Computability Of Julia Sets
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Computability Of Julia Sets
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Author : Mark Braverman
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-02-08
Computability Of Julia Sets written by Mark Braverman and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-02-08 with Computers categories.
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.
Turing S Legacy
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Author : Rod Downey
language : en
Publisher: Cambridge University Press
Release Date : 2014-05
Turing S Legacy written by Rod Downey and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-05 with Biography & Autobiography categories.
A collection of essays celebrating the influence of Alan Turing's work in logic, computer science and related areas.
Computability And Complexity Of Julia Sets
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Author : Mark Braverman
language : en
Publisher:
Release Date : 2008
Computability And Complexity Of Julia Sets written by Mark Braverman and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Computer science categories.
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.
Handbook Of Computability And Complexity In Analysis
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Author : Vasco Brattka
language : en
Publisher: Springer Nature
Release Date : 2021-06-04
Handbook Of Computability And Complexity In Analysis written by Vasco Brattka and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-06-04 with Computers categories.
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.
Proceedings Of The Acm Symposium On Theory Of Computing
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Author :
language : en
Publisher:
Release Date : 2007
Proceedings Of The Acm Symposium On Theory Of Computing written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with Formal languages categories.
Computability Theory And Its Applications
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Author : Peter Cholak
language : en
Publisher: American Mathematical Soc.
Release Date : 2000
Computability Theory And Its Applications written by Peter Cholak and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Mathematics categories.
This collection of articles presents a snapshot of the status of computability theory at the end of the millennium and a list of fruitful directions for future research. The papers represent the works of experts in the field who were invited speakers at the AMS-IMS-SIAM 1999 Summer Conference on Computability Theory and Applications, which focused on open problems in computability theory and on some related areas in which the ideas, methods, and/or results of computability theory play a role. Some presentations are narrowly focused; others cover a wider area. Topics included from "pure" computability theory are the computably enumerable degrees (M. Lerman), the computably enumerable sets (P. Cholak, R. Soare), definability issues in the c.e. and Turing degrees (A. Nies, R. Shore) and other degree structures (M. Arslanov, S. Badaev and S. Goncharov, P. Odifreddi, A. Sorbi). The topics involving relations between computability and other areas of logic and mathematics are reverse mathematics and proof theory (D. Cenzer and C. Jockusch, C. Chong and Y. Yang, H. Friedman and S. Simpson), set theory (R. Dougherty and A. Kechris, M. Groszek, T. Slaman) and computable mathematics and model theory (K. Ambos-Spies and A. Kucera, R. Downey and J. Remmel, S. Goncharov and B. Khoussainov, J. Knight, M. Peretyat'kin, A. Shlapentokh).
The Best Writing On Mathematics 2014
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Author : Mircea Pitici
language : en
Publisher: Princeton University Press
Release Date : 2015
The Best Writing On Mathematics 2014 written by Mircea Pitici and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015 with Mathematics categories.
An anthology of the year's finest writing on mathematics from around the world, featuring promising new voices as well as some of the foremost names in mathematics.
Recent Developments In Fractal Geometry And Dynamical Systems
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Author : Sangita Jha
language : en
Publisher: American Mathematical Society
Release Date : 2024-04-18
Recent Developments In Fractal Geometry And Dynamical Systems written by Sangita Jha and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-04-18 with Mathematics categories.
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.
Models And Computability
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Author : S. Barry Cooper
language : en
Publisher: Cambridge University Press
Release Date : 1999-06-17
Models And Computability written by S. Barry Cooper and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999-06-17 with Computers categories.
Second of two volumes providing a comprehensive guide to the current state of mathematical logic.
Computability In Context
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Author : S. Barry Cooper
language : en
Publisher: World Scientific
Release Date : 2011
Computability In Context written by S. Barry Cooper and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Computers categories.
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.