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This exceptional book is concerned with the application of fractals and chaos, as well as other concepts from nonlinear dynamics to biomedical phenomena. Herein we seek to communicate the excitement being experienced by scientists upon making application of these concepts within the life sciences. Mathematical concepts are introduced using biomedical data sets and the phenomena being explained take precedence over the mathematics.In this new edition what has withstood the test of time has been updated and modernized; speculations that were not borne out have been expunged and the breakthroughs

Chaotic behavior in systems. --- Fractals. --- Medicine --- Physiology --- Health Workforce --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Dimension theory (Topology) --- Chaos in systems --- Chaos theory --- Chaotic motion in systems --- Differentiable dynamical systems --- Dynamics --- Nonlinear theories --- System theory --- Mathematical models.

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Transient friction effects determine the behavior of a wide class of mechatronic systems. Classic examples are squealing brakes, stiction in robotic arms, or stick-slip in linear drives. To properly design and understand mechatronic systems of this type, good quantitative models of transient friction effects are of primary interest. The theory developed in this book approaches this problem bottom-up, by deriving the behavior of macroscopic friction surfaces from the microscopic surface physics. The model is based on two assumptions: First, rough surfaces are inherently fractal, exhibiting roughness on a wide range of scales. Second, transient friction effects are caused by creep enlargement of the real area of contact between two bodies. This work demonstrates the results of extensive Finite Element analyses of the creep behavior of surface asperities, and proposes a generalized multi-scale area iteration for calculating the time-dependent real contact between two bodies. The toolset is then demonstrated both for the reproduction of a variety of experimental results on transient friction as well as for system simulations of two example systems.

Fractals. --- Friction. --- Friction --- Materials --- Mechanical Engineering --- Engineering & Applied Sciences --- Mechanical Engineering - General --- Simulation methods --- Creep --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Engineering. --- Continuum mechanics. --- Mechatronics. --- Thin films. --- Continuum Mechanics and Mechanics of Materials. --- Surfaces and Interfaces, Thin Films. --- Surfaces. --- Dimension theory (Topology) --- Mechanics --- Bearings (Machinery) --- Tribology --- Mechanics. --- Mechanics, Applied. --- Surfaces (Physics). --- Solid Mechanics. --- Physics --- Surface chemistry --- Surfaces (Technology) --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Dynamics --- Quantum theory --- Materials—Surfaces. --- Films, Thin --- Solid film --- Solid state electronics --- Solids --- Coatings --- Thick films --- Mechanical engineering --- Microelectronics --- Microelectromechanical systems

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Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings; that is, one-dimensional drums with fractal boundary. This second edition of Fractal Geometry, Complex Dimensions and Zeta Functions will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, complex analysis, distribution theory, and mathematical physics. The significant studies and problems illuminated in this work may be used in a classroom setting at the graduate level. Key Features include: The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings · Complex dimensions of a fractal string are studied in detail, and used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra · Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal ·  Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula ·  The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field.

Fractals. --- Functions, Zeta. --- Geometry, Riemannian. --- Mathematics. --- Fractals --- Functions, Zeta --- Geometry, Riemannian --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Geometry --- Number theory. --- Zeta functions --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Number study --- Numbers, Theory of --- Riemann geometry --- Riemannian geometry --- Dynamics. --- Ergodic theory. --- Functional analysis. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Measure theory. --- Partial differential equations. --- Number Theory. --- Measure and Integration. --- Partial Differential Equations. --- Dynamical Systems and Ergodic Theory. --- Global Analysis and Analysis on Manifolds. --- Functional Analysis. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Partial differential equations --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Geometry, Differential --- Topology --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Math --- Science --- Dimension theory (Topology) --- Generalized spaces --- Geometry, Non-Euclidean --- Semi-Riemannian geometry --- Differential equations, partial. --- Differentiable dynamical systems. --- Global analysis. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics