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 (4.0 / 5.0) Formulated in 1859, the Riemann Hypothesis is the most celebrated and multifaceted open problem in mathematics. In essence, it states that the primes are distributed as harmoniously as possible--or, equivalently, that the Riemann zeros are located on a single vertical line, called the critical line. In this book, the author proposes a new approach to understand and possibly solve the Riemann Hypothesis. His reformulation builds upon earlier (joint) work on complex fractal dimensions and the vibrations of fractal strings, combined with string theory and noncommutative geometry. Accordingly, it relies on the new notion of a fractal membrane or quantized fractal string, along with the modular flow on the associated moduli space of fractal membranes. Conjecturally, under the action of the modular flow, the spacetime geometries become increasingly symmetric and crystal-like, hence, arithmetic. Correspondingly, the zeros of the associated zeta functions eventually condense onto the critical line, towards which they are attracted, thereby explaining why the Riemann Hypothesis must be true. Written with a diverse audience in mind, this unique book is suitable for graduate students, experts and nonexperts alike, with an interest in number theory, analysis, dynamical systems, arithmetic, fractal or noncommutative geometry, and mathematical or theoretical physics.
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$53.32 |
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 (5.0 / 5.0) The power to change things lies within us. Presented in this book is a theory of how shifts in oneself can have profound shifts in corporations, markets, systems and the world. The fractal theory introduced in this book indicates how these complex structures can be holistically perceived and correspondingly shifted. It presents the ideas through reader-friendly figures and tables for better understanding.
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$23.09 |
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 (2.0 / 5.0) We are surrounded by order that-until now-physics has been unable to explain.
The spread of veins in the back of our hands mirrors the spread of branches on a tree; fern fronds bear a resemblance to the outline of fjords; the best-loved classical music echoes the patterns of our heartbeats. The theory of Universality is using fractal patterns to explain much of the world around us. Could it be that the same laws that govern systems in their critical states also govern some of the most unpredictable events such as earthquakes, avalanches, the growth of cities and stock market crashes-even the way businesses are run and the way fashions come and go? Is there a common principle, a universal affinity that binds us to the forces of nature? A consensus is emerging on how complex structures grow and sustain themselves; phenomena that were once thought to be unique now appear to have a great deal in common. Mark Ward examines these theories, explores how they fit into an age-long quest to discover how the universe works, delves into their possible limitations and asks what we can do with this new knowledge. While identifying patterns does not mean that we can always predict what will happen next, some of the trends scientists are noticing prove that life is not a series of random events. Universality deepens our understanding of natural phenomena and our place in the physical world. We are surrounded by order that-until now-physics has been unable to explain. The spread of veins in the back of our hands mirrors the spread of branches on a tree; fern fronds bear a resemblance to the outline of fjords; the best-loved classical music echoes the patterns of our heartbeats. The theory of Universality is using fractal patterns to explain much of the world around us. Could it be that the same laws that govern systems in their critical states also govern some of the most unpredictable events such as earthquakes, avalanches, the growth of cities and stock market crashes-even the way businesses are run and the way fashions come and go? Is there a common principle, a universal affinity that binds us to the forces of nature? A consensus is emerging on how complex structures grow and sustain themselves; phenomena that were once thought to be unique now appear to have a great deal in common. Mark Ward examines these theories, explores how they fit into an age-long quest to discover how the universe works, delves into their possible limitations and asks what we can do with this new knowledge. While identifying patterns does not mean that we can always predict what will happen next, some of the trends scientists are noticing prove that life is not a series of random events. Universality deepens our understanding of natural phenomena and our place in the physical world.
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$9.50 |
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Almost all real systems are nonlinear. For a nonlinear system the superposition principle breaks down: The system's response is not proportional to the stimulus it receives; the whole is more than the sum of its parts. The three parts of this book contains the basics of nonlinear science, with applications in physics. Part I contains an overview of fractals, chaos, solitons, pattern formation, cellular automata and complex systems. In Part II, 14 reviews and essays by pioneers, as well as 10 research articles are reprinted. Part III collects 17 students projects, with computer algorithms for simulation models included. The book can be used for self-study, as a textbook for a one-semester course, or as supplement to other courses in linear of nonlinear systems. The reader should have some knowledge in introductory college physics. No mathematics beyond calculus and no computer literacy are assumed.
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$50.59 |
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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. Key Features: - The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings - Complex dimensions of a fractal string, defined as the poles of an associated zeta function, are studied in detail, then 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 dynamic zeta functions associated with a fractal - Examples of such formulas include Prime Orbit Theorem with error term for self-similar flows, and a 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 Throughout new results are examined. The final chapter gives a new definition of fractality as the presence of nonreal complex dimensions with positive real parts. The significant studies and problems illuminated in this work may be used in a classroom setting at the graduate level. Fractal Geometry, Complex Dimensions and Zeta Functions will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics.
