Thesis (MSc) -- University of Stellenbosch, 2000. / ENGLISH ABSTRACT: A discrete dynamical system is a pair (X, cf;) comprising a non-empty set X and a map cf; : X ---+ X. A study is made of the effect of repeated application of cf; on X, whereby points and subsets of X are classified according to their behaviour under iteration. These subsets include the JULIA and FATOU sets of the map and the sets of periodic and preperiodic points, and many interesting questions arise in the study of their properties. Such questions have been extensively studied in the case of complex dynamics, but much recent work has focussed on non-archimedean dynamical systems, when X is projective space over some field equipped with a non-archimedean metric. This work has uncovered many parallels to complex dynamics alongside more striking differences. In this thesis, various aspects of the theory of non-archimedean dynamics are presented, with particular reference to JULIA and FATOU sets and the relationship between good reduction of a map and the empty JULIA set. We also discuss questions of the finiteness of the sets of periodic points in special contexts. / AFRIKAANSE OPSOMMING: 'n Paar (X, <jJ) bestaande uit 'n nie-leë versameling X tesame met 'n afbeelding <jJ: X -+ X vorm 'n diskrete dinamiese sisteem. In die bestudering van so 'n sisteem lê die klem op die uitwerking op elemente van X van herhaalde toepassing van <jJ op die versameling. Elemente en subversamelings van X word geklasifiseer volgens dinamiese kriteria en op hierdie wyse ontstaan die JULIA en FATOU versamelings van die afbeelding en die versamelings van periodiese en preperiodiese punte. Interessante vrae oor die eienskappe van hierdie versamelings kom na vore. In die geval van komplekse dinamika is sulke vrae reeds deeglik bestudeer, maar onlangse werk is op nie-archimediese dinamiese sisteme gedoen, waar X 'n projektiewe ruimte is oor 'n liggaam wat met 'n nie-archimediese norm toegerus is. Hierdie werk het baie ooreenkomste maar ook treffende verskille met die komplekse dinamika uitgewys. In hierdie tesis word daar ondersoek oor verskeie aspekte van die teorie van nie-archimediese dinamika ingestel, in besonder met betrekking tot die JULIA en FATOU versamelings en die verband tussen goeie reduksie van 'n afbeelding en die leë JULIA versameling. Vrae oor die eindigheid van versamelings van periodiese punte in spesiale kontekste word ook aangebied.

Modeling complex gene interacting systems as Boolean networks lead to a significant simplification of computational investigation. This can be achieved by discretization of the expression level to ON or OFF states and classifying the interactions to inhibitory and activating. In this respect, Boolean functions are responsible for the evolution of the binary elements of the Boolean networks. In this thesis, we investigate the mostly used Boolean functions in modeling gene regulatory networks. Moreover, we introduce a new type of function with strong inhibitory namely the veto function. Our computational and analytic studies on the verity of the networks capable of constructing the same State Transition Graph lead to define a new concept namely the “degeneracy” of Boolean functions. We further derive analytically the sensitivity of the Boolean functions to perturbations. It turns out that the veto function forms the most robust dynamics. Furthermore, we verify the applicability of veto function to model the yeast cell cycle networks. In particular, we show that in an intracellular signal transduction network [Helikar et al, PNAS (2008)], the functions with veto are over-represented by a factor exceeding the over-representation of threshold functions and canalyzing functions in the same system. The statistics of the connections of the functional networks are studied in detail. Finally, we look at a different scale of biological phenomena using a binary model. We propose a simple correlation-based model to describe the pattern formation of Fly eye. Specifically, we model two different procedures of Fly eye formation, and provide a generic approach for Fly eye simulation.

The purpose of this thesis is to generalize Filippov's operation, and to get more useful results. It includes two main parts: The C-Filippov operation for the finite and countable cases and the Filippov operation with different measures. In the first chapter, we give brief information about the importance of Filippov's operation, our goal and the ideas behind our generalizations. In the second chapter, we give some sufficient background notes. In the third chapter, we introduce the Filippov operation, explain how to calculate the Filippov of a function and give some sufficient properties of it. In the fourth chapter, we introduce a generalization of the Filippov operation, the C-Filippov, and give some of its properties which we need for the next chapter. In the fifth chapter, in the first main part, we discuss some properties of the C-Filippov for special cases and observe the differences and common properties between the Filippov and C-Filippov operations. Finally, in the sixth chapter, we present the other generalization of the Filippov operation which is Filippov with different measures. We observe the properties of the corresponding Filippovs when we know the relationship between the measures. We finish the thesis by summarizing our work and discussing future work.

