“…Periodic Points Iterates of Functions Fixed Points Periodic Points Families of Functions The Quadratic Family Bifurcations Period-3 Points The Schwarzian Derivative One-Dimensional Chaos Chaos Transitivity and Strong Chaos Conjugacy Cantor Sets Two-Dimensional Chaos Review of Matrices Dynamics of Linear FunctionsNonlinear Maps The Hénon Map The Horseshoe Map Systems of Differential Equations Review of Systems of Differential Equations Almost Linearity The Pendulum The Lorenz System Introduction to Fractals Self-Similarity The Sierpiński Gasket and Other "Monsters"Space-Filling Curves Similarity and Capacity DimensionsLyapunov Dimension Calculating Fractal Dimensions of Objects Creating Fractals Sets Metric Spaces The Hausdorff Metric Contractions and Affine Functions Iterated Function SystemsAlgorithms for Drawing Fractals Complex Fractals: Julia Sets and the Mandelbrot Set Complex Numbers and Functions Julia Sets The Mandelbrot Set Computer Programs Answers to Selected Exercises References Index.…”
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“…Finally, the last series of lectures, delivered by Thierry Giordano, is devoted to the theory of topological orbit equivalence, with particular attention to the classification of minimal actions by finitely generated abelian groups on the Cantor set.…”
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“…Second, the isoenergetic curves in the space of momenta \vec \varkappa corresponding to these eigenfunctions have the form of slightly distorted circles with holes (Cantor type structure). A new method of multiscale analysis in the momentum space is developed to prove these results. …”
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“…Moreover, experimental measurements are also carried out for a particular type of binary sequence, the Cantor ternary set, validating the theoretical model and demonstrating that the distribution of transparent Fresnel regions becomes a critical parameter in applications requiring directional emitters.…”
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“…Current models of anti-viral and anti-tumor immunity rest solidly on Steinman’s discovery of DCs, but also rely on two seemingly unrelated phenomena, also reported in the mid-1970s: the discoveries of “help” for cytolytic T cell responses by Cantor and Boyse in 1974 and “cross-priming” by Bevan in 1976. …”
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“…Integration on a product space, change-of-variables formulas as well as the construction and study of classical Cantor sets are treated in detail. Classical convolution inequalities, such as Young's inequality and Hardy-Littlewood-Sobolev inequality, are proven. …”
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“…Number Systems The Real Numbers The Complex Numbers Sequences Convergence of Sequences Subsequences Limsup and Liminf Some Special Sequences Series of Numbers Convergence of Series Elementary Convergence Tests Advanced Convergence Tests Some Special Series Operations on Series Basic Topology Open and Closed Sets Further Properties of Open and Closed Sets Compact Sets The Cantor SetConnected and Disconnected Sets Perfect Sets Limits and Continuity of Functions Basic Properties of the Limit of a Function Continuous Functions Topological Properties and Continuity Classifying Discontinuities and Monotonicity Differentiation of Functions The Concept of Derivative The Mean Value Theorem and Applications More on the Theory of Differentiation The Integral Partitions and the Concept of Integral Properties of the Riemann IntegralSequences and Series of Functions Convergence of a Sequence of Functions More on Uniform Convergence Series of Functions The Weierstrass Approximation Theorem Elementary Transcendental Functions Power Series More on Power Series: Convergence Issues The Exponential and Trigonometric Functions Logarithms and Powers of Real Numbers Appendix I: Elementary Number Systems Appendix II: Logic and Set Theory Table of Notation Glossary Bibliography Index.…”
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“…The notion of dimension itself presented a basic problem to the pioneers of topology. Cantor asked if dimension was a topological feature of Euclidean space. …”
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“…In addition, several topics from real analysis and elementary topology which are not usually covered in undergraduate courses (totally disconnected spaces and Cantor sets, points of discontinuity of maps and the Baire Category Theorem, surjectivity of isometries of compact metric spaces) are also included in the book. …”
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“…Also serving as an introduction to the works and thought of major mathematicians and philosophers, from Plato and Aristotle to Cantor, Dedekind, Frege, Husserl and Weyl, this book will be of interest to a wide variety of readers, from scholars with a general interest in the philosophy or mathematics to philosophers and mathematicians themselves.…”
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“…Results show that MEA is susceptible to HE, but is stronger than as-rolled and 3D-printed Cantor alloy and stainless steel. The HE susceptibility of MEA decreases with increasing current density. …”
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“…Simulation results show acoustic focusing profiles for Fresnel, Fibonacci and Cantor zone plates operating at a wavelength of 1.5 mm without any distortion.…”
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“…Some of the turning points considered are the rise of the axiomatic method (most famously in Euclid), and the subsequent major changes in it (for example, by David Hilbert); the “wedding,” via analytic geometry, of algebra and geometry; the “taming” of the infinitely small and the infinitely large; the passages from algebra to algebras, from geometry to geometries, and from arithmetic to arithmetics; and the revolutions in the late nineteenth and early twentieth centuries that resulted from Georg Cantor’s creation of transfinite set theory. The origin of each turning point is discussed, along with the mathematicians involved and some of the mathematics that resulted. …”
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“…In the first part, the classical results of Alexandroff, Cantor, Hausdorff, Helly, Peano, Radon, Tietze and Urysohn illustrate the theories of metric, topological and normed spaces. …”
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“…This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, allowing us to unify and substantially generalize a large number of existing results on the geometry of random Cantor sets and measures, as well as obtaining many new ones. …”
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“…Along the way, the reader is introduced to Cantor's transfinite numbers, axiomatic set theory, Turing machines, and Gödel's incompleteness theorems. …”
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“…This collection of stories touches upon many genres: Normed Trek is a clever and witty Alice-in-Wonderland-type narrative set in the realm of mathematical analysis, The Cantor Trilogy is a dystopia about the consequences of relying upon computer-based mathematical proofs, In Search of Future Time bears the flavor of Tales from Arabian Nights set in the future, and – last but not least - Murder on the Einstein Express is a short, non-technical primer on probabilities and modern classical physics, disguised as a detective story. …”
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“…The example of a function in exactly one of the trace spaces is given by a characteristic function of a suitably chosen Cantor set.…”
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