Abstract This work is devoted to the theory of joint interpretation of multimethod geophysical data and its application to the solution of real geophysical inverse problems. The targets of such joint interpretation can be geological bodies with an established dependence between various physical properties that cause anomalies in several geophysical fields (geophysical multiresponse). The establishing of the relationship connecting the various physical properties is therefore a necessary first step in any joint interpretation procedure. Bodies for which the established relationship between physical properties is violated (single-response bodies) can be targets of separate interpretations. The probabilistic (Bayesian) approach provides the necessary formalism for addressing the problem of the joint inversion of multimethod geophysical data, which can be non-linear and have a non-unique solution. Analysis of the lower limit of resolution of the non-linear problem of joint inversion using the definition of e-entropy demonstrates that joint inversion of multimethod geophysical data can reduce non-uniqueness in real geophysical inverse problems. The question can be formulated as a multiobjective optimisation problem (MOP), enabling the numerical methods of this theory to be employed for the purpose of geophysical data inversion and for developing computer algorithms capable of solving highly non-linear problems. An example of such a problem is magnetotelluric impedance tensor inversion with the aim of obtaining a 3-D resistivity distribution. An additional area of application for multiobjective optimisation can be the combination of various types of uncertain information (probabilistic and non-probabilistic) in a common inversion scheme applicable to geophysical inverse problems. It is demonstrated how the relationship between seismic velocity and density can be used to construct an algorithm for the joint interpretation of gravity and seismic wide-angle reflection and refraction data. The relationship between the elastic and electrical properties of rocks, which is a necessary condition for the joint inversion of data obtained by seismic and electromagnetic methods, can be established for solid- liquid rock mixtures using theoretical modelling of the elastic and electrical properties of rocks with a fractal microstructure and from analyses of petrophysical data and borehole log data.

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The aim of this research study is to investigate different practical ways in which fractal geometry and self-similarity can be applied to art music, with reference to music composition and analysis. This specific topic was chosen because there are many misconceptions in the field of fractal and self-similar music. Analyses of previous research as well as the music analysis of several compositions from different composers in different genres were the main methods for conducting the research. Although the dissertation restates much of the existing research on the topic, it is (to the researcher‟s knowledge) one of the first academic works that summarises the many different facets of fractal geometry and music. Fractal and self-similar shapes are evident in nature and art dating back to the 16th century, despite the fact that the mathematics behind fractals was only defined in 1975 by the French mathematician, Benoit B. Mandelbrot. Mathematics has been a source of inspiration to composers and musicologists for many centuries and fractal geometry has also infiltrated the works of composers in the past 30 years. The search for fractal and self-similar structures in music composed prior to 1975 may lead to a different perspective on the way in which music is analysed. Basic concepts and prerequisites of fractals were deliberately simplified in this research in order to collect useful information that musicians can use in composition and analysis. These include subjects such as self-similarity, fractal dimensionality and scaling. Fractal shapes with their defining properties were also illustrated because their structures have been likened to those in some music compositions. This research may enable musicians to incorporate mathematical properties of fractal geometry and self-similarity into original compositions. It may also provide new ways to view the use of motifs and themes in the structural analysis of music. / Dissertation (MMus)--University of Pretoria, 2014. / lk2014 / Music / MMus / Unrestricted

Doctor of Philosophy / Department of Physics / Christopher M. Sorensen / We demonstrated that an aerosol can gel. This gelation was then used for a one-step method to produce an ultralow density porous carbon or silica material. This material was named an aerosol gel because it was made via gelation of particles in the aerosol phase. The carbon and silica aerosol gels had high specific surface area (200 – 350 sq m/g for carbon and 300 – 500 sq m/g for silica) and an extremely low density (2.5 – 6.0 mg/cm[superscript3]), properties similar to conventional aerogels. Key aspects to form a gel from an aerosol are large volume fraction, ca. 10[superscript-4] or greater, and small primary particle size, 50 nm or smaller, so that the gel time is fast compared to other characteristic times. Next we report the results of a study of the cluster morphology and kinetics of a dense aggregating aerosol system using the small angle light scattering technique. The soot particles started as individual monomers, ca. 38 nm radius, grew to bigger clusters with time and finally stopped evolving after spanning a network across the whole system volume. This spanning is aerosol gelation. The gelled system showed a hybrid morphology with a lower fractal dimension at length scales of a micron or smaller and a higher fractal dimension at length scales greater than a micron. The study of the kinetics of the aggregating system showed that when the system gelled, the aggregation kernel homogeneity attained a value 0.4 or higher. The magnitude of the aggregation kernel showed an increase with increasing volume fraction. We also used image analysis technique to study the cluster morphology. From the digitized pictures of soot clusters the cluster morphology was determined by two different methods: structure factor and perimeter analysis. We find a hybrid, superaggregate morphology characterized by a fractal dimension of D[subscript f] nearly equal to 1.8 between the monomer size, ca. 50 nm, and 1 micron and D[subscript f] nearly equal to 2.6 at larger length scales up to [similar to] 10 micron. The superaggregate morphology is a consequence of late stage aggregation in a cluster dense regime near a gel point.

