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. This thesis consists of an introduction and a summary, followed by three papers. Paper I, Anders Nilsson, Dimensions and Projections: An Overview and Relevant Examples, 2006. Manuscript. Paper II, Anders Nilsson and Peter Wingren, Homogeneity and Non-coincidence of Hausdorff- and Box Dimensions for Subsets of ℝn, 2006. Submitted. Paper III, Anders Nilsson and Fredrik Georgsson, Projective Properties of Fractal Sets, 2006. To be published in Chaos, Solitons and Fractals. 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. The second paper concerns sets for which different types of dimensions give different values. Given three arbitrary and different numbers in (0,n), a compact set in ℝn 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. 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 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>

We consider random fractals generated by random recursive constructions, prove zero-one laws concerning their dimensions and find their packing and Minkowski dimensions. Also we investigate the packing measure in corresponding dimension. For a class of random distribution functions we prove that their packing and Hausdorff dimensions coincide.

京都大学 / 0048 / 新制・課程博士 / 博士(人間・環境学) / 甲第22534号 / 人博第937号 / 新制||人||223(附属図書館) / 2019||人博||937(吉田南総合図書館) / 京都大学大学院人間・環境学研究科共生人間学専攻 / (主査)教授 角 大輝, 教授 上木 直昌, 准教授 木坂 正史 / 学位規則第4条第1項該当 / Doctor of Human and Environmental Studies / Kyoto University / DFAM

The thesis consists of four main chapters. The first chapter includes an introduction to inhomogeneous self-similar sets and measures. In particular, we show that these sets and measures are natural generalizations of the well known self-similar sets and measures. We then investigate the structure of these sets and measures. In the second chapter we study various fractal dimensions (Hausdorff, packing and box dimensions) of inhomogeneous self-similar sets and compare our results with the well-known results for (ordinary) self-similar sets. In the third chapter we investigate the L {q} spectra and the Renyi dimensions of inhomogeneous self-similar measures and prove that new multifractal phenomena, not exhibited by (ordinary) self-similar measures, appear in the inhomogeneous case. Namely, we show that inhomogeneous self-similar measures may have phase transitions which is in sharp contrast to the behaviour of the L {q} spectra of (ordinary) self-similar measures satisfying the Open Set Condition. Then we study the significantly more difficult problem of computing the multifractal spectra of inhomogeneous self-similar measures. We show that the multifractal spectra of inhomogeneous self-similar measures may be non-concave which is again in sharp contrast to the behaviour of the multifractal spectra of (ordinary) self-similar measures satisfying the Open Set Condition. Then we present a number of applications of our results. Many of them are related to the notoriously difficult problem of computing (or simply obtaining non-trivial bounds) for the multifractal spectra of self-similar measures not satisfying the Open Set Condition. More precisely, we will show that our results provide a systematic approach to obtain non-trivial bounds (and in some cases even exact values) for the multifractal spectra of several large and interesting classes of self-similar measures not satisfying the Open Set Condition. In the fourth chapter we investigate the asymptotic behaviour of the Fourier transforms of inhomogeneous self-similar measures and again we present a number of applications of our results, in particular to non-linear self-similar measures.

<p>Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics.</p><p>In this thesis we take a look at some basic measure theory needed to introduce certain definitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these definitions are done and we investigate when they coincide. With these tools different fractals are studied and compared.</p><p>A key idea in this thesis has been to sum up different names and definitions referring to similar concepts.</p>

Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics. In this thesis we take a look at some basic measure theory needed to introduce certain definitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these definitions are done and we investigate when they coincide. With these tools different fractals are studied and compared. A key idea in this thesis has been to sum up different names and definitions referring to similar concepts.