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Computability Of Julia Sets
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Book Synopsis Computability of Julia Sets by : Mark Braverman
Download or read book Computability of Julia Sets written by Mark Braverman and published by Springer Science & Business Media. This book was released on 2009-02-08 with total page 158 pages. Available in PDF, EPUB and Kindle. Book excerpt: Among all computer-generated mathematical images, Julia sets of rational maps occupy one of the most prominent positions. Their beauty and complexity can be fascinating. They also hold a deep mathematical content. Computational hardness of Julia sets is the main subject of this book. By definition, a computable set in the plane can be visualized on a computer screen with an arbitrarily high magnification. There are countless programs to draw Julia sets. Yet, as the authors have discovered, it is possible to constructively produce examples of quadratic polynomials, whose Julia sets are not computable. This result is striking - it says that while a dynamical system can be described numerically with an arbitrary precision, the picture of the dynamics cannot be visualized. The book summarizes the present knowledge (most of it from the authors' own work) about the computational properties of Julia sets in a self-contained way. It is accessible to experts and students with interest in theoretical computer science or dynamical systems.
Book Synopsis Computability and Complexity of Julia Sets by : Mark Braverman
Download or read book Computability and Complexity of Julia Sets written by Mark Braverman and published by . This book was released on 2008 with total page 390 pages. Available in PDF, EPUB and Kindle. Book excerpt: The field of study in the thesis is Real Computation, and Computability and Complexity in Dynamical Systems. The thesis consists of two parts. The first part is devoted to results on general computation of functions and sets over the reals, while in the second part results about computability and complexity in Complex Dynamics are presented. Studying computability and complexity over the reals is important for understanding the relationship between nature and computing, and for providing theoretical backing to fundamental problems in Numerical Analysis. In the first part of the thesis results unifying two most commonly used models of computation are presented. One is the model of Computable Analysis that is based on rational approximations of continuous objects, such as real functions and sets [Grz55, Ko91, Wei00]. The other is the Blum-Shub-Smale (BSS) model that is based on precise algebraic operations [BCSS98]. Insights obtained through this connection are then used to extend the computational complexity notion to some discontinuous functions. In the second part of the thesis, results on the computability and complexity of Julia sets are presented. Julia sets arise in one-dimensional complex dynamics. They have been intensely studied in the past 100 years, and since the 1980s numerous programs have been written to produce images of the sets, both for research and for their aesthetic value. We have studied computational properties of Julia sets, obtaining a virtually complete classification of the computational properties of Julia sets as well as complexity results, that complement the previous empirical work on producing their images.
Book Synopsis Handbook of Computability and Complexity in Analysis by : Vasco Brattka
Download or read book Handbook of Computability and Complexity in Analysis written by Vasco Brattka and published by Springer Nature. This book was released on 2021-06-04 with total page 427 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computable analysis is the modern theory of computability and complexity in analysis that arose out of Turing's seminal work in the 1930s. This was motivated by questions such as: which real numbers and real number functions are computable, and which mathematical tasks in analysis can be solved by algorithmic means? Nowadays this theory has many different facets that embrace topics from computability theory, algorithmic randomness, computational complexity, dynamical systems, fractals, and analog computers, up to logic, descriptive set theory, constructivism, and reverse mathematics. In recent decades computable analysis has invaded many branches of analysis, and researchers have studied computability and complexity questions arising from real and complex analysis, functional analysis, and the theory of differential equations, up to (geometric) measure theory and topology. This handbook represents the first coherent cross-section through most active research topics on the more theoretical side of the field. It contains 11 chapters grouped into parts on computability in analysis; complexity, dynamics, and randomness; and constructivity, logic, and descriptive complexity. All chapters are written by leading experts working at the cutting edge of the respective topic. Researchers and graduate students in the areas of theoretical computer science and mathematical logic will find systematic introductions into many branches of computable analysis, and a wealth of information and references that will help them to navigate the modern research literature in this field.
