Fractals And Universal Spaces In Dimension Theory
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| Author | : Stephen Lipscomb |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 259 |
| Release | : 2008-10-28 |
| Genre | : Mathematics |
| ISBN | : 0387854940 |
Historically, for metric spaces the quest for universal spaces in dimension theory spanned approximately a century of mathematical research. The history breaks naturally into two periods - the classical (separable metric) and the modern (not-necessarily separable metric). The classical theory is now well documented in several books. This monograph is the first book to unify the modern theory from 1960-2007. Like the classical theory, the modern theory fundamentally involves the unit interval. Unique features include: * The use of graphics to illustrate the fractal view of these spaces; * Lucid coverage of a range of topics including point-set topology and mapping theory, fractal geometry, and algebraic topology; * A final chapter contains surveys and provides historical context for related research that includes other imbedding theorems, graph theory, and closed imbeddings; * Each chapter contains a comment section that provides historical context with references that serve as a bridge to the literature. This monograph will be useful to topologists, to mathematicians working in fractal geometry, and to historians of mathematics. Being the first monograph to focus on the connection between generalized fractals and universal spaces in dimension theory, it will be a natural text for graduate seminars or self-study - the interested reader will find many relevant open problems which will create further research into these topics.
| Author | : Stephen Leon Lipscomb |
| Publisher | : Springer |
| Total Pages | : 191 |
| Release | : 2014-10-13 |
| Genre | : Mathematics |
| ISBN | : 3319062549 |
To see objects that live in the fourth dimension we humans would need to add a fourth dimension to our three-dimensional vision. An example of such an object that lives in the fourth dimension is a hyper-sphere or “3-sphere.” The quest to imagine the elusive 3-sphere has deep historical roots: medieval poet Dante Alighieri used a 3-sphere to convey his allegorical vision of the Christian afterlife in his Divine Comedy. In 1917, Albert Einstein visualized the universe as a 3-sphere, describing this imagery as “the place where the reader’s imagination boggles. Nobody can imagine this thing.” Over time, however, understanding of the concept of a dimension evolved. By 2003, a researcher had successfully rendered into human vision the structure of a 4-web (think of an ever increasingly-dense spider’s web). In this text, Stephen Lipscomb takes his innovative dimension theory research a step further, using the 4-web to reveal a new partial image of a 3-sphere. Illustrations support the reader’s understanding of the mathematics behind this process. Lipscomb describes a computer program that can produce partial images of a 3-sphere and suggests methods of discerning other fourth-dimensional objects that may serve as the basis for future artwork.
| Author | : Andreas M. Hinz |
| Publisher | : Birkhäuser |
| Total Pages | : 469 |
| Release | : 2018-04-17 |
| Genre | : Mathematics |
| ISBN | : 3319737791 |
The solitaire game “The Tower of Hanoi" was invented in the 19th century by the French number theorist Édouard Lucas. The book presents its mathematical theory and offers a survey of the historical development from predecessors up to recent research. In addition to long-standing myths, it provides a detailed overview of the essential mathematical facts with complete proofs, and also includes unpublished material, e.g., on some captivating integer sequences. The main objects of research today are the so-called Hanoi graphs and the related Sierpiński graphs. Acknowledging the great popularity of the topic in computer science, algorithms, together with their correctness proofs, form an essential part of the book. In view of the most important practical applications, namely in physics, network theory and cognitive (neuro)psychology, the book also addresses other structures related to the Tower of Hanoi and its variants. The updated second edition includes, for the first time in English, the breakthrough reached with the solution of the “The Reve's Puzzle" in 2014. This is a special case of the famed Frame-Stewart conjecture which is still open after more than 75 years. Enriched with elaborate illustrations, connections to other puzzles and challenges for the reader in the form of (solved) exercises as well as problems for further exploration, this book is enjoyable reading for students, educators, game enthusiasts and researchers alike. Excerpts from reviews of the first edition: “The book is an unusual, but very welcome, form of mathematical writing: recreational mathematics taken seriously and serious mathematics treated historically. I don’t hesitate to recommend this book to students, professional research mathematicians, teachers, and to readers of popular mathematics who enjoy more technical expository detail.” Chris Sangwin, The Mathematical Intelligencer 37(4) (2015) 87f. “The book demonstrates that the Tower of Hanoi has a very rich mathematical structure, and as soon as we tweak the parameters we surprisingly quickly find ourselves in the realm of open problems.” László Kozma, ACM SIGACT News 45(3) (2014) 34ff. “Each time I open the book I discover a renewed interest in the Tower of Hanoi. I am sure that this will be the case for all readers.” Jean-Paul Allouche, Newsletter of the European Mathematical Society 93 (2014) 56.
| Author | : Laurent Nottale |
| Publisher | : World Scientific |
| Total Pages | : 349 |
| Release | : 1993-04-13 |
| Genre | : Science |
| ISBN | : 9814520217 |
This is the first detailed account of a new approach to microphysics based on two leading ideas: (i) the explicit dependence of physical laws on scale encountered in quantum physics, is the manifestation of a fundamental principle of nature, scale relativity. This generalizes Einstein's principle of (motion) relativity to scale transformations; (ii) the mathematical achievement of this principle needs the introduction of a nondifferentiable space-time varying with resolution, i.e. characterized by its fractal properties.The author discusses in detail reactualization of the principle of relativity and its application to scale transformations, physical laws which are explicitly scale dependent, and fractals as a new geometric description of space-time.
