This thesis examines the effects on demand, supply, imports, and prices of partial easing of sanitary and phytosanitary (SPS) barriers to trade in the U.S. market in the case of Mexican avocados. The SPS Agreement plays a role in the avocado market studied here through its implications for negotiations between countries that have not utilized the formal channels of the WTO for resolving disputes. A quarantine in place from 1914 until very recently banned entry of Mexican avocados into the U.S. market on grounds of risk of pest infestation. Since the early 1970s this quarantine has been a cause of dispute between the Mexican and U.S. governments, resulting in elaborate evaluations of possible pest risks and risk mitigation procedures that might be carried out. However, after the initiation of negotiations for the North American Free Trade Agreement (NAFTA) in 1991, the import ban was partially eased in 1995 allowing Mexico access to the Northeastern part of the U.S. during four winter months. After three years of successfully exporting without any pest outbreaks, Mexico requested increased access to an additional part of the U.S. market, which it was granted in 2001. This study develops a partial equilibrium trade model to investigate the effects that this increased access will have on the avocados markets. Hypothesized further increases in access are described, and their potential effects are evaluated as well. / Master of Science

We present an investigation of two non-equilibrium systems: spatial many-species predator-prey games and systems of interacting magnetic skyrmions. We numerically study two predator-prey systems characterized by nested pattern formations. We first consider a six species game in which spiral patterns spontaneously form within coarsening domains. Through a systematic investigation of relevant correlation functions, the interface width, and other quantities, we show that the non-trivial in-domain dynamics affect the coarsening process and the interfacial properties. The exponents which govern domain growth, aging, and interface fluctuations differ from those expected from curvature driven coarsening. The response to perturbations of the reaction rates is also studied. Furthermore, we introduce a nine species model characterized by nested spiral pattern formations. Quantitative evidence of the existence of two length and time scales associated to the spiral levels is presented in the form of correlation lengths and a temporal Fourier analysis of the species densities. A generalized interaction scheme is proposed for dynamically generated hierarchies. Magnetic skyrmions are particle-like spin configurations found in certain chiral magnets. We study the effect of the Magnus force on the relaxation dynamics through Langevin molecular dynamics simulations. The Magnus force enhances the disorder of the system at high noise strengths while we observe a dynamic regime with slow decaying correlations at low noise strengths. The different regimes are characterized by changes in the aging exponent. In general, the Magnus force accelerates the approach to the steady state. In the presence of quenched disorder, we find that the relaxation dynamics are more robust in systems with a strong Magnus force. We also examine periodically driven skyrmion systems and show that a transition from reversible to irreversible flow exists in the presence of attractive defects. The Magnus force enhances the irreversible regime in this case. The work on predator-prey systems was supported by the U.S. National Science Foundation through Grant No. DMR-1606814 whereas the work on skyrmions was supported by the US Department of Energy, Office of Basic Energy Sciences (DOE-BES), under Grant No. DE-FG02-09ER46613. / Doctor of Philosophy / We present an investigation of two non-equilibrium systems: spatial many-species predator- prey games and systems of interacting magnetic skyrmions. We numerically study two predator-prey systems characterized by nested pattern formations. We first consider a six species game in which spiral patterns spontaneously form within coarsening domains. Through a systematic investigation of relevant correlation functions, the interface width, and other quantities, we show that the non-trivial in-domain dynamics affect the coarsening process and, to a greater extent, properties at the interface between competing groups of species. The exponents which govern domain growth, aging, and interface fluctuations are shown to differ from those expected in typical games of competition. We also study the change of the system due to a perturbation of the reaction rates, which could represent an abrupt change in the environment. Furthermore, we introduce a nine species model characterized by the emergence of nested spiral pattern formations. Quantitative evidence of the existence of two distinct spiral levels is presented. We also propose a generalized interaction scheme for dynamically generated spiral hierarchies. Magnetic skyrmions are particle-like spin configurations found in certain chiral magnets. We study the effect of the Magnus force on the dynamic properties of skyrmion systems through particle-based simulations. The Magnus force enhances the disorder of the system at high noise strengths while accelerating the formation of the triangular lattice at low noise strengths. We find that, in general, the Magnus force accelerates the approach to the steady state. In the presence of randomly placed attractive pinning sites, we find that a strong Magnus force can prevent caging effects and allow skyrmions to more easily move around pinning sites. We also examine periodically driven skyrmion systems and show that a transition from reversible to irreversible flow exists in the presence of attractive defects. The Magnus force is shown to enhance the irreversible regime in this case. The work on predator-prey systems was supported by the U.S. National Science Foundation through Grant No. DMR-1606814 whereas the work on skyrmions was supported by the US Department of Energy, Office of Basic Energy Sciences (DOE-BES), under Grant No. DE-FG02-09ER46613.