Book Synopsis Computability and Complexity by : Neil D. Jones
Download or read book Computability and Complexity written by Neil D. Jones and published by MIT Press. This book was released on 1997 with total page 494 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computability and complexity theory should be of central concern to practitioners as well as theorists. Unfortunately, however, the field is known for its impenetrability. Neil Jones's goal as an educator and author is to build a bridge between computability and complexity theory and other areas of computer science, especially programming. In a shift away from the Turing machine- and G�del number-oriented classical approaches, Jones uses concepts familiar from programming languages to make computability and complexity more accessible to computer scientists and more applicable to practical programming problems. According to Jones, the fields of computability and complexity theory, as well as programming languages and semantics, have a great deal to offer each other. Computability and complexity theory have a breadth, depth, and generality not often seen in programming languages. The programming language community, meanwhile, has a firm grasp of algorithm design, presentation, and implementation. In addition, programming languages sometimes provide computational models that are more realistic in certain crucial aspects than traditional models. New results in the book include a proof that constant time factors do matter for its programming-oriented model of computation. (In contrast, Turing machines have a counterintuitive "constant speedup" property: that almost any program can be made to run faster, by any amount. Its proof involves techniques irrelevant to practice.) Further results include simple characterizations in programming terms of the central complexity classes PTIME and LOGSPACE, and a new approach to complete problems for NLOGSPACE, PTIME, NPTIME, and PSPACE, uniformly based on Boolean programs. Foundations of Computing series
Book Synopsis The Computational Beauty of Nature by : Gary William Flake
Download or read book The Computational Beauty of Nature written by Gary William Flake and published by MIT Press. This book was released on 2000-01-27 with total page 524 pages. Available in PDF, EPUB and Kindle. Book excerpt: Gary William Flake develops in depth the simple idea that recurrent rules can produce rich and complicated behaviors. In this book Gary William Flake develops in depth the simple idea that recurrent rules can produce rich and complicated behaviors. Distinguishing "agents" (e.g., molecules, cells, animals, and species) from their interactions (e.g., chemical reactions, immune system responses, sexual reproduction, and evolution), Flake argues that it is the computational properties of interactions that account for much of what we think of as "beautiful" and "interesting." From this basic thesis, Flake explores what he considers to be today's four most interesting computational topics: fractals, chaos, complex systems, and adaptation. Each of the book's parts can be read independently, enabling even the casual reader to understand and work with the basic equations and programs. Yet the parts are bound together by the theme of the computer as a laboratory and a metaphor for understanding the universe. The inspired reader will experiment further with the ideas presented to create fractal landscapes, chaotic systems, artificial life forms, genetic algorithms, and artificial neural networks.
Book Synopsis Recent Developments in Fractal Geometry and Dynamical Systems by : Sangita Jha
Download or read book Recent Developments in Fractal Geometry and Dynamical Systems written by Sangita Jha and published by American Mathematical Society. This book was released on 2024-04-18 with total page 270 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the virtual AMS Special Session on Fractal Geometry and Dynamical Systems, held from May 14–15, 2022. The content covers a wide range of topics. It includes nonautonomous dynamics of complex polynomials, theory and applications of polymorphisms, topological and geometric problems related to dynamical systems, and also covers fractal dimensions, including the Hausdorff dimension of fractal interpolation functions. Furthermore, the book contains a discussion of self-similar measures as well as the theory of IFS measures associated with Bratteli diagrams. This book is suitable for graduate students interested in fractal theory, researchers interested in fractal geometry and dynamical systems, and anyone interested in the application of fractals in science and engineering. This book also offers a valuable resource for researchers working on applications of fractals in different fields.
Book Synopsis Computable Structure Theory by : Antonio Montalbán
Download or read book Computable Structure Theory written by Antonio Montalbán and published by Cambridge University Press. This book was released on 2021-06-24 with total page 214 pages. Available in PDF, EPUB and Kindle. Book excerpt: In mathematics, we know there are some concepts - objects, constructions, structures, proofs - that are more complex and difficult to describe than others. Computable structure theory quantifies and studies the complexity of mathematical structures, structures such as graphs, groups, and orderings. Written by a contemporary expert in the subject, this is the first full monograph on computable structure theory in 20 years. Aimed at graduate students and researchers in mathematical logic, it brings new results of the author together with many older results that were previously scattered across the literature and presents them all in a coherent framework, making it easier for the reader to learn the main results and techniques in the area for application in their own research. This volume focuses on countable structures whose complexity can be measured within arithmetic; a forthcoming second volume will study structures beyond arithmetic.
