Analysis of mathematical models of microstrip transmission lines

Solution methods for eigenvalue problems nonlinear in the spectral parameter are considered for model problems of the theory of microstrip transmission lines. The spectral properties of the Fredholm operator function F() in Sobolev weighting classes are considered. A numerical method is proposed for determining the approximate characteristic values of F(). We prove theorems on nonemptiness, discreteness, and factorization of the spectrum and on convergence of the approximate spectra and characteristic values to the exact quantities.Translated from Metody Matematicheskogo Modelirovaniya, Avtomatizatsiya Obrabotki Nablyudenii i Ikh Primeneniya, pp. 175–198, 1986.     

Stochastic field processes in the mathematical modelling of damped transmission line vibrations

Wind-excited vibrations in the frequency range of 10 to 50 Hz due to vortex shedding often cause fatigue failures in the cables of overhead transmission lines. Damping devices, such as the Stockbridge dampers, have been in use for a long time for supressing these vibrations. The dampers are conveniently modelled by means of their driving point impedance, measured in the lab over the frequency range under consideration. The cables can be modelled as strings with additional small bending stiffness. The main problem in modelling the vibrations does however lay in the aerodynamic forces, which usually are approximated by the forces acting on a rigid cylinder in planar flow. In the present paper, the wind forces are represented by stochastic processes with arbitrary crosscorrelation in space; the case of a Kármán vortex street on a rigid cylinder in planar flow is contained as a limit case in this approach. The authors believe that this new view of the problem may yield useful results, particularly also concerning the reliability of the lines and the probability of fatigue damages.     

A comparison of mathematical models for a recycling cascade in glycogenolysis

Mathematical models are derived which describe a cyclic-AMP initiated recycling enzyme cascade for the activation of glycogen phosphorylase. The components of the cascade are protein kinase, phosphorylase kinase, phosphorylase kinase phosphatase, phosphorylase, and phosphorylase phosphatase. Two classes of models are formulated and compared. The models differ only in the molecular hypothesis for activation of protein kinase by cyclic-AMP; in one case the assumption leads to second order kinetics, in the other third order kinetics are indicated. Amplification and a fast response time are the prime features of both classes of models for the activation of phosphorylase. The role of free calcium in optimizing the activation and deactivation of the cascade is discussed.     

Historical evolution and mathematical models: A sociocultural algorithm

Though there are a lot of approaches to the problem of sociocultural evolution most of them are only one-sided, i.e., they deal only with either social or cultural processes. With few exceptions they are also only informal theories with no formal rigour. In this article we propose a theoretical model which considers both sides of the problem, that is the mutual interdependence of the evolution of social structures and of the culture of a society. A mathematical model, the sociocultural algorithm (SCA), based on these theoretical considerations maps several of the dynamic characteristics of sociocultural evolution, suggesting that universal principles underlie the dynamics of historical evolution.     

A comparison of mathematical programming modeling systems

We compare three mathematical programming modeling languages, GAMS, OMNI and MathPro. To understand the properties of these languages, we formulate four linear programs in each language. The formulations are representative of the kinds of model structures one encounters in practice. Each of the languages focuses on a different view of linear programs. GAMS approximates algebra, OMNI uses the activity view and MathPro uses a block schematic. We summarize our experiences with the languages and suggest areas for further enhancement.     

Phase- and sequence-domain static models equivalent to long double-circuit transmission lines

The application of equivalent rather than nominal models of transmission lines in various areas of power system studies ensures the accuracy of results when the line length and hence distributed-parameter effect is significant. In this research, the static modeling of long double-circuit lines is comprehensively addressed. Equivalent models are analytically derived for them in different possible operating configurations, incorporating both mutual induction and capacitive coupling between the circuits. This is achieved by finding the solution to the differential equations characterizing the phase voltages and currents of the line circuits. Hence, the equivalent models are first derived in the phase domain, and then the equations are transformed into the sequence domain to obtain equivalent models of sequence networks. The accuracy of the models derived is validated via simulation studies in PSCAD/EMTDC.     