Melting and vaporization of solids occur very often in nature and in man-made processes. Many analytical and numerical solutions exist for solving the temperature field in the liquid and solid regions, but inaccuracies persist in tracking the phase change interfaces and the numerical solution of the temperature field is usually cumbersome. The Boundary Element Method is proposed as an accurate, efficient way to solve for the temperature field and the interface positions in a phase change problem involving combined melting and vaporization. When comparing to specific one-dimensional test cases, accurate results arc obtained when using a sufficiently small time step. A comparison is made to existing data from a laser drilling experiment. The anticipated physical effects which occur on semi-infinite and finite domains arc confirmed. Consequently, this method can be used to model natural and industrial phenomena involving phase change. / Master of Science

M.S.

We study game theoretic ecological models with cyclic competition in the case where the strategies can be mixed or pure. For both projects, reported in [49] and [50], we employ Monte Carlo simulations to study finite systems. In chapter 3 the results of a previously published paper [49] are presented and expanded upon, where we study the extinction time of four cyclically competing species on different lattice structures using Lotka-Volterra dynamics. We find that the extinction time of a well mixed system goes linearly with respect to the system size and that the probability distribution approximately takes the shape of a shifted exponential. However, this is not true for when spatial structure is added to the model. In that case we find that instead the probability distribution takes on a non-trivial shape with two characteristic slopes and that the mean goes as a power law with an exponent greater than one. This is attributed to neutral species pairs, species who do not interact, forming domains and coarsening. In chapter 4 the results of [50] are reported and expanded, where we allow agents to choose cyclically competing strategies out of a distribution. We first study the case of three strategies and find through both simulation and mean field equations that the probability distributions of the agents synchronize and oscillate with time in the limit where the agents probability distributions can be approximated as continuous. However, when we simulate the system on a one-dimensional lattice and the probability distributions are small and discretized, it is found that there is a drastic transition in stability, where the average extinction time of a strategy goes from being a power law with respect to system size to an exponential. This transition can also be observed in space time images with the emergence of tile patterns. We also look into the case of four cyclically competing strategies and find results similar to that of [49], such as the coarsening of neutral domains. However, the transition from power law to exponential for the average extinction time seen for three strategies is not observed, but we do find a transition from one power law to another with a different slope. This work was supported by the United States National Science Foundation through grants DMR-0904999 and DMR-1205309. / Ph. D.

The agricultural negotiations in the World Trade Organization's (WTO) Doha Development Agenda (DDA) are calling for a specific Special Safeguard Mechanism (SSM) for developing countries that will protect agricultural producers from import surges or price declines, and could potentially add stability to domestic markets. While most of the parameters of this SSM have been decided upon, the DDA negotiations faltered on the issue of whether or not developing nations should be allowed to exceed their pre-Doha bound tariff rates when invoking the SSM. For developing countries, tariffs on agricultural products are an important policy tool to support domestic prices and protect their smallholder producers from global market shifts. Tariffs, however, distort world prices and create global welfare losses. The purpose of this thesis is to assess the impacts of the SSM on global prices and welfare using a non-spatial, synthetic, stochastic, global, partial equilibrium model of the world soybean market. The SSM is assessed in concert with the currently proposed DDA tariff cutting formulas since the additional duties allowed under the SSM are proportional to prevailing bound tariff levels. This study asserts that the SSM actually decreases global price and welfare stability, decreasing world prices of the commodities on which an SSM is placed, though positively affects tariff revenues for those particular commodities. While the SSM may offer a short-term solution for developing countries, its long-term outlook as a price stabilization tool is a not credible argument. / Master of Science

