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functional analysis --- quantum algebra --- nuclear physics --- thermodynamic --- algorithms --- geometry --- Mathematical analysis --- Thermodynamics --- Fractals --- Fractales --- Fractals. --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Dimension theory (Topology)
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This monograph gives a state-of-the-art and accessible treatment of a new general higher-dimensional theory of complex dimensions, valid for arbitrary bounded subsets of Euclidean spaces, as well as for their natural generalization, relative fractal drums. It provides a significant extension of the existing theory of zeta functions for fractal strings to fractal sets and arbitrary bounded sets in Euclidean spaces of any dimension. Two new classes of fractal zeta functions are introduced, namely, the distance and tube zeta functions of bounded sets, and their key properties are investigated. The theory is developed step-by-step at a slow pace, and every step is well motivated by numerous examples, historical remarks and comments, relating the objects under investigation to other concepts. Special emphasis is placed on the study of complex dimensions of bounded sets and their connections with the notions of Minkowski content and Minkowski measurability, as well as on fractal tube formulas. It is shown for the first time that essential singularities of fractal zeta functions can naturally emerge for various classes of fractal sets and have a significant geometric effect. The theory developed in this book leads naturally to a new definition of fractality, expressed in terms of the existence of underlying geometric oscillations or, equivalently, in terms of the existence of nonreal complex dimensions. The connections to previous extensive work of the first author and his collaborators on geometric zeta functions of fractal strings are clearly explained. Many concepts are discussed for the first time, making the book a rich source of new thoughts and ideas to be developed further. The book contains a large number of open problems and describes many possible directions for further research. The beginning chapters may be used as a part of a course on fractal geometry. The primary readership is aimed at graduate students and researchers working in Fractal Geometry and other related fields, such as Complex Analysis, Dynamical Systems, Geometric Measure Theory, Harmonic Analysis, Mathematical Physics, Analytic Number Theory and the Spectral Theory of Elliptic Differential Operators. The book should be accessible to nonexperts and newcomers to the field.
Mathematics. --- Measure theory. --- Number theory. --- Mathematical physics. --- Number Theory. --- Measure and Integration. --- Mathematical Physics. --- Functions, Zeta. --- Fractals. --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Zeta functions --- Dimension theory (Topology) --- Math --- Science --- Number study --- Numbers, Theory of --- Algebra --- Physical mathematics --- Physics --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Mathematics
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This book provides a short, hands-on introduction to the science of complexity using simple computational models of natural complex systems-with models and exercises drawn from physics, chemistry, geology, and biology. By working through the models and engaging in additional computational explorations suggested at the end of each chapter, readers very quickly develop an understanding of how complex structures and behaviors can emerge in natural phenomena as diverse as avalanches, forest fires, earthquakes, chemical reactions, animal flocks, and epidemic diseases.Natural Complexity provides the necessary topical background, complete source codes in Python, and detailed explanations for all computational models. Ideal for undergraduates, beginning graduate students, and researchers in the physical and natural sciences, this unique handbook requires no advanced mathematical knowledge or programming skills and is suitable for self-learners with a working knowledge of precalculus and high-school physics.Self-contained and accessible, Natural Complexity enables readers to identify and quantify common underlying structural and dynamical patterns shared by the various systems and phenomena it examines, so that they can form their own answers to the questions of what natural complexity is and how it arises.
Complexity (Philosophy) --- Physics --- Computational complexity. --- Complexity, Computational --- Electronic data processing --- Machine theory --- Philosophy --- Emergence (Philosophy) --- Methodology. --- Burridge-Knopoff stick-slip model. --- Gutenberg-Richter law. --- Johannes Kepler. --- Olami-Feder-Christensen model. --- Python code. --- accretion. --- active flockers. --- agents. --- automobile traffic. --- avalanches. --- cells. --- cellular automata. --- chaos. --- clusters. --- complex behavior. --- complex structure. --- complex system. --- complexity. --- computational model. --- computer program. --- contagious diseases. --- criticality. --- diffusion-limited aggregation. --- earthquake forecasting. --- earthquakes. --- emergence. --- emergent behavior. --- emergent structure. --- epidemic spread. --- epidemic surges. --- excitable system. --- flocking. --- forest fires. --- fractal clusters. --- fractal geometry. --- growth. --- hodgepodge machine. --- infection rate. --- iterated growth. --- lattice. --- lichens. --- natural complex system. --- natural complexity. --- natural order. --- natural phenomena. --- nature. --- open dissipative system. --- panic. --- passive flockers. --- pattern formation. --- percolation threshold. --- percolation. --- phase transition. --- planetary motion. --- power-law. --- random walk. --- randomness. --- repulsion. --- rule-based growth. --- sandpile. --- scale invariance. --- segregation. --- self-organization. --- self-organized criticality. --- self-propulsion. --- self-similarity. --- simple rules. --- small-world network. --- solar flares. --- spaghetti. --- spatiotemporal pattern. --- spiral. --- tagging algorithm. --- traffic jams. --- waves. --- wildfire management.
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