<p>This thesis concerns dimensions and projections of sets that could be described as fractals. The background is applied problems regarding analysis of human tissue. One way to characterize such complicated structures is to estimate the dimension. The existence of different types of dimensions makes it important to know about their properties and relations to each other. Furthermore, since medical images often are constructed by x-ray, it is natural to study projections.</p><p>This thesis consists of an introduction and a summary, followed by three papers.</p><p>Paper I, Anders Nilsson, Dimensions and Projections: An Overview and Relevant Examples, 2006. Manuscript.</p><p>Paper II, Anders Nilsson and Peter Wingren, Homogeneity and Non-coincidence of Hausdorff- and Box Dimensions for Subsets of ℝ<i>n</i>, 2006. Submitted.</p><p>Paper III, Anders Nilsson and Fredrik Georgsson, Projective Properties of Fractal Sets, 2006. To be published in Chaos, Solitons and Fractals.</p><p>The first paper is an overview of dimensions and projections, together with illustrative examples constructed by the author. Some of the most frequently used types of dimensions are defined, i.e. Hausdorff dimension, lower and upper box dimension, and packing dimension. Some of their properties are shown, and how they are related to each other. Furthermore, theoretical results concerning projections are presented, as well as a computer experiment involving projections and estimations of box dimension.</p><p>The second paper concerns sets for which different types of dimensions give different values. Given three arbitrary and different numbers in (0,<i>n</i>), a compact set in ℝ<i>n</i> is constructed with these numbers as its Hausdorff dimension, lower box dimension and upper box dimension. Most important in this construction, is that the resulted set is homogeneous in the sense that these dimension properties also hold for every non-empty and relatively open subset.</p><p>The third paper is about sets in space and their projections onto planes. Connections between the dimensions of the orthogonal projections and the dimension of the original set are discussed, as well as the connection between orthogonal projection and the type of projection corresponding to realistic x-ray. It is shown that the estimated box dimension of the orthogonal projected set and the realistic projected set can, for all practical purposes, be considered equal.</p>

This thesis consists of an introduction and a summary, followed by two papers, both of them on the topic of function spaces on fractals. Paper I: Andreas Brodin, Pointwise Convergence of Haar type Wavelets on Self-Similar Sets, Manuscript. Paper II: Andreas Brodin, Regularization of Wavelet Expansion characterizes Besov Spaces on Fractals, Manuscript. Properties of wavelets, originally constructed by A. Jonsson, are studied in both papers. The wavelets are piecewise polynomial functions on self-similar fractal sets. In Paper I, pointwise convergence of partial sums of the wavelet expansion is investigated. On a specific fractal set, the Sierpinski gasket, pointwise convergence of the partial sums is shown by calculating the kernel explicitly, when the wavelets are piecewise constant functions. For more general self-similar fractals, pointwise convergence of the partial sums and their derivatives, in case the expanded function has higher regularity, is shown using a different technique based on Markov's inequality. A. Jonsson has shown that on a class of totally disconnected self-similar sets it is possible to characterize Besov spaces by means of the magnitude of the coefficients in the wavelet expansion of a function. M. Bodin has extended his results to a class of graph directed self-similar sets introduced by Mauldin and Williams. Unfortunately, these results only holds for fractals such that the sets in the first generation of the fractal are disjoint. In Paper II we are able to characterize Besov spaces on a class of fractals not necessarily sharing this condition by making the wavelet expansion smooth. We create continuous regularizations of the partial sums of the wavelet expansion and show that properties of these regularizations can be used to characterize Besov spaces.

Accurate estimates for the molecular transport coefficients are critical to predicting the movement of gases in geological media. Here I present a novel methodology for using prompt gamma activation analysis to measure the effective diffusivity of noble gases in a porous medium. I also present a model to estimate the connectivity parameter of a soil from measurements of its saturated conductivity, macro porosity, and pore volume and pore surface fractal dimensions. Experiments with argon or xenon diffusing through a nitrogen saturated geological media were conducted. The noble gas concentration variations at its source were measured using prompt gamma activation analysis and later compared to a numerical diffusion model to estimate the effective diffusion coefficient. Numerical simulations using the estimated diffusivity and the experimental argon data produced results with a correlation parameter R² = 0.98. However, neglecting transport mechanisms other than diffusion largely under-predicted the xenon depletion rates observed during the first hours of experiment. To explain these results, a second model was developed which included the effect of pressure gradients and bulk convection that might arise from the faster molecular migration of the light species in a non-equimolar system and gravitational currents. Finally, the fractal model developed for this dissertation was used to estimate the connectivity parameters and walking fractal dimension of a group of geological samples that were previously characterized. This model successfully predicted positive connectivity factors and walking fractal dimensions between two and three for every sample analyzed. / text