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$19.22 |
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 (4.5 / 5.0) Jane Smythe, a math professor specializing in fractal geometry, is shocked to learn that three professors with the same speciality have died amid mysterious circumstances. That's where Pepper Keane, an ex-Marine turned PI with an encyclopedic knowledge of rock 'n' roll, comes in. He finds himself attracted to Professor Smythe and is determined to discover the root of these incidents. At first, he can't find any evidence that the three dead mathematicians even knew each other. But Keane, with the help of his hacker best friend and exercise guru brother, continues to dig. Suspects begin to appear and then multiply as they race through the rocky terrain of Colorado to Mexico, Boston, and Nebraska - with the main suspect an FBI agent who is also Keane's worst enemy.
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$8.50 |
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(4.5 / 5.0) Benoit Mandelbrot discovered what is now called the M-Set in the early seventies and coined the term fractal to describe the geometry behind it. The power and the beauty of fractals were only capable of being seen with the advent of computers, which become psychedelic windows on the infinite when using simple fractal equations. In 1992 Nigel Lesmoir-Gordon made the TV documentary, The Colors of Infinity about the Mandelbrot Set and fractals, which has since been seen right round the world. Nigel s enthusiasm brought together a dream team of contributors for the film who all now contribute to the book tracking how fractals have developed since the film was made. Sir Arthur C Clarke presented the film and in the book gives a lucidly simple account of the mathematics of the M-Set. Benoit Mandelbrot, the Belgian mathematician explains how it began. Professor Michael Barnsley, the computer graphics researcher who developed fractal image compression technology, explains the applications of the breakthroughs. Professor Ian Stewart, author of Does God Play Dice? adds his insights into the beautifully simple equation that gives birth to fractals. Two of the most interesting applications of fractal geometry, reflected by the two new contributors to the book, are to the Internet and to the Stock Market. Dr Gary Flake, Chief Technology Officer at Overture, the leading provider of commercial search on the Internet and just taken over by Yahoo for 1.6 billion dollars, discusses the profoundly fractal nature of the Web in his article: The Self-ish Web. Robert Prechter Jr is President of Elliott Wave International, Inc. and founder of the Socionomics Institute. His latest title is Socionomics: The Science of History and Social Prediction (2003). He writes about how fractals can help us understand the oscillations of stock markets. In the back of the book is a DVD of the original documentary with soundtrack by David Gilmour of Pink Floyd PLUS a 30-minute fractal animation to the music of members of Quintessence.
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$89.00 |
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(4.5 / 5.0) The authors present an unusual attempt to publicize the field of Complex Dynamics, an exciting mathematical discipline of respectable tradition that recently sprang into new life under the impact of modern computer graphics. Where previous generations of scientists had to develop their own inner eye to perceive the abstract aesthetics of their work, the astonding pictures assembled here invite the reader to share in a new mathematical experience, to revel in the charm of fractal frontiers. 184 illustrations in 211 parts, 88 in color.
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$32.94 |
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This book is for the student to gain a basic understanding o fpattern grading. While the basic concepts of grading are simple, there are intircacies of grading which take professional expertise and many years of epxerience to learn. In this book, the student will learn the essentials of grading patterns.
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$45.00 |
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Differential Equations on Fractals opens the door to understanding the recently developed area of analysis on fractals, focusing on the construction of a Laplacian on the Sierpinski gasket and related fractals. Written in a lively and informal style, with lots of intriguing exercises on all levels of difficulty, the book is accessible to advanced undergraduates, graduate students, and mathematicians who seek an understanding of analysis on fractals. Robert Strichartz takes the reader to the frontiers of research, starting with carefully motivated examples and constructions. One of the great accomplishments of geometric analysis in the nineteenth and twentieth centuries was the development of the theory of Laplacians on smooth manifolds. But what happens when the underlying space is rough? Fractals provide models of rough spaces that nevertheless have a strong structure, specifically self-similarity. Exploiting this structure, researchers in probability theory in the 1980s were able to prove the existence of Brownian motion, and therefore of a Laplacian, on certain fractals. An explicit analytic construction was provided in 1989 by Jun Kigami. Differential Equations on Fractals explains Kigami's construction, shows why it is natural and important, and unfolds many of the interesting consequences that have recently been discovered. This book can be used as a self-study guide for students interested in fractal analysis, or as a textbook for a special topics course.
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$27.00 |