A low-order discrete dynamical system (DDS) for finite-rate chemistry of H2-air combustion is derived in 3D. Fourier series with a single wavevector are employed to represent dependent variables of subgrid-scale (SGS) behaviors for applications to large-eddy simulation (LES). A Galerkin approximation is applied to the governing equations for comprising the DDS. Regime maps are employed to aid qualitative determination of useful values for bifurcation parameters of the DDS. Both isotropic and anisotropic assumptions are employed when constructing regime maps and studying bifurcation parameters sequences. For H2-air reactions, two reduced chemical mechanisms are studied via the DDS. As input to the DDS, physical quantities from experimental turbulent flow are used. Numerical solutions consisting of time series of velocities, species mass fractions, temperature, and the sum of mass fractions are analyzed. Numerical solutions are compared with experimental data at selected spatial locations within the experimental flame to check whether this model is suitable for an entire flame field. The comparisons show the DDS can mimic turbulent combustion behaviors in a qualitative sense, and the time-averaged computed results of some species are quantitatively close to experimental data.

Elite swimmers can be distinguished from novice swimmers by freestyle stroke technique. Elite swimmers move through multiple coordination modes, increases in stroke lengths, stroke rates, and body roll allowing for a more symmetrical stroke and increased speed compared with novice swimmer during 100m freestyle. Coaches strive to improve swimmers’ performance by providing feedback about stroke technique, mostly from the pool deck where view of the full stroke cycle is obstructed by the water. Tools to assess swimming are often expensive and require extra training, which does not provide a pragmatic solution. A dryland rotational swimbench would provide a means to evaluate freestyle swimming. The aim of the present study is to evaluate the sensory motor system of elite and novice level swimmers by comparing kinematic, coordinative structures and spatial-temporal characteristics of freestyle stroke on a dryland swimbench with a rotational component. Thirty elite and novice collegiate and masters swimmers were instrumented with reflective markers bilaterally on the upper extremity and torso. A series of four ten second trials of freestyle sprint swimming were performed on the swimbench. Repeated measures were used for statistical analysis for comparison between and within groups. Bonferroni corrections were used as post-hoc analysis. Results indicated no significant difference between elite and novice swimmers’ sensory-motor system, kinematics or spatio-temporal systems on a rotational swimbench. Similarities could be accounted for by swimmers perceiving a novel task due to differences in sensory feedback, and mechanical limitations of the bench. It is noteworthy that catch-up/opposition coordination are more common than superposition which provides support for the swimbench providing a more similar representation to in water swimming.

Cette thèse concerne le problème d'identification robuste H indice infini de données harmoniques sur une bande limitée de fréquence, généralisation plus réaliste du problème d'identification robuste H indice infini étudié ces dernières années notamment par Gu, Helmiki, Jacobson, Kargonekhar, Mäkilä, Nett et Partington. L'introduction, en dehors de cette bande, d'un comportement de référence et d'un gabarit rend possible une adaptation des algorithmes classiques en deux étapes, La solution du problème posé est alors donnée par la résolution d'un problème extrémal borné après une première étape d'interpolation robuste des données sur un arc du cercle unité. Cependant, la solution ainsi calculée est typiquement discontinue. La principale contribution de ce travail à l'identification fréquentielle robuste consiste à montrer qu'il est possible de prendre en compte le caractère local des données en fréquence et garantir l'appartenance de la solution à l'algèbre du disque. Un algorithme est donné et sa mise en œuvre numérique est détaillée. Le choix du comportement en dehors de la bande considérée pose plus généralement le problème de complétion analytique borné dans H indice p. Nous le résolvons dans H indice 2 et l'utiliserons pour vérifier la validité de l'hypothèse de linéarité du système.

