Filters








27,343 Hits in 2.4 sec

Constructing Non-Computable Julia Sets [article]

Mark Braverman, Michael Yampolsky
2006 arXiv   pre-print
As a consequence, we constructively produce quadratic polynomials with non-computable Julia sets.  ...  We completely characterize the conformal radii of Siegel disks in the family P_θ(z)=e^2π iθz+z^2, corresponding to computable parameters θ.  ...  Constructing non-computable Julia sets Theorem 4.1 provides us with a tool for constructing explicit parameters c for which J z 2 +c is non-computable.  ... 
arXiv:math/0604371v1 fatcat:mrybqz4tivhjjjpsuylxjlfdgu

Constructing non-computable Julia sets

Mark Braverman, Michael Yampolsky
2007 Proceedings of the thirty-ninth annual ACM symposium on Theory of computing - STOC '07  
We show how to construct a specific polynomial with a non-computable Julia set.  ...  The proof was non-constructive, and indeed there were doubts as to whether specific examples of parameters with non-computable Julia sets could be constructed.  ...  Acknowledgments We would like to thank John Milnor for posing the question of computability of filled Julia sets to us.  ... 
doi:10.1145/1250790.1250893 dblp:conf/stoc/BravermanY07 fatcat:s5xwtsfxbfhhvmxyxquexhljzi

Constructing Locally Connected Non-Computable Julia Sets

Mark Braverman, Michael Yampolsky
2009 Communications in Mathematical Physics  
We constructively produce parameter values for Siegel quadratics for which the Julia sets are non-computable, yet locally connected.  ...  A locally connected quadratic Siegel Julia set has a simple explicit topological model.  ...  A passage to the limit, carefully made, produces a parameter with a non-computable Julia set.  ... 
doi:10.1007/s00220-009-0858-5 fatcat:ce4ramq3azbchi367dmaf7vd3i

Research Synergy of Multi-Stage Mathematical and Informational Tasks on Fractal Geometry

Valery S. Sekovanov, Eugeny I. Smirnov
2017 US-China Education Review. A  
The Julia sets of quadratic polynomials are considered first and the algorithms of constructing the Julia sets are proposed.  ...  Attention to the filling Julia sets is given. The algorithms of constructing the Julia sets of rational functions are considered and the examples are given.  ...  Carrying out of mathematical and computer experiments; 3. Performance of laboratory work on mathematics; 4. Decision of non-standard mathematical problems; 5.  ... 
doi:10.17265/2161-623x/2017.08.003 fatcat:rjzqbt4gvfd2tgpc7dqkliv4gm

Almost every real quadratic polynomial has a poly-time computable Julia set [article]

Artem Dudko, Michael Yampolsky
2017 arXiv   pre-print
We prove that Collet-Eckmann rational maps have poly-time computable Julia sets. As a consequence, almost all real quadratic Julia sets are poly-time.  ...  The phenomenon of non-computability is quite rare, and "most" quadratic Julia sets are computable.  ...  The Julia set has non-empty interior if and only if it is equal to all ofĈ.  ... 
arXiv:1702.05768v2 fatcat:yggkxuwztjefjdfkxwt5ozlycq

On computability of Julia sets: answers to questions of Milnor and Shub [article]

Mark Braverman, Michael Yampolsky
2006 arXiv   pre-print
In this note we give answers to questions posed to us by J.Milnor and M.Shub, which shed further light on the structure of non-computable Julia sets.  ...  For a given polynomial p(z) we construct a machine computing the corresponding filled Julia set K p . We will use the following combinatorial information about p in the construction.  ...  This time, instead of filling in a non-computable Julia set, we will make a "fuzzy" picture of it by letting the parameter c vary in a neighborhood.  ... 
arXiv:math/0604175v1 fatcat:tijjyywdafhn3gwb7kenw7tjpa

Computability and Complexity of Julia Sets

Kota Hiratsuka, Yuzuru Sato, Zin Arai
2014 IEICE Proceeding Series  
In this review paper, we introduce a recently developed computability theory for Julia sets in complex dynamical systems by Braverman and Yampolsky [3].  ...  Turing introduced the notion of computability in 1936, various theories of real number computation have been studied [1][10][13].  ...  The class NP is also defined as a set of problems, which is polynomial time computable using a non-deterministic Turing machine.  ... 
doi:10.15248/proc.2.2 fatcat:ewbeetz6bffopnmxduqxfh26qq

Computability and complexity of Julia sets: a review

Kota Hiratsuka, Yuzuru Sato, Zin Arai
2014 Nonlinear Theory and Its Applications IEICE  
In this review paper, we introduce a recently developed computability theory for Julia sets in complex dynamical systems by Braverman and Yampolsky [3].  ...  Turing introduced the notion of computability in 1936, various theories of real number computation have been studied [1, 10, 13].  ...  The class NP is also defined as a set of problems, which is polynomial time computable using a non-deterministic Turing machine.  ... 
doi:10.1587/nolta.5.410 fatcat:rjexh5kffngsbjbfs62w2hwwti

Non-Computable Impressions of Computable External Rays of Quadratic Polynomials

Ilia Binder, Cristobal Rojas, Michael Yampolsky
2014 Communications in Mathematical Physics  
We discuss computability of impressions of prime ends of compact sets. In particular, we construct quadratic Julia sets which possess explicitly described non-computable impressions.  ...  Non locally connected computable Siegel Julia sets as well as Cremer Julia sets (which are always not locally connected and always computable [4] ) may potentially contain non-computable impressions.  ...  This non-computability phenomenon is quite subtle. In particular, the filled Julia set K c is computable [8] , and, moreover, the harmonic measure ω c of the Julia set is computable [3] .  ... 
doi:10.1007/s00220-014-2218-3 fatcat:ambirwf3pfbfdmobmi4wxxkyf4

RELATIVE SUPERIOR MANDELBROT AND JULIA SETS FOR INTEGER AND NON-INTEGER VALUES

Rajeshri Rana .
2012 International Journal of Research in Engineering and Technology  
Superior Julia Sets: We generate Relative Superior Julia sets.  ...  integer and non-integer  Geometrical analysis of the Relative Superior Julia sets of values of n have been presented by Gujar et al. [6, 7] along inverse function for non integer values  ... 
doi:10.15623/ijret.2012.0102011 fatcat:vquhqotdonadhkognufenq5aoy

Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays [article]

Alberto Paoluzzi, Vadim Shapiro, Antonio DiCarlo, Giorgio Scorzelli, Elia Onofri
2020 arXiv   pre-print
The computational evaluation of every possible solid expression, usually denoted as CSG (Constructive Solid Geometry), is reduced to an equivalent logical expression of a finite set algebra over the cells  ...  This method is implemented in Julia using sparse arrays.  ...  We have: : (+, A Computational examples This section provides the reader with some examples of solid modeling programming style with Julia and its sparse and non-sparse arrays.  ... 
arXiv:1910.11848v3 fatcat:3okqlmuh2ncihjq2ha2vvk3ikm

Computability of Julia Sets

M. Braverman, M. Yampolsky
2008 Moscow Mathematical Journal  
In this paper we settle most of the open questions on algorithmic computability of Julia sets