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Fractals are a relatively recent development in mathematics that show promise as a foundation for models of complex systems like natural porous media. One important issue that has not been thoroughly explored is the affect of different algorithms commonly used to generate random fractal porous media on their properties and processes within them. The heterogeneous method can lead to large, uncontrolled variations in porosity. It is proposed that use of the homogeneous algorithm might lead to more reproducible applications. Computer codes that will make it easier for researchers to experiment with fractal models are provided. In Chapter 2, the application of percolation theory and fractal modeling to porous media are combined to investigate percolation in prefractal porous media. Percolation thresholds are estimated for the pore space of homogeneous random 2-dimensional prefractals as a function of the fractal scale invariance ratio b and iteration level i. Percolation in prefractals occurs through large pores connected by small pores. The thresholds increased beyond the 0.5927 porosity expected in Bernoulli (uncorrelated) networks. The thresholds increase with both b (a finite size effect) and i. The results allow the prediction of the onset of percolation in models of prefractal porous media. Only a limited range of parameters has been explored, but extrapolations allow the critical fractal dimension to be estimated for many b and i values. Extrapolation to infinite iterations suggests there may be a critical fractal dimension of the solid at which the pore space percolates. The extrapolated value is close to 1.89 -- the well-known fractal dimension of percolation clusters in 2-dimensional Bernoulli networks. The results of Chapters 1 and 2 are synthesized in an application to soil water retention in Chapter 3.

The main focus of the present study is to identify and integrate the factors affecting the degree of irregularity of five continental divide traces, as expressed by their fractal characteristics measured by the divider method. The factors studied are climate, relief and tectonic environment. The second objective of this study is to determine the relationship between uplift rates and divide trace fractal dimension. Analysis of the results suggests that the degree of irregularity of continental divide traces at fine scale (approximately 10-70 km of resolution) is strongly affected by both climate and tectonics. It is found that control of the factors is generally weaker at coarse scale (above approximately 70 km of resolution). Generic relief should be ranked below both climate and tectonic environment as a factor affecting the complexity of continental divide traces. In terms of the second objective, the fractal dimension at fine scales follows a weakly inverse relationship with uplift. At coarse scale, there is stronger inverse relationship between uplift rate and fractal dimension. / Introduction -- Methodology -- Geomorphic environment -- Evaluation of results -- Significance of control factors -- Conclusion. / Department of Geological Sciences

We investigate dynamical processes on random and regular fractals. The (static) problem of percolation in the semi-infinite plane introduces many pertinent ideas including real space renormalisation group (RSRG) fugacity transformations and scaling forms. We study the percolation probability to determine the surface critical behaviour and to establish exponent relations. The fugacity approach is generalised to study random walks on diffusion-limited aggregates (DLA). Using regular and random models, we calculate the walk dimensionality and demonstrate that it is consistent with a conjecture by Aharony and Stauffer. It is shown that the kinetically grown DLA is in a distinct dynamic universality class to lattice animals. Similarly, the speculation of Helman-Coniglio-Tsallis regarding diffusion on self-avoiding walks (SAWs) is shown to be incorrect. The results are corroborated by an exact enumeration analysis of the internal structure of SAWs. A 'spin' and field theoretic Hamiltonian formulation for the conformational and resistance properties of random walks is presented. We consider Gaussian random walks, SAWs, spiral SAWs and valence walks. We express resistive susceptibilities as correlation functions and hence e-expansions are calculated for the resistance exponents. For SAWs, the local crosslinks are shown to be irrelevant and we calculate corrections to scaling. A scaling description is introduced into an equation-of-motion method in order to study spin wave damping in d-dimensional isotropic Heisenberg ferro-, antiferro- and ferri- magnets near pc . Dynamic scaling is shown to be obeyed by the Lorentzian spin wave response function and lifetime. The ensemble of finite clusters and multicritical behaviour is also treated. In contrast, the relaxational dynamics of the dilute Anisotropic Heisenberg model is shown to violate conventional dynamic scaling near the percolation bicritical point but satisfies instead a singular scaling behaviour arising from activation of Bloch walls over percolation cluster energy barriers.

