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Fractal analysis has entered a new era. The applications to different areas of knowledge have been surprising. Benoit Mandelbrot, creator of fractal geometry, would have been surprised by the use of fractal analysis presented in this book. Here we present the use of fractal geometry, in particular, fractal analysis in two sciences: health sciences and social sciences and humanities. Part 1 is Health Science. In it, we present the latest advances in cardiovascular signs, kidney images to determine cancer growth, EEG signals, magnetoencephalography signals, and photosensitive epilepsy. We show how it is possible to produce ultrasonic lenses or even sound focusing. In Part 2, we present the use of fractal analysis in social sciences and humanities. It includes anthropology, hierarchical scaling, human settlements, language, fractal dimension of different cultures, cultural traits, and Mesoamerican complexity. And in Part 3, we present a few useful tools for fractal analysis, such as graphs and correlation, self-affine and self-similar graphs, and correlation function. It is impossible to picture today's research without fractal geometry.

Fractal analysis. --- Fractal geometric analysis --- Geometric analysis --- Medicine --- Public Health --- Health Care --- Health Sciences

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Fractal analysis has entered a new era. The applications to different areas of knowledge have been surprising. Let us begin with the fractional calculus-fractal geometry relationship, which allows for modeling with extreme precision of phenomena such as diffusion in porous media with fractional partial differential equations in fractal objects. Where the order of the equation is the same as the fractal dimension, this allows us to make calculations with enormous precision in diffusion phenomena-particularly in the oil industry, for new spillage prevention. Main applications to industry, design of fractal antennas to receive all frequencies and that is used in all cell phones, spacecraft, radars, image processing, measure, porosity, turbulence, scattering theory. Benoit Mandelbrot, creator of fractal geometry, would have been surprised by the use of fractal analysis presented in this book: ""Part I: Petroleum Industry and Numerical Analysis""; ""Part II: Fractal Antennas, Spacecraft, Radars, Image Processing, and Measure""; and ""Part III: Scattering Theory, Porosity, and Turbulence."" It's impossible to picture today's research without fractal analysis.

Fractal analysis. --- Fractal geometric analysis --- Geometric analysis --- Physical Sciences --- Engineering and Technology --- Dynamical Systems Theory --- Mathematics --- Applied Mathematics

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The aim of this book is to show some applications of fractal analysis in the fields of sciences. The first chapter introduces the readers to the book, while the second chapter shows the methods and challenges of fractal analysis of time-series data sets. The third chapter demonstrates fractal geometry as an attractive choice for miniaturized planar microwave filter design. The fourth chapter presents fractal antennas for wearable applications. The objective of the fifth chapter is to show some Parrondian games in discrete dynamic systems, while the last chapter reveals fractal structures of carbon nanotube system arrays.

Fractal analysis. --- Fractals. --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Dimension theory (Topology) --- Fractal geometric analysis --- Geometric analysis --- Physical Sciences --- Engineering and Technology --- Dynamical Systems Theory --- Mathematics --- Applied Mathematics

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This volume contains lecture notes on some topics in geometric analysis, a growing mathematical subject which uses analytical techniques, mostly of partial differential equations, to treat problems in differential geometry and mathematical physics. The presentation of the material should be rather accessible to non-experts in the field, since the presentation is didactic in nature. The reader will be provided with a survey containing some of the most exciting topics in the field, with a series of techniques used to treat such problems.

Geometric analysis --- Differential equations, Partial --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Geometric analysis PDEs (Geometric partial differential equations) --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Partial differential equations. --- Physics. --- Analysis. --- Partial Differential Equations. --- Mathematical Methods in Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Partial differential equations --- 517.1 Mathematical analysis --- Mathematical analysis --- Math --- Science --- Geometry --- Global analysis (Mathematics). --- Differential equations, partial. --- Mathematical physics. --- Physical mathematics --- Physics --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Cetraro <2007>

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This volume discusses the basic geometric contents of an image and presents a tree data structure to handle those contents efficiently. The nodes of the tree are derived from connected components of level sets of the intensity, while the edges represent inclusion information. Grain filters, morphological operators simplifying these geometric contents, are analyzed and several applications to image comparison and registration, and to edge and corner detection, are presented. The mathematically inclined reader may be most interested in Chapters 2 to 6, which generalize the topological Morse description to continuous or semicontinuous functions, while mathematical morphologists may more closely consider grain filters in Chapter 3. Computer scientists will find algorithmic considerations in Chapters 6 and 7, the full justification of which may be found in Chapters 2 and 4 respectively. Lastly, all readers can learn more about the motivation for this work in the image processing applications presented in Chapter 8.

Image processing --- Applied Physics --- Computer Science --- Engineering & Applied Sciences --- Mathematical models --- Geometric analysis. --- Topographic maps. --- Mathematical models. --- Hypsographic maps --- Hypsometric maps --- Maps, Topographic --- Relief maps --- Topo maps --- Topographical maps --- Geometric analysis PDEs (Geometric partial differential equations) --- Pictorial data processing --- Picture processing --- Processing, Image --- Computer science. --- Coding theory. --- Computers. --- Information theory. --- Topology. --- Discrete mathematics. --- Combinatorics. --- Computer Science. --- Coding and Information Theory. --- Discrete Mathematics. --- Information and Communication, Circuits. --- Computing Methodologies. --- Combinatorics --- Algebra --- Mathematical analysis --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Communication theory --- Communication --- Cybernetics --- Automatic computers --- Automatic data processors --- Computer hardware --- Computing machines (Computers) --- Electronic brains --- Electronic calculating-machines --- Electronic computers --- Hardware, Computer --- Computer systems --- Machine theory --- Calculators --- Cyberspace --- Data compression (Telecommunication) --- Digital electronics --- Information theory --- Signal theory (Telecommunication) --- Computer programming --- Informatics --- Science --- Mathematics --- Maps --- Imaging systems --- Optical data processing --- Mathematics. --- Artificial intelligence. --- Artificial Intelligence. --- AI (Artificial intelligence) --- Artificial thinking --- Intellectronics --- Intelligence, Artificial --- Intelligent machines --- Machine intelligence --- Thinking, Artificial --- Bionics --- Cognitive science --- Digital computer simulation --- Logic machines --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- Math --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical --- Numerical analysis