Both complex dynamics and the theory of reproducing kernel Hilbert spaces have found widespread application over the last few decades. Although complex dynamics started over a century ago, the gravity of it's importance was only recently realized due to B.B. Mandelbrot's work in the 1980's. B.B. Mandelbrot demonstrated to the world that fractals, which are chaotic patterns containing a high degree of self-similarity, often times serve as better models to nature than conventional smooth models. The theory of reproducing kernel Hilbert spaces also having started over a century ago, didn't pick up until N. Aronszajn's classic was written in 1950. Since then, the theory has found widespread application to fields including machine learning, quantum mechanics, and harmonic analysis. In the paper, Infinite Product Representations of Kernel Functions and Iterated Function Systems, the authors, D. Alpay, P. Jorgensen, I. Lewkowicz, and I. Martiziano, show how a kernel function can be constructed on an attracting set of an iterated function system. Furthermore, they show that when certain conditions are met, one can construct an orthonormal basis of the associated Hilbert space via certain pull-back and multiplier operators. In this thesis we take for our iterated function system, the family of iterates of a given rational map. Thus we investigate for which rational maps their kernel construction holds as well as their orthornormal basis construction. We are able to show that the kernel construction applies to any rational map conjugate to a polynomial with an attracting fixed point at 0. Within such rational maps, we are able to find a family of polynomials for which the orthonormal basis construction holds. It is then natural to ask how the orthonormal basis changes as the polynomial within a given family varies. We are able to determine for certain families of polynomials, that the dynamics of the corresponding orthonormal basis is well behaved. Finally, we conclude with some possible avenues of future investigation.

Philosophiae Doctor - PhD / Conventional partial differential equations under the classical Black-Scholes approach have been extensively explored over the past few decades in solving option pricing problems. However, the underlying Efficient Market Hypothesis (EMH) of classical economic theory neglects the effects of memory in asset return series, though memory has long been observed in a number financial data. With advancements in computational methodologies, it has now become possible to model different real life physical phenomenons using complex approaches such as, fractional differential equations (FDEs). Fractional models are generalised models which based on literature have been found appropriate for explaining memory effects observed in a number of financial markets including the stock market. The use of fractional model has thus recently taken over the context of academic literatures and debates on financial modelling. / 2023-12-02

In this work, Paper-based inkjet-printed Ultra-wide band (UWB) fractal antennas are presented. Three new designs, a combined UWB fractal monopole based on the fourth order Koch Snowflake fractal which utilizes a Sierpinski Gasket fractal for ink reduction, a Cantor-based fractal antenna which performs a larger bandwidth compared to previously published UWB Cantor fractal monopole antenna, and a 3D loop fractal antenna which attains miniaturization, impedance matching and multiband characteristics. It is shown that fractals prove to be a successful method of reducing fabrication cost in inkjet printed antennas while retaining or enhancing printed antenna performance.

The master’s thesis deals with computational fluid dynamics (CFD) of two orifices, that have different shapes of holes but similar cross-sectional flow areas. The first of them is orifice with circular-shaped hole, which is used for maintenance free measurement of flow. The second one is orifice with fractal-shaped hole, inspired by von Koch snow-flake. This thesis follows bachelor thesis, in which was experimentally examined, that fractal-shaped orifices have better hydraulic properties (hydraulic losses and lower pressure pulsations) than circle-shaped one. The main target is to confirm this conclusion based on experiment, this time using CFD with various types of turbulence modelling ap-proaches. Both single phase (cavitation free) and multiphase numerical simulations were realized. Each model was compared from perspective of hydraulic and dynamic charac-teristics.

abstract: Fractal analyses examine variability in a time series to look for temporal structure or pattern that reveals the underlying processes of a complex system. Although fractal property has been found in many signals in biological systems, how it relates to behavioral performance and what it implies about the complex system under scrutiny are still open questions. In this series of experiments, fractal property, movement kinematics, and behavioral performance were measured on participants performing a reciprocal tapping task. In Experiment 1, the results indicated that the alpha value from detrended fluctuation analysis (DFA) reflected deteriorating performance when visual feedback delay was introduced into the reciprocal tapping task. This finding suggests that this fractal index is sensitive to performance level in a movement task. In Experiment 2, the sensitivity of DFA alpha to the coupling strength between sub-processes within a system was examined by manipulation of task space visibility. The results showed that DFA alpha was not influenced by disruption of subsystems coupling strength. In Experiment 3, the sensitivity of DFA alpha to the level of adaptivity in a system under constraints was examined. Manipulation of the level of adaptivity was not successful, leading to inconclusive results to this question. / Dissertation/Thesis / Masters Thesis Psychology 2019

Indiana University-Purdue University Indianapolis (IUPUI) / Given a parametrized family of Iterated Function System (IFS) we give sufficient conditions for a parameter on the boundary of the connectedness locus, M, to be accessible from the complement of M. Moreover, we provide a few examples of such parameters and describe how they are connected to Misiurewicz parameter in the Mandelbrot set, i.e. the connectedness locus of the quadratic family z^2+c.