Studies of corn root morphology, canopy description, light and nutrient relationships, have focused on conventional corn hybrids. We are now extending these studies to other corn types with contrasting canopy and root architectures. Field and greenhouse experiments were carried out in order to characterize root morphology, N status in the plant and its relationship with yield and yield components, canopy architecture and light interception of these genotypes. The indoor experiments investigated root morphology and how N affects it. Root fractal geometry and its relationship with standard measured root variables were investigated. The field research, at two sites and over two growing seasons, examined (i) maize canopy architecture with regard to light interception and (ii) nitrogen effects on grain yield of different maize genotypes. Four genotypic types were included: (i) Leafy reduced-stature, Lfy1rd1 (LRS), (ii) non Leafy-reduced stature, lfyrd1 (NLRS), (iii) Leafy normal stature, Lfy1Rd1 (LNS), and (iv) conventional commercial hybrids, lfy1Rd1. Pioneer 3905 served as the check hybrid for late maturity, and Pioneer 3979, the check for early maturity. The work allowed development of following methods: (i) root sampling for measurement of large root systems, (ii) staining to enhance root contrast for measurement with a scanner-based software system, (iii) sample size determination for SPAD meter readings, and (iv) the design and construction of a mobile and multi-strata device for measurement of light interception. Data were collected for mathematical characterization of canopies (i.e. leaf angle, co-ordinates of the maximum height of the leaf, co-ordinates of the leaf tip), plant N status (SPAD meter readings), light interception, yield and grain yield components. Conventional hybrids generally showed greater root length and surface area than their leafy genotypic counterparts at early developmental stages (i.e. up to 15 days from emergence). However, Leafy geno

Aquest treball tracta, en primer lloc, sobre les possibilitats d'aplicació de la teoria del caos a l'estudi de les organitzacions i el canvi organitzatiu, a la seva comprensió i gestió des d'una nova perspectiva. La part teòrica és una introducció als principals conceptes implicats en el què es coneix com a teoria de la complexitat, centrant-se principalment en el concepte de caos determinista i el d'autoorganització sense oblidar, però, la seva relació amb altres conceptes importants com els de règim dinàmic, no linealitat i fractals i la seva aplicacio a les ciències socials i a la teoria de l'organització en particular.La part empírica és el resultat d'un estudi fet en organitzacions reals utilitzant metodologia quantitativa i qualitativa. Quantitativament prenent la grandària com a variable important en l'evolució de les organitzacions i, aplicant un model de creixement no lineal (equació de Verhulst) clàssic en el tractament de dades des de la teoria del caos. Qualitativament, utilitzant l'entrevista per a la identificació d'indicadors en organitzacions immerses en règims dinàmics diferents pel que fa a la variable grandària. Les estratègies de recerca utilitzades són doncs, l'estudi i comparació de casos i la "Grounded Theory Methodology". En segon lloc, aquest treball pretén ajudar a construir què signifiquen conceptes com caos, autoorganització, fractals, atractors... quan els apliquem a la vida i evolució de les organitzacions, mitjançant la interrelació entre teoria i resultats obtinguts. Hi ha doncs, un intent de desenvolupament teòric, en un marc paradigmàtic emergent en la teoria de l'organització.