Increasing dynamics in power systems on account of renewable integration, electric vehicle penetration and rising demands have resulted in the exploration of energy storage for potential solutions. Recent technology- and industry-driven developments have led to a drastic decrease in costs of these storages, further advocating their usage. This thesis compiles the author's research on optimal integration of energy storage. Unpredictability is modelled using random variables favouring the need of stochastic optimization algorithms such as Lyapunov optimization and stochastic approximation. Moreover, consumer interactions in a competitive environment implore the need of topics from game theory. The concept of Nash equilibrium is introduced and methods to identify such equilibrium points are laid down. Utilizing these notions, two research contributions are made. Firstly, a strategy for controlling heterogeneous energy storage units operating at different timescales is put forth. They strategy is consequently employed optimally for arbitrage in an electricity market consisting of day-ahead and real-time pricing. Secondly, energy storages owned by consumers connected to different nodes of a power distribution grid are coordinated in a competitive market. A generalized Nash equilibrium problem is formulated for their participation in arbitrage and energy balancing, which is then solved using a novel emph{weighted} Lyapunov approach. In both cases, we design real-time algorithms with provable suboptimality guarantees in terms of the original centralized and equilibrium problems. The algorithms are tested on realistic scenarios comprising of actual data from electricity markets corroborating the analytical findings. / Master of Science / Modern power system, which is responsible for generation and transport of electricity, is witnessing a lot of changes such as the increased adoption of wind and solar energy, promotion of electric vehicles, and ever increasing consumer demands. Amidst such developments, energy storage devices like batteries are being propagated as a necessary addition to the power system for its safe operation. This has been further supported by the decrease in prices of these devices over time. An effective assimilation of energy storage however, requires extensive mathematical studies on account of unpredictable renewable generation and consumer demands.This thesis focuses upon the preceding concern. To this note, two novel research contributions are made. In the first, an individual consumer is modeled who wishes to reduce his/her energy costs by simultaneously employing energy storages belonging to different technologies. In the latter, a more challenging multi-consumer interaction is reviewed where multiple end consumers wish to reduce costs while competing against each other over limited resources. In either of the cases, efficient algorithms are designed that are shown to produce desirable results over real-life data and have mathematically provable performance guarantees.

Yes / Three equations of state with a group contribution model for binary interaction parameters were employed to calculate the vapor-liquid equilibria of synthetic and real natural gas mixtures with heavy fractions. In order to estimate the binary interaction parameters, critical temperatures, critical pressures and acentric factors of binary constituents of the mixture are required. The binary interaction parameter model also accounts for temperature. To perform phase equilibrium calculations, the heavy fractions were first discretized into 12 Single Carbon Numbers (SCN) using generalized molecular weights. Then, using the generalized molecular weights and specific gravities, the SCN were characterized. Afterwards, phase equilibrium calculations were performed employing a set of (nc + 1) equations where nc stands for the number of known components plus 12 SCN. The equations were solved iteratively using Newton's method. Predictions indicate that the use of binary interaction parameters for highly sour natural gas mixtures is quite important and must not be avoided. For sweet natural gas mixtures, the use of binary interaction parameters is less remarkable, however.

In 1867, James Clerk Maxwell introduced a thought experiment involving a micro-scopic being (observer) capable of making precise measurements of microscopic quantitiesthrough observation of the micro-dynamics in a thermodynamic system. This observerlater became known as Maxwell’s demon due to its devious impact on thermodynamics,particularly the perceived violation of the second law. Subsequently, Leo Szilard pro-posed a machine, the so-called Szilard Machine, which, by utilising a Maxwell’s demon,successfully extracts work from thermal fluctuations in a closed system, seemingly vio-lating the second law. This thesis re-evaluates the second law of thermodynamics in the context of the Szi-lard Machine and Maxwell demons. The study explores the intersection of informationtheory and thermal physics, both theoretically and practically, with the aid of MonteCarlo simulations. The results indicate that machines with information feedback control,such as those utilising a Maxwell demon, challenge classical statements of the second lawof thermodynamics. This is because classical formulations, such as Clausius’ and Kelvin’sstatements, do not account for the entropic content of information. Simulations of thesefeedback processes, in conjunction with the detailed fluctuation theorem, provide a basisfor understanding feedback processes in so-called information machines. Ultimately, thesecond law of thermodynamics is upheld by an alternative statement endorsed by thedetailed fluctuation theorem.