Book Synopsis Hilbert’s Tenth Problem: An Introduction to Logic, Number Theory, and Computability by : M. Ram Murty
Download or read book Hilbert’s Tenth Problem: An Introduction to Logic, Number Theory, and Computability written by M. Ram Murty and published by American Mathematical Soc.. This book was released on 2019-05-09 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: Hilbert's tenth problem is one of 23 problems proposed by David Hilbert in 1900 at the International Congress of Mathematicians in Paris. These problems gave focus for the exponential development of mathematical thought over the following century. The tenth problem asked for a general algorithm to determine if a given Diophantine equation has a solution in integers. It was finally resolved in a series of papers written by Julia Robinson, Martin Davis, Hilary Putnam, and finally Yuri Matiyasevich in 1970. They showed that no such algorithm exists. This book is an exposition of this remarkable achievement. Often, the solution to a famous problem involves formidable background. Surprisingly, the solution of Hilbert's tenth problem does not. What is needed is only some elementary number theory and rudimentary logic. In this book, the authors present the complete proof along with the romantic history that goes with it. Along the way, the reader is introduced to Cantor's transfinite numbers, axiomatic set theory, Turing machines, and Gödel's incompleteness theorems. Copious exercises are included at the end of each chapter to guide the student gently on this ascent. For the advanced student, the final chapter highlights recent developments and suggests future directions. The book is suitable for undergraduates and graduate students. It is essentially self-contained.
Book Synopsis Computable Analysis by : Klaus Weihrauch
Download or read book Computable Analysis written by Klaus Weihrauch and published by Springer Science & Business Media. This book was released on 2000-09-14 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: Merging fundamental concepts of analysis and recursion theory to a new exciting theory, this book provides a solid fundament for studying various aspects of computability and complexity in analysis. It is the result of an introductory course given for several years and is written in a style suitable for graduate-level and senior students in computer science and mathematics. Many examples illustrate the new concepts while numerous exercises of varying difficulty extend the material and stimulate readers to work actively on the text.
Book Synopsis Computability in Context by : S. Barry Cooper
Download or read book Computability in Context written by S. Barry Cooper and published by World Scientific. This book was released on 2011 with total page 419 pages. Available in PDF, EPUB and Kindle. Book excerpt: Addresses various ways computability and theoretical computer science enable scientists and philosophers to deal with mathematical and real-world issues. This book covers problems related to logic, mathematics, physical processes, real computation and learning theory.
Book Synopsis Theory and Formal Methods 1993 by : Geoffrey Burn
Download or read book Theory and Formal Methods 1993 written by Geoffrey Burn and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 333 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the fIrst workshop held by the Theory and Formal Methods Section ofthe Imperial College Department of Computing. It contains papers from almost every member of the Section, from our long-term academic visitors, and from those who have recently left us. The papers fall into four broad areas: • semantics • concurrency • logic • specification with some papers spanning a number of disciplines. The subject material varies from work on mathematical foundations to practical applications of this theory, expressing the Section's commitment to both the foundations of computer science, and the application of theory to real computing problems. In preparing the workshop and these proceedings, care was taken to ensure that there were papers overviewing a field, as well as ones whose primary aim was to present new scientifIc results. This had a dual purpose: to bring our Section members up to speed in some of the areas being worked on by the Section; and to provide the reader of the proceedings not only with a good introduction to many of the specifIc areas being investigated by the Section, but also with details of some of our latest results. All the papers presented at the workshop were revised following comments made by the workshop participants, and all were subsequently reviewed by at least two people before producing the fInal versions contained in this volume.
Book Synopsis Complex Dynamics by : Lennart Carleson
Download or read book Complex Dynamics written by Lennart Carleson and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 181 pages. Available in PDF, EPUB and Kindle. Book excerpt: A discussion of the properties of conformal mappings in the complex plane, closely related to the study of fractals and chaos. Indeed, the book ends in a detailed study of the famous Mandelbrot set, which describes very general properties of such mappings. Focusing on the analytic side of this contemporary subject, the text was developed from a course taught over several semesters and aims to help students and instructors to familiarize themselves with complex dynamics. Topics covered include: conformal and quasi-conformal mappings, fixed points and conjugations, basic rational iteration, classification of periodic components, critical points and expanding maps, some applications of conformal mappings, the local geometry of the Fatou set, and quadratic polynomials and the Mandelbrot set.
Book Synopsis The Best Writing on Mathematics 2014 by : Mircea Pitici
Download or read book The Best Writing on Mathematics 2014 written by Mircea Pitici and published by Princeton University Press. This book was released on 2014-11-23 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: Collects essays on mathematics, from the importance of big data and the varying forms of Klein bottles to fountain design and the possible shape of the universe.