A mathematical framework for stochastic climate models

There has been a recent burst of activity in the atmosphere‐ocean sciences community in utilizing stable linear Langevin stochastic models for the unresolved degrees of freedom in stochastic climate prediction. Here a systematic mathematical strategy for stochastic climate modeling is developed, and some of the new phenomena in the resulting equations for the climate variables alone are explored. The new phenomena include the emergence of both unstable linear Langevin stochastic models for the climate mean variables and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances. All of these phenomena are derived from a systematic self‐consistent mathematical framework for eliminating the unresolved stochastic modes that is mathematically rigorous in a suitable asymptotic limit. The theory is illustrated for general quadratically nonlinear equations where the explicit nature of the stochastic climate modeling procedure can be elucidated. The feasibility of the approach is demonstrated for the truncated equations for barotropic flow with topography. Explicit concrete examples with the new phenomena are presented for the stochastically forced three‐mode interaction equations. The conjecture of Smith and Waleffe [Phys. Fluids 11 (1999), 1608–1622] for stochastically forced three‐wave resonant equations in a suitable regime of damping and forcing is solved as a byproduct of the approach. Examples of idealized climate models arising from the highly inhomogeneous equilibrium statistical mechanics for geophysical flows are also utilized to demonstrate self‐consistency of the mathematical approach with the predictions of equilibrium statistical mechanics. In particular, for these examples, the reduced stochastic modeling procedure for the climate variables alone is designed to reproduce both the climate mean and the energy spectrum of the climate variables. © 2001 John Wiley & Sons, Inc.     

Prediction,measurement and comparison of fluid-structure interaction using mathematical and experimental models

Without presenting all the associated mathematics this paper considers the combined use of theoretical and experimental methods in the investigation of the fluid structure interaction of rigid and articulated structures. The structures considered are the rigid body form of the Lancaster Flexible Bag (LFB) and an articulated Cockerell raft system.In the case of the LFB the Salvesen-Tuck-Faltinsen strip theory is shown to predict vertical and horizontal bending moments which are generally consistent with measured trends (almost) irrespective of whether 2D-based or 3D-based calculated hydrodynamic active and reactive coefficients are used. The non-repeatability of some of the measured results used in the comparison caused some concern and the additional required model testing is discussed.In direct comparison application difficulties were encountered when using theoretical analyses to assist with the organization of the experiments to investigate the hydrodynamic loading of articulated systems. The finite element program PAFEC 75 was used to predict the vibration frequencies of the framework used to support the Cockerell raft model and the actuator system used to force oscillate the model. The 3D diffraction programme MATTHEW provided predictions of pressure distributions over the raft to assist with sensible positioning of the pressure transducers on the physical model. The difficulties encountered in applying these prediction tools are discussed, also the current research programme is reported.     

A revisit of spinning disk models. Part II: linear transverse vibrations

In this paper, two recently derived linear models for the transverse vibrations of a spinning plate are considered. The disk is modelled as a pure plate with no membrane effects. Furthermore, the effect of the rotary inertia of the plate is taken into account. The first model is based on the assumption of linear (Kirchhoff) strains. The second model is based on the assumption of non-linear (von Karman) strains. The merits of both models are considered and their predictions are compared with those of the traditional linear model.     

Atmospheric diffusion: some new mathematical models

The paper reviews some recent theoretical work on the statistical properties of the distribution of concentration of an atmospheric pollutant. This work is based on elementary physics. There is emphasis on the behaviour of the moments, particularly on the collapse of the skewness-kurtosis plot onto a parabola. Some of the consequences for modelling the probability density function of concentration are discussed, and examples of models are given.Dedicated to Professor J. Crank on the occasion of his 80th birthday     

A survey of some mathematical programming models in transportation

Transportation problems constitute a fertile domain for the application of mathematical programming models and nonlinear optimization techniques, distribution problems, entropy models, traffic assigment problems and many others are good examples of this assertion. This paper provides a summary overview of the main modeling approaches in transportation and the related optimization models, symmetric and asymmetric, and an overview on the state-of-the-art of the origindestination adjustment problems and the related bilevel optimization methods.     

A review of mathematical models in economic environmental problems

The paper presents a review of mathematical models used in economic analysis of environmental problems. This area of research combines macroeconomic models of growth, as dependent on capital, labour, resources, etc., with environmental models describing such phenomena like natural resources exhaustion or pollution accumulation and degradation. In simpler cases the models can be treated analytically and the utility function can be optimized using, e.g., such tools as the maximum principle. In more complicated cases calculation of the optimal environmental policies requires a computer solution. This revised version was published online in June 2006 with corrections to the Cover Date.     