Thesis (MEng)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Thermodynamic models have been investigated extensively since Johannes van der Waals first devised a mathematical relation capable of predicting both vapour and liquid phases for a mixture at equilibrium. With the advent of modern computing power, these equations of state have gone from their humble empirical beginnings to the comprehensive and fundamentally derived models we have today. One such physically sound model is the Statistical Associating Fluid Theory (SAFT) family of equations, derived from the molecular perturbation theories of the 1980’s. The relative youth of this thermodynamic framework has meant that much work has gone into modification and optimisation of the model recently. The variants of particular interest to this work are the simplified perturbed chain SAFT equations with the Jog & Chapman (sPC-SAFTJC) and Gross & Vrabec (sPC-SAFTGV) polar terms. Each of the polar terms supports one adjustable polar parameter that relates to the quantity of polar segments in the reference fluid but not necessarily its position in the carbon chain. The strength of polar interactions is known to decrease as the functional group moves away from the terminal methyl group and the effects of steric hindrance increase. Thus, in question here is whether the models can account for the change in polar interactions associated with the changing position of the polar group, by only adjusting the values of the existing pure component parameters; that is, in lieu of a position specific parameter. The carbonyl group in ketone molecules is one such polar group, and it is this homologous series that is the focus of this study. The decrease in polar interactions as the carbonyl group in a ketone molecule shifts centrally is apparent from the lower boiling points of the isomers where the polar group is central as compared to those where the functional group is nearer the terminal methyl group. The effect of this functional group shift on binary phase behaviour has not previously been assessed for any system however, as the lack of experimental data attests. Thus, experiments had to be conducted to generate phase equilibrium data for systems comprising each structural isomer of a mid-length ketone with a common second component with no functionality. This limitation was imposed to isolate the cause of experimentally observed phenomena to the shifting polar group alone. The generated data could then be appropriately modelled using the polar sPC-SAFT variants and the capabilities of each model, as outlined above, assessed. To this end, isobaric binary vapour-liquid equilibrium data were measured for 2-, 3- & 4-heptanone with three separate normal alkanes of similar length (n-octane, n-nonane & n-decane) at 40kPa. The apparatus used was a dynamic Gillespie VLE still with temperature and pressure accuracies of 0.03°C and 1.6mbar respectively. Equipment verification was achieved through the reproduction of experimental data for the ethanol/1-butanol system at 1.013bar. The vapour and liquid samples for all nine systems were analysed by gas chromatography with a maximum compositional error of ±0.022 mole fraction. All reported data were found to be thermodynamically consistent using both the L/W and McDermott-Ellis consistency tests. When paired with a common n-alkane, all three structural heptanone isomers displayed similar qualitative trends in phase behaviour. Minimum boiling azeotropes were measured in all nine systems; in the high alkane region for n-octane systems (~98 mole%), the equal concentration region for n-nonane systems (34 mole% to 53 mole%) and in the very dilute n-alkane region for n-decane systems (~3 mole%). The n-nonane systems in particular highlighted the effect of shifting functional group, with completely separate phase envelopes away from the pure alkane composition space evident in a particularly small temperature range. Modelling was performed using in-house developed software, with pure component parameters generated for each system using five different regression procedures. The first was traditional fitting of the segment diameter (σ), segment number (m), segment energy (є/k) and the respective polar parameter (xp, np) to DIPPR correlations of pure component saturated vapour pressure, liquid density and the heat of vaporisation. The latter four procedures included the fixing of the polar parameter according to functional group correlations and the three instances of including the binary VLE data set for each of the three alkanes considered