Book Synopsis Fractals, Wavelets, and their Applications by : Christoph Bandt
Download or read book Fractals, Wavelets, and their Applications written by Christoph Bandt and published by Springer. This book was released on 2014-09-27 with total page 508 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fractals and wavelets are emerging areas of mathematics with many common factors which can be used to develop new technologies. This volume contains the selected contributions from the lectures and plenary and invited talks given at the International Workshop and Conference on Fractals and Wavelets held at Rajagiri School of Engineering and Technology, India from November 9-12, 2013. Written by experts, the contributions hope to inspire and motivate researchers working in this area. They provide more insight into the areas of fractals, self similarity, iterated function systems, wavelets and the applications of both fractals and wavelets. This volume will be useful for the beginners as well as experts in the fields of fractals and wavelets.
Author :Mark Braverman Publisher :Library and Archives Canada = Bibliothèque et Archives Canada ISBN 13 :9780612952645 Total Pages :180 pages Book Rating :4.9/5 (526 download)
Book Synopsis Computational Complexity of Euclidean Sets [microform] : Hyperbolic Julia Sets are Poly-time Computable by : Mark Braverman
Download or read book Computational Complexity of Euclidean Sets [microform] : Hyperbolic Julia Sets are Poly-time Computable written by Mark Braverman and published by Library and Archives Canada = Bibliothèque et Archives Canada. This book was released on 2004 with total page 180 pages. Available in PDF, EPUB and Kindle. Book excerpt: We apply the concepts developed to show that hyperbolic Julia sets are polynomial time computable. This result is a significant generalization of the result in [RW03], where polynomial time computability has been shown for a restricted type of hyperbolic Julia sets. We investigate different definitions of the computability and complexity of sets in Rk, and establish new connections between these definitions. This allows us to connect the computability of real functions and real sets in a new way. We show that equivalence of some of the definitions corresponds to equivalence between famous complexity classes. The model we use is mostly consistent with [Wei00].
Book Synopsis Holomorphic Dynamics and Renormalization by : Mikhail Lyubich
Download or read book Holomorphic Dynamics and Renormalization written by Mikhail Lyubich and published by American Mathematical Soc.. This book was released on 2008 with total page 408 pages. Available in PDF, EPUB and Kindle. Book excerpt: Collects papers that reflect some of the directions of research in two closely related fields: Complex Dynamics and Renormalization in Dynamical Systems. This title contains papers that introduces the reader to this fascinating world and a related area of transcendental dynamics. It also includes open problems and computer simulations.
Book Synopsis What Can Be Computed? by : John MacCormick
Download or read book What Can Be Computed? written by John MacCormick and published by Princeton University Press. This book was released on 2018-05-01 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: An accessible and rigorous textbook for introducing undergraduates to computer science theory What Can Be Computed? is a uniquely accessible yet rigorous introduction to the most profound ideas at the heart of computer science. Crafted specifically for undergraduates who are studying the subject for the first time, and requiring minimal prerequisites, the book focuses on the essential fundamentals of computer science theory and features a practical approach that uses real computer programs (Python and Java) and encourages active experimentation. It is also ideal for self-study and reference. The book covers the standard topics in the theory of computation, including Turing machines and finite automata, universal computation, nondeterminism, Turing and Karp reductions, undecidability, time-complexity classes such as P and NP, and NP-completeness, including the Cook-Levin Theorem. But the book also provides a broader view of computer science and its historical development, with discussions of Turing's original 1936 computing machines, the connections between undecidability and Gödel's incompleteness theorem, and Karp's famous set of twenty-one NP-complete problems. Throughout, the book recasts traditional computer science concepts by considering how computer programs are used to solve real problems. Standard theorems are stated and proven with full mathematical rigor, but motivation and understanding are enhanced by considering concrete implementations. The book's examples and other content allow readers to view demonstrations of—and to experiment with—a wide selection of the topics it covers. The result is an ideal text for an introduction to the theory of computation. An accessible and rigorous introduction to the essential fundamentals of computer science theory, written specifically for undergraduates taking introduction to the theory of computation Features a practical, interactive approach using real computer programs (Python in the text, with forthcoming Java alternatives online) to enhance motivation and understanding Gives equal emphasis to computability and complexity Includes special topics that demonstrate the profound nature of key ideas in the theory of computation Lecture slides and Python programs are available at whatcanbecomputed.com
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