Parametric scaling of mathematical models

Parametric scaling, the process of extrapolation of a modelling result to new parametric conditions, is often required in model optimization, and can be important if the effects of parametric uncertainty on model predictions are to be quantified. Knowledge of the functional relationship between the model solution (y) and the system parameters (α) may also provide insight into the physical system underlying the model. This paper examines strategies for parametric scaling, assuming that only the nominal model solution y(α) and the associated parametric sensitivity coefficients (?y/?α, ?²y/?α², etc.) are known. The truncated Taylor series is shown to be a poor choice for parametric scaling, when y has known bounds. Alternate formulae are proposed which ‘build-in’ the constraints on y, thus expanding the parametric region in which the extrapolation may be valid. In the case where y has a temporal as well as a parametric dependence, the extrapolation may be further improved by removing from the Taylor series coefficients the ‘secular’ components, which refer to changes in the time scale of y(t), not to changes in y as a function of α.     

Applications of mathematical programming models

A sizeable proportion of manufacturing expenses can be attributed to facility layout and material handling. Facility layout decisions involve designing the arrangement of elements in manufacturing systems. Among the most critical material handling decisions in this area are the arrangement and design of material flow patterns. This survey article reviews loop based facility planning and material handling decisions for trip based material handling equipment with an emphasis on unit load automated guided vehicles. The article examines issues related with facility design, material handling design, and fleet sizing and operating.     

The transmission of meningococcal infection: a mathematical study

The most common type of bacterial meningitis in the developed countries is caused by Neisseria meningitidis bacteria, which cause meningococcal meningitis. Case fatality rate can be between 3% and 10% in developed countries and as high as 20% in African countries. During epidemics in sub-Saharan countries, the so-called meningitis belt, the case fatality rate can peak to 70% or higher. Some people who have bacterial meningitis experience some form of after effects: epilepsy, damaged eyesight, hearing loss, brain damage. There is no immunity after infection. Approximately on average 10% of the population at any time carry the germs for days, weeks, or months. Carriers can infect other individuals by close contacts, even though they do not become ill themselves. An age-structured mathematical model is formulated that enables the understanding of the dynamics of the infection transmission. The model is used to study the conditions for the stability of the disease-free steady state (which imply extinction of the disease) and the existence of an endemic state (which leads to persistence of the disease in the population). The results of the model are applied to identify the contribution of the carriers to the transmission of the disease. Final epidemiological conclusions are given.     

Simulation of slotted transmission lines

Translated from Aktual'nye Voprosy Prikladnoi Matematiki, pp. 127–138, 1989.     

ROI-based procedures for progressive transmission of digital images: A comparison

Nowadays, problems arise when handling large-sized images (i.e. medical image such as Computed Tomographies or satellite images) of 10, 50, 100 or more Megabytes, due to the amount of time required for transmitting and displaying, this time being even worse when a narrow bandwidth transmission medium is involved (i.e. dial-up or mobile network), because the receiver must wait until the entire image has arrived. To solve this issue, progressive transmission schemes are used. These schemes allow the image sender to encode the image data in such a way that it is possible for the receiver to perform a reconstruction of the original image from the very beginning of transmission. Despite this reconstruction being, of course, partial, it is possible to improve the reconstruction on the fly, as more and more data of the original image are received. There are many progressive transmission methods available, such as it planes, TSVQ, DPCM, and, more recently, matrix polynomial interpolation, Discrete Cosine Transform (DCT, used in JPEG) and wavelets (used in JPEG 2000). However, none of them is well suited, or perform poorly, when, in addition to progressive transmission, we want to include also ROIs (Region Of Interest) handling. In the progressive transmission of ROIs, we want not only to reconstruct the image as we receive image data, but also to be able to select which part or parts of the emerging image we think are relevant and want to receive first, and which part or parts are of no interest. In this context we present an algorithm for lossy adaptive encoding based on singular value decomposition (SVD). This algorithm turns out to be well suited for progressive transmission and ROI selection of 2D and 3D images, as it is able to avoid redundancy in data transmission and does not require any sort of data recodification, even if we select arbitrary ROIs on the fly. We compare the performing of SVD with DCT and wavelets and show the results.     

New mathematical models of porous flow

A new, viable mathematical approach to the analysis of porous flow is developed. Liquids and solids are modelled as sets of particles which interact in accordance with both long range (gravity) and short range (molecular type) forces. The resulting nonlinear, n-body problem is resolved numerically on a digital computer. A variety of fundamental porous flow type problems, solved on the University of Texas at Arlington IBM 370-155, are described and analysed.