Complex systems are abundant in our natural environment. In linear systems, the equations of their dynamics can be very difficult to solve, but if they cannot be described with a single characteristic scale, at least they can be described by a set of few characteristic scales that are totally decoupled from each other. However, this takes on a completely different flavour in non-linear systems, where scales are coupled and appropriate multiscale analysis is in order. This is the case of complex systems and, more particularly, scale invariant systems. In these, the approach to their solution is different, and it usually involves a multiscale basis. In this context, wavelets are one of the most used representation paradigms.The research context of complex systems and, particularly, scale invariant systems and multifractals has been in constant evolution over the last few years. Theoretical advances, either statistical (stochastic processes and probability distributions) or geometrical (function analysis and measure theory), along with fancy signal-processing algorithms suited to scale invariant data (and additionally handling aliasing, discretization and other artefacts of experimental data), have originated new tools for multifractal characterization of systems. While ten years ago the only methods available were statistical, by the start of this thesis project, development of geometrical methods had begun (most notably, the microcanonical multifractal formalism (MMF)). Geometrical methods have a clear advantage over statistical methods: they characterize each point of the system and thus they permit new applications such as reconstruction and prediction of signals, i.e., not only statistical characterization. Additionally, geometrical methods provide statistical characterization with much less need of data than statistical methods.In the present thesis, we have worked on the generalization and improvement of MMF, as well as its applications to the inference and forecasting of systems that follow a cascade process. In particular, we have described applications to two very different systems: stock-market series and ocean turbulence. The representation of the signal as a microcanonical cascade plays a crucial role in these applications. This representation can be achieved with one particular wavelet called optimal wavelet. The most relevant theoretical achievements are the regularization of diverging multifractal measures, the establishment of the bridge between multiplicative variables in microcanonical cascade processes and local singularity exponents, and the design of accurate and robust measure of wavelet optimality for a given dataset. To achieve this, we have introduced a new formalism, that of microcanonical cascades, that marries the cascade formalisms with MMF.Regarding the developed applications, on stock-market time series, we have inferred the distribution of future returns conditioned by the cascade and we have shown that a prediction based on this inference improves that of an ARIMA model. From the distribution of future returns, future volatility and value-at-risk can be reliably forecasted. On ocean data we have characterized dynamical aspects from optimal wavelet cascade analysis. In particular, we have observed that anomalies in the cascade of sea surface temperature show particular points of heat transfer between structures at different scales in the zones of wind-driven currents, also in the gyres.Both understanding -- combined with appropriate modelling -- of dynamics and design of inference/forecasting algorithms have crucial importance for the anticipation of changes in natural phenomena. In this context, the chain formed by the three steps followed during the thesis, namely multifractal characterization first, then obtaining of the optimal wavelet and finally design of inference algorithms, summarizes the direction we have followed to tackle the study of econometric time series and ocean maps.KEYWORDS: statistical physics, nonlinear dynamics, fractals, cascade processes / Fenòmens naturals tant diversos com són la turbulència, les sèries economètriques i el camp magnètic heliocèntric tenen en comú el fet que són multifractals. La recent concepció de nous models que tenen en compte la presència de l'estructura multifractal estant permetent millorar la comprensió d'aquests fenòmens. D'igual manera, s'està explotant la capacitat d'aquests models en problemes com la codificació de dades amb mínim de redundància, la inferència de dades no disponibles i la previsió de l'evolució futura de la dinàmica del sistema.Malgrat aquests avenços, el disseny d'algorismes d'anàlisi multifractal de dades reals continua essent un repte important al qual fins ara s'ha donat resposta únicament de manera molt limitada. La presència de forats de dades, la discretització pròpia de les dades digitals, la presència de soroll i la de correlacions de llarg abast són dificultats comunes que necessiten ser tractades de forma curosa, mitjançant mètodes sofisticats, sobretot si es té en compte que les variables multifractals són intrínsecament irregulars a totes les escales i suavitzar-les implica modificar-ne les propietats.En aquesta tesi doctoral donem resposta als problemes esmentats per mitjà d'un formalisme multifractal basat en cascades multiplicatives microcanòniques. Mostrem que, si la base de representació de la cascada s'escull de forma adequada, les possibilitats d'inferència de la cascada milloren notablement i a més a més ho fan d'una forma robusta. En aquest sentit, mostrarem l'aplicació d'un model de cascada microcanònica per a la predicció de la distribució de cotització futura en sèries borsàries. Mostrarem també una altra aplicació d'un model de cascada microcanònica per a la detecció de transferències de calor entre escales a la superfície oceànica i com això permet identificar les zones de girs oceànics, les de ressorgiment d'aigües profundes i els corrents originats pels vents elisis.

Thesis (Ph.D.)--Australian National University, 2000.

Thesis (M.F.A.)--Rochester Institute of Technology, 2008. / Typescript.

Thesis (Ph. D.)--West Virginia University, 1999. / Title from document title page. Document formatted into pages; contains xxiii, 337 p. : ill. (some col.). Vita. Includes abstract. Includes bibliographical references (p. 289-296).

Thesis (Ph. D.)--University of Missouri-Columbia, 1997. / Typescript. Vita. Includes bibliographical references (leaves 151-159). Also available on the Internet.