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This book provides a generalised approach to fractal dimension theory from the standpoint of asymmetric topology by employing the concept of a fractal structure. The fractal dimension is the main invariant of a fractal set, and provides useful information regarding the irregularities it presents when examined at a suitable level of detail. New theoretical models for calculating the fractal dimension of any subset with respect to a fractal structure are posed to generalise both the Hausdorff and box-counting dimensions. Some specific results for self-similar sets are also proved. Unlike classical fractal dimensions, these new models can be used with empirical applications of fractal dimension including non-Euclidean contexts. In addition, the book applies these fractal dimensions to explore long-memory in financial markets. In particular, novel results linking both fractal dimension and the Hurst exponent are provided. As such, the book provides a number of algorithms for properly calculating the self-similarity exponent of a wide range of processes, including (fractional) Brownian motion and Lévy stable processes. The algorithms also make it possible to analyse long-memory in real stocks and international indexes. This book is addressed to those researchers interested in fractal geometry, self-similarity patterns, and computational applications involving fractal dimension and Hurst exponent.

Fractal analysis. --- Fractal geometric analysis --- Geometric analysis --- Differentiable dynamical systems. --- Topology. --- Mathematics. --- Distribution (Probability theory. --- Algorithms. --- Dynamical Systems and Ergodic Theory. --- Measure and Integration. --- Probability Theory and Stochastic Processes. --- Mathematical Applications in Computer Science. --- Algorism --- Algebra --- Arithmetic --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Math --- Science --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Foundations --- Dynamics. --- Ergodic theory. --- Measure theory. --- Probabilities. --- Computer science—Mathematics. --- Computer mathematics. --- Computer mathematics --- Electronic data processing --- Mathematics --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics

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This work gathers a selection of outstanding papers presented at the 25th Conference on Differential Equations and Applications / 15th Conference on Applied Mathematics, held in Cartagena, Spain, in June 2017. It supports further research into both ordinary and partial differential equations, numerical analysis, dynamical systems, control and optimization, trending topics in numerical linear algebra, and the applications of mathematics to industry. The book includes 14 peer-reviewed contributions and mainly addresses researchers interested in the applications of mathematics, especially in science and engineering. It will also greatly benefit PhD students in applied mathematics, engineering and physics.

Partial differential equations --- Differential equations --- Numerical methods of optimisation --- Operational research. Game theory --- Mathematics --- Applied physical engineering --- Engineering sciences. Technology --- Computer. Automation --- differentiaalvergelijkingen --- ICT (informatie- en communicatietechnieken) --- toegepaste wiskunde --- automatisering --- economie --- informatica --- externe fixatie (geneeskunde --- wiskunde --- ingenieurswetenschappen