in this work. When applied to the nine binary ketone-alkane systems measured in this work, excellent predictions of the experimental data were in evidence in most cases and only small binary interaction parameters were necessary to correlate the data where pure predictions were poor. The performance of the parameter sets based on the fixing of the polar parameter and the inclusion of VLE data were consistent and of a high quality for both models, with near identical parameters generated in all four cases for each of the nine systems. The parameter sets generated in this fashion were shown to be applicable not only to the systems measured in this work, but also successfully predicted the independently measured experimental data of the n-hexane/4-heptanone system. It was thus concluded that either of these regression alternatives are viable for the generation of accurate component parameters, and the choice of VLE data set included is trivial. The pure predictions of the sPC-SAFTGV model were generally better than its sPC-SAFTJC counterpart, particularly in the case of the traditionally regressed parameter sets. sPC-SAFTGV displayed constant qualitative agreement with the experimental data for each of the heptanone isomers with a given n-alkane. The quality of the predictions of sPC-SAFTJC, however, worsened significantly as the polar interactions diminished from 2- to 4-heptanone, with no predictions even possible for the least polar isomer. This was attributed to the different perturbation theories used in the development of these terms, but a more detailed study would be necessary to confirm this. This work thus shows an apparent inability of the sPC-SAFTJC equation of state to account for the decreasing polar interactions associated with the carbonyl group in a ketone molecule shifting centrally, while sPC-SAFTGV produces qualitatively good fits for all three isomers. These flaws can be overcome through the incorporation of VLE data in the regression procedure if such data is available, or otherwise through the use of group specific correlations for fixing the polar parameter value. / AFRIKAANSE OPSOMMING: Sedert Johannes van der Waals die eerste wiskundige verhouding ontwikkel het wat beide die damp- en vloeibare fases van 'n mengsel by ewewig kon voorspel, is die veld van termodinamiese modellering al deeglik ondersoek. Na die koms van die moderne rekenaars het hierdie vergelykings van hul nederige empiriese wortels gegroei tot die omvattende, fundamentele modelle wat ons vandag het. Een so 'n fundamenteel gebaseerde familie van vergelykings is die ‘Statistical Associating Fluid Theory’ (SAFT) modelle, wat afgelei is vanaf molekulêre versteuringsteorieë, ontwikkel in die 1980s. Hierdie relatiewe jong modelle het in die afgelope ruk aansienlike aanpassing en optimering ondervind. Modelvariante van besondere belang tot hierdie werk, is die vereenvoudigde versteurde ketting of ‘simplified perturbed chain’ SAFT vergelykings, met Jog & Chapman (sPC- SAFTJC) en Gross & Vrabec (sPC- SAFTGV) polêre terme. In die sPC-SAFT toestandsvergelyking word elkeen van die polêre terme ondersteun deur een polêre veranderlike. Hierdie veranderlike is afhanklik van die aantal polêre segmente in die verwysingsvloeistof, maar nie noodwendig hul posisie in die koolstofketting nie. Daarteen is dit bekend dat die polêre interaksies tussen molekules swakker word soos die polêre groep wegbeweeg van die terminale metielgroep, en steriese hindernis ʼn groter rol begin speel. Dus is die vraag of die model die verandering in die polêre interaksie, as gevolg van veranderende posisie van die polêre groep, kan voorspel deur in plek van ʼn posisie afhanklike parameter, slegs ʼn aanpassing van die polêre waardes van die suiwer komponente te maak. Die karbonielgroep in ketoon molekules is een so 'n polêre groep, en ʼn homoloë reeks ketone word in hierdie studie ondersoek. Die afname in die polêre interaksie soos wat die karbonielgroep in 'n ketoon molekule weg skuif vanaf die terminale metiel groep is sigbaar deur die afname in kookpunt van die verskillende isomere. Hierdie effek van die funksionele groepsposisie op binêre fasegedrag is nog nie voorheen vir enige stelsels geëvalueer nie en geen eksperimentele data is vrylik beskikbaar nie. Om hierdie tekortkoming in die literatuur aan te spreek, is eksperimentele fase ewewig data gemeet. ʼn Reeks stelsels is ondersoek wat elk bestaan uit ʼn struktuurisomeer van ʼn mid-lengte ketoon en ʼn tweede komponent met geen funksionele bydrae. Eksperimente is so opgestel om die effek van die skuiwende polêre