<p>During the past few years a vast amount of research has been done to increase the understanding of the complex ecosystem of the Baltic Sea. Modelling and simulations are important tools to increase knowledge of the system. A suitable model must be simple to use and the parameters and variables needed in the model must be easy to access.</p><p>In this paper a dynamical mass-balance model, CoastMab, which is validated for smaller coastal areas, has been used to predict concentrations and transports of phosphorus in three large coastal areas - the Gulf of Finland, the Gulf of Gdansk and the Gulf of Riga. CoastMab uses ordinary differential equations to regulate inflow, outflow and internal flows. To reflect seasonal variations in temperature and different types of flows the model has a temporal resolution of a month. The main purposes of this paper have been to evaluate CoastMab, predict the concentrations and the transports of phosphorus in each coastal area and to analyse how much and why the results differ from empirical data.</p><p>The performed simulations show the importance of a correct calculation of the wave base. This is due to its influence on the division between surface and deep water as well as the division of areas of accumulation and areas of erosion and transport. The retention times of surface and deep water calculated by the model also is of great importance because of their direct influence on the flow of water between the study area and the sea outside the area. This together with the concentration of phosphorus determines the flow of phosphorus in and out of the study area.</p><p>The dynamical mass-balance model is considered to work well in the three studied areas even though the areas are outside the model domain in several respects. Even if the predictions of the phosphorus concentration in the gulf of Riga was less accurate than in the other study areas, the prediction was inside the interval of standard deviation that has been calculated from empirical data. The major flow of phosphorus in the model was in the cross section between the Baltic proper and the modelled coastal area.</p><p>Since the model is easy to use and requires only a small number of obligatory input variables it would be appealing to continue development of the model to handle coastal areas of the same size and larger than the study areas in this paper.</p> / <p>Under senare år har mycket forskning utförts för att öka förståelsen av det komplexa ekosystem Östersjön utgör. Viktiga redskap för att öka förståelsen för systemet är modellering och simulering. För att en modell ska vara lämplig att använda är det viktigt att den inte är för komplicerad och att de parametrar och variabler som används i modellen är lättillgängliga.</p><p>I detta arbete har en dynamisk massbalansmodell, CoastMab, som är validerad för mindre kustområden använts för att modellera fosforkoncentrationer och fosforflöden i tre större kustområden - Finska viken, Gdanskbukten och Rigabukten. CoastMab reglerar genom ordinära differentialekvationer inflöden, utflöden och interna flöden. För att kunna hantera säsongsvariationer i temperatur och olika typer av flöden har modellen en temporär upplösning på en månad. Syftet med arbetet har främst varit att se hur väl CoastMab predikterar fosforkoncentrationen och fosforflöden inom respektive område, samt att analysera hur mycket och varför prediktionerna skiljer sig från empiriska data.</p><p>De simuleringar som utfördes visade att djupet på den teoretiska vågbasen, som modellen beräknar, är av stor betydelse eftersom den i sin tur bestämmer fördelningen yt- och djupvatten samt fördelningen av ackumulationsbottnar och erosions- och transportbottnar. De utbytestider för yt- och djupvatten som modellen beräknar är också av största vikt eftersom de har en direkt inverkan på vattenflödet mellan studerat område och havet utanför, vilket i sin tur tillsammans med fosforkoncentrationen bestämmer fosforflöden in och ut ur det studerade området.</p><p>Den dynamiska massbalansmodellen kan anses fungera väl för samtliga tre studerade områden, trots att de ligger utanför modellens domän i flera avseenden. Även om prediktionen av fosforkoncentrationen i Rigabukten blev sämre än för de två övriga områdena, låg prediktionen inom det intervall för standardavvikelsen som beräknats utifrån empiriska data. De största fosforflödena i modellen förekom för samtliga områden i gränssnittet mellan egentliga Östersjön och det modellerade kustområdet.</p><p>Enkelheten i att använda modellen och det låga antalet obligatoriska drivvariabler gör att den är väl värd att utveckla för att hantera områden av samma eller större storlek än de nu studerade områdena.</p>

We conduct a broad investigation about stellar mass and population diagnostics in order to formulate novel constraints related to the formation and evolution of galaxies from a nearby cluster environment. Our work is powered by the use of stellar population models which transform galaxy colours and/or absorption line strengths into estimates of its stellar properties. As input to such models, we assemble an extensive compilation of age and chemical abundance information for Galactic globular clusters. This compilation allows a confident expansion of these models into new regions of parameter space that promise to refine our knowledge of galactic chemical evolution. We then draw upon a state-of-the-art spectroscopic and photometric survey of the Virgo galaxy cluster in order to constrain spatial variations of the stellar ages, metallicities, and masses within its member galaxies, and their dynamical masses. We interpret these data in the context of the histories of star formation, chemical enrichment, and stellar mass assembly to formulate a broad picture of the build-up of this cluster’s content over time. In it, the giant early-type galaxies formed through highly dissipational processes at early times that built up most of their stellar mass and drew significant amounts of dark matter within their optical radii. Conversely, dwarf early-types experienced environmental processes that quenched their star formation during either the early stages of cluster assembly or upon infall at later times. Somewhat perplexing is our finding that the internal dynamics of these galaxies are largely explained by their stellar masses. Lastly, Virgo spirals also suffer from their dense environment, through ram pressure stripping and/or tidal harrassment. In addition to quenching, these effects leave an imprint on their internal dynamical evolution too. Late-type spirals exhibit evidence of having ejected significant amounts of baryons from their inner regions, likely via energetic feedback events. Rich as our picture of the history of the Virgo cluster has become, real progress in our understanding of this system will truly benefit from future high-resolution cosmological and hydrodynamic simulations of this environment. Such simulations are still in their infancy, but the data assembled here should soon provide their most direct validation. / Thesis (Ph.D, Physics, Engineering Physics and Astronomy) -- Queen's University, 2013-09-30 23:32:48.575

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This thesis investigates the possibility of approximating stationary solutions of stochastic differential equations using numerical methods. We consider a particular class of stochastic differential equations, which are known to generate random dynamical systems. The existence of stochastic stationary solution is proved using global attractor approach. Euler's numerical method, applied to the stochastic differential equation, is proved to generate a discrete random dynamical system. The existence of stationary solution is proved again using global attractor approach. At last we prove that the approximate stationary point converges in mean-square sense to the exact one as the time step of the numerical scheme diminishes.