| Author | : Pertti Mattila |
| Publisher | : Cambridge University Press |
| Total Pages | : 360 |
| Release | : 1999-02-25 |
| Genre | : Mathematics |
| ISBN | : 9780521655958 |
This book studies the geometric properties of general sets and measures in euclidean space.
| Author | : Heping Xie |
| Publisher | : CRC Press |
| Total Pages | : 464 |
| Release | : 2020-12-17 |
| Genre | : Nature |
| ISBN | : 1000099660 |
Important developments in the progress of the theory of rock mechanics during recent years are based on fractals and damage mechanics. The concept of fractals has proved to be a useful way of describing the statistics of naturally occurring geometrics. Natural objects, from mountains and coastlines to clouds and forests, are found to have boundaries best described as fractals. Fluid flow through jointed rock masses and clusterings of earthquakes are found to follow fractal patterns in time and space. Fracturing in rocks at all scales, from the microscale (microcracks) to the continental scale (megafaults), can lead to fractal structures. The process of diagenesis and pore geometry of sedimentary rock can be quantitatively described by fractals, etc. The book is mainly concerned with these developments, as related to fractal descriptions of fragmentations, damage and fracture of rocks, rock burst, joint roughness, rock porosity and permeability, rock grain growth, rock and soil particles, shear slips, fluid flow through jointed rocks, faults, earthquake clustering, and so on. The prime concerns of the book are to give a simple account of the basic concepts, methods of fractal geometry, and their applications to rock mechanics, geology, and seismology, and also to discuss damage mechanics of rocks and its application to mining engineering. The book can be used as a textbook for graduate students, by university teachers to prepare courses and seminars, and by active scientists who want to become familiar with a fascinating new field.
| Author | : R. A. Feddes |
| Publisher | : Cambridge University Press |
| Total Pages | : 198 |
| Release | : 1995-08-24 |
| Genre | : Science |
| ISBN | : 0521495083 |
This book presents the integrated contributions of hydrologists, meteorologists and ecologists to the first IHP/IAHS George Kovacs Colloquium in connection with the study of global hydrology and climate change. The atmospherical, hydrological and terrestrial components of the Earth's systems operate on different time and space scales. Resolving these scaling incongruities, as well as understanding and modelling the complex interaction of land surface processes at the different scales, represent a major challenge for hydrologists, ecologists and meteorological scientists alike. This book deals with time and space scale variations with reference to several topics including: soil water balance; ecosystems and interaction of flow systems; and macroscale hydrological modelling. This book will be of great use to researchers, engineers and forecasters with an interest in space and time scale variability.
| Author | : Robert J. Daverman |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 496 |
| Release | : 2009-10-14 |
| Genre | : Mathematics |
| ISBN | : 0821836978 |
A topological embedding is a homeomorphism of one space onto a subspace of another. The book analyzes how and when objects like polyhedra or manifolds embed in a given higher-dimensional manifold. The main problem is to determine when two topological embeddings of the same object are equivalent in the sense of differing only by a homeomorphism of the ambient manifold. Knot theory is the special case of spheres smoothly embedded in spheres; in this book, much more general spaces and much more general embeddings are considered. A key aspect of the main problem is taming: when is a topological embedding of a polyhedron equivalent to a piecewise linear embedding? A central theme of the book is the fundamental role played by local homotopy properties of the complement in answering this taming question. The book begins with a fresh description of the various classic examples of wild embeddings (i.e., embeddings inequivalent to piecewise linear embeddings). Engulfing, the fundamental tool of the subject, is developed next. After that, the study of embeddings is organized by codimension (the difference between the ambient dimension and the dimension of the embedded space). In all codimensions greater than two, topological embeddings of compacta are approximated by nicer embeddings, nice embeddings of polyhedra are tamed, topological embeddings of polyhedra are approximated by piecewise linear embeddings, and piecewise linear embeddings are locally unknotted. Complete details of the codimension-three proofs, including the requisite piecewise linear tools, are provided. The treatment of codimension-two embeddings includes a self-contained, elementary exposition of the algebraic invariants needed to construct counterexamples to the approximation and existence of embeddings. The treatment of codimension-one embeddings includes the locally flat approximation theorem for manifolds as well as the characterization of local flatness in terms of local homotopy properties.
| Author | : K.P. Hart |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 898 |
| Release | : 2013-12-11 |
| Genre | : Mathematics |
| ISBN | : 946239024X |
The book presents surveys describing recent developments in most of the primary subfields of General Topology, and its applications to Algebra and Analysis during the last decade, following the previous editions (North Holland, 1992 and 2002). The book was prepared in connection with the Prague Topological Symposium, held in 2011. During the last 10 years the focus in General Topology changed and therefore the selection of topics differs from that chosen in 2002. The following areas experienced significant developments: Fractals, Coarse Geometry/Topology, Dimension Theory, Set Theoretic Topology and Dynamical Systems.
| Author | : Portugali, Juval |
| Publisher | : Edward Elgar Publishing |
| Total Pages | : 456 |
| Release | : 2021-09-16 |
| Genre | : Social Science |
| ISBN | : 1789900123 |
Written by some of the founders of complexity theory and complexity theories of cities (CTC), this Handbook expertly guides the reader through over forty years of intertwined developments: the emergence of general theories of complex self-organized systems and the consequent emergence of CTC.