groep op die fasegedrag te isoleer en kwalitatief te ondersoek. Die gegenereerde data is dan gemodelleer met behulp van die polêre sPC- SAFT variante, soos hierbo gespesifiseer, en die vermoëns van elke model is beoordeel. Isobariese binêre fase ewewig data is by 40kPa gemeet vir damp-vloeistof stelsels bestaande uit 2, 3 & 4 heptanoon, gemeng met drie verskillende normaal alkane van vergelykbare kettinglengte (n-oktaan, n-nonaan & n-dekaan). Die apparaat wat gebruik was is 'n dinamiese Gillespie VLE eenheid met temperatuur- en drukakkuraatheid van 0,03°C en 1.6mbar, onderskeidelik. Die akkuraatheid van die toerusting is bevestig deur eksperimentele data vir ʼn etanol/1-butanol stelsel by 1.013bar te reproduseer. Die damp en vloeibare monsters vir al nege stelsels is ontleed deur gaschromatografie met 'n maksimum komposisionele fout van ± 0,022 (molfraksie). Alle data is as termodinamies konsekwent gevind deur van beide die L/W en McDermott-Ellis konsekwentheidstoetse gebruik te maak. Mengsels van die drie strukturele isomere van heptanoon met ʼn gemene n-alkaan het tydens eksperimente soortgelyke kwalitatiewe tendense in fasegedrag getoon. Gedurende eksperimente is die lae kookpunt asiotrope gemeet vir al nege stelsels. Die asiotrope verskyn in die hoë alkaan konsentrasies (~98 mol%) vir n-oktaan stelsels, medium konsentrasies (34 mol% tot 53 mol%) vir n-nonaan stelsels en baie verdunde konsentrasies (~ 3 mol%) vir n-dekaan stelsels. Die n-nonaan stelsels beeld veral die effek van die verskuiwing van die funksionele groep uit, met diskrete fasegrense wat duidelik apart staan van die suiwer alkaan ruimte, binne ʼn klein temperatuurverskil. Modellering van die stelsels is uitgevoer met behulp van sagteware wat in-huis ontwikkel is. Suiwer komponent data is gegenereer vir elke stelsel deur van vyf verskillende regressie prosedures gebruik te maak. Die eerste is die tradisionele passing van die segment deursnee (σ), segment nommer (m), segment energie (є/k) en die onderskeie polêre parameters (xp, np) op DIPPR korrelasies van die suiwer komponent versadigde dampdruk, vloeistof digtheid en die hitte van verdamping. Die oorblywende vier prosedures sluit in die bepaling van die polêre parameter deur funksionelegroep korrelasies, en drie gevalle waar die binêre VLE data vir elk van die drie alkane ingesluit is. Deur hierdie prosedures op die modellering van die nege binêre ketoon/alkaan stelsels toe te pas, is uitstekende passings van die eksperimentele data verkry met slegs baie klein binêre interaksie parameters nodig waar voorspellings minder akkuraat was. Die prestasie van die parameter stelle, gebaseer op die bepaling van die polêre parameter en die insluiting van VLE data, is konsekwent en van 'n hoë gehalte vir albei modelle, met 'n byna identiese parameters gegenereer in al vier gevalle vir elk van die nege stelsels. Die parameter stelle wat op hierdie metode gegenereer is, is nie net toepaslik gevind op eksperimentele data gemeet in hierdie werk nie, maar ook op onafhanklike data vir die n-heksaan/4-heptanoon stelsel. Daar is tot die gevolgtrekking gekom dat beide van die regressie alternatiewe lewensvatbaar is vir die generasie van akkurate suiwer komponent parameters, en dat die insluiting van die VLE data triviaal is. Die suiwer sPC - SAFTGV voorspelling was oor die algemeen beter as die suiwer sPC- SAFTJC model met die voorspelling van data, veral in die geval van passings gedoen met parameters verkry vanaf tradisionele regressie metodes. sPC- SAFTGV het ʼn voortdurende, kwalitatiewe ooreenkoms met eksperimentele data getoon vir elk van die nege stelsels. Daarteen het voorspellings deur sPC- SAFTJC beduidend verswak soos die polêre interaksies afgeneem het vanaf 2- na 4- heptanoon, met geen akkurate voorspelling moontlik vir die minste polêre isomeer nie. Die verskynsel kan toegeskryf word aan die verskil in versteuringsteorieë wat gebruik word in die ontwikkeling van die onderskeie toestandsvergelykings, maar ʼn meer in-diepte ondersoek is nodig om hierdie teorie te bevestig. Hierdie werk toon dus 'n skynbare onvermoë van die sPC - SAFTJC toestandsvergelyking om die verandering in polêre interaksie, as gevolg van die veranderende posisie van die polêre groep, vir die karbonielgroep in ʼn ketoon te voorspel, terwyl die sPC-SAFTGV toestandsvergelyking goeie kwalitatiewe passings vir al drie isomere bied. Hierdie tekortkominge kan oorkom word deur VLE data, indien beskikbaar, in die regressie prosedure in te sluit, of deur die gebruik van groep spesifieke korrelasies vir die aanpassing van die polêre parameter.