a discrete form, to adapt it for running on digital computers. The model covers the whole range of traffic conditions from free flow to heavy congestion. Section III analyses the defects of the basic model, and develops an extended model able to cope with traffic bottlenecks caused by changes in the number of lanes in a highway and by merging of on-ramp flows. Section IV shows how to calibrate the parameters of the extended model to achieve optimal agreement with observed traffic flows. It then uses the calibrated model for a quantitative comparison between the model's performance and the real road traffic flow phenomena, and for evaluating its accuracy. Section v presents some conclusions: there was good agreement between theory and observations, and the bottleneck phenomena were modelled fairly realistically. Future work should include tests of the extended model with more data sets from different traffic situations and different sections of intercity motorways. Appendix a gives a sample of motorway measurement data, and appendix b gives a FORTRAN program listing of the simulation model.

Steady state multiplicity analysis of lumped-parameter systems described by a set of algebraic equations

Abstract: The mathematical models of many lumped-parameter chemically reacting systems consist of a set of algebraic equations which cannot be reduced explicitly to a single equation. We describe here the Liapunov-Schmidt procedure which reduces the prediction of the local multiplicity features of a system of algebraic equations to the analysis of the features of a single equation, even though the original set of equations cannot be reduced to a single equation. This reduction enables the systematic analysis of a class of problems which could not be handled previously. Several examples illustrate the application of this procedure. A special reduction procedure is presented for systems described by a set of polynomial equations. The reduction in these cases is easier to carry out and enables prediction of the global multiplicity features of the system.

Principles of semiconductor laser modelling

Abstract: The basic equations for analysis of semiconductor lasers are presented and discussed in detail. Numerical and analytical methods for solution of these equations, as well as various approximations and simplifications, are described. Various models for the static properties are reviewed, with emphasis on models including the field distribution. Some models for the dynamic properties are also discussed.

Some Simple Models for Continuous Variate Time Series.

Abstract: : A survey is given of recently developed mathematical models for continuous variate non-Gaussian time series. The emphasis is on marginally specific models with given correlation structure. Exponential, Gamma, Weibull, Laplace, Beta, and Mixed Exponential models are considered for the marginal distributions of the stationary time series. Most of the models are random coefficient, additive linear models. Some discussion of the meaning of autoregression and linearity is given, as well as suggestions for higher-order linear residual analysis for nonGaussian models. (Author)

Analytical Well Models for Reservoir Simulation

Abstract: The computation of flowing well bottom-hole pressure from the pressure of the block containing the well, or the computation of well flow rate when the flowing bottom-hole well pressure is specified, are important considerations in reservoir simulation. While this problem has been addressed by several authors, some important aspects of the problem ae not adequately treated in the literature. An analytical method is presented for computing the well block factors (constants of the productivity index) for a well coated anywhere in a square or rectangular block. Equations for well geometric factors and well fraction constants are given for grid blocks of various types, containing a single well, encountered in reservoir simulation studies. The equations given can be used for both block-centered and point-distributed grids in 5- and 9-point (2-dimensional) finite difference formulations. However, the radial flow assumption utilized in deriving the equations is not always strictly valid. For most practical situations it provides an adequate approximation for near well flow. 12 ref.).