Dendrites are tree-like topological spaces, and in this thesis, the physical characteristics of various random fractal versions of this type of set are investigated. This work will contribute to the development of analysis on fractals, an area which has grown considerably over the last twenty years. First, a collection of random self-similar dendrites is constructed, and their Hausdorff dimension is calculated. Previous results determining this quantity for random self-similar structures have often relied on the scaling factors being bounded uniformly away from zero. However, using a percolative argument, and taking advantage of the tree-like structure of the sets considered here, it is shown that this condition is not necessary; a simple condition on the tail of the distribution of the scaling factors at zero is all that is assumed. The scaling factors of these recursively defined structures form what is known as a multiplicative cascade, and results about the height of this random object are also obtained. With important physical and probabilistic applications, the heat equation has justifiably received a substantial amount of attention in a variety of settings. For certain types of fractals, it has become clear that a key factor in estimating the heat kernel is the volume growth with respect to the resistance metric on the space. In particular, uniform polynomial volume growth, which occurs for many deterministic self-similar fractals, immediately implies uniform (on-diagonal) heat kernel behaviour. However, in the random fractal setting, this is frequently not the case, and volume fluctuations are often observed. Motivated by this, an analysis of how volume fluctuations lead to corresponding heat kernel fluctuations for measure-metric spaces equipped with a resistance form is conducted here. These results apply to the aforementioned random self-similar dendrites, amongst other examples. The continuum random tree (CRT) of Aldous is an important random example of a measure-metric space, and fits naturally into the framework of the previous paragraph. In this thesis, quenched (almost-sure) volume growth asymptotics for the CRT are deduced, which show that the behaviour in almost-every realisation is not uniform. Applying the results introduced above, these yield heat kernel bounds for the CRT, demonstrating that heat kernel fluctuations occur almost-surely. Finally, a new representation of the CRT as a random self-similar dendrite is presented.

The above-ground part of a plant has an important contribution to plant development and yield production. Physiological activities of a plant canopy highly correlate to morphology of plant vegetation. Obviously, leaf area index is a good indicator for leaf area, but does not provide any information about the spatial architecture of plant canopy. With the development of fractal theory, a quantitative toot is now available for the investigation of complex objects and shapes such as plant structure. Vegetation structure of corn ( Zea mays L.) and soybean (Glycine max. (L.) Merr.] plants might be affected by the plant population density (low, normal, high) of each crop and corn-soybean intercropping. Skeletonized leaf-off images provided acceptable information to estimate the fractal dimension of the soybean plant 2-dimensionally, using the box-counting method. Fractal dimension varied among soybean treatments, with rankings: low > normal > intercrop > high, in the overall mean and normal ≈ intercrop ≈ low > high, in the slope of time plots. An adjustment of field corn plants to treatments, by changing the orientation of the plane of developed leaves with respect to the row, was observed. Thus, the fractal dimension of corn plant skeletal images from each of two sides, side I (parallel to row) and side 2 (perpendicular to row), was analyzed. On the basis of overall means of fractal dimension, treatments were ranked as: high > normal ≈ intercrop ≈ low for side 1 and intercrop > low ≈ normal > high for side 2. In both cases of soybean and corn plants, leaf area index, plant height and number of leaves (only in case of soybean plant) increased over the experiment for all the treatments, indicating a positive correlation with fractal dimension. In contrast, light penetration decreased during crop development, indicating a negative correlation with fractal dimension. Furthermore, a modified version of the Beer-Lambert equation, in which fractal dimension mu

Thesis (M.S.) in Physics--University of Maine, 2007. / Includes vita. Includes bibliographical references (leaves 64-65).

Thesis (Ph. D.)--University of Washington, 1999. / Vita. Includes bibliographical references (p. [297]-311).

Thesis (Ph. D.)--University of North Texas, May, 2009. / Title from title page display. Includes bibliographical references.

Thesis (M.S.)--University of Missouri-Columbia, 2004. / Typescript. Includes bibliographical references (leaves 80-85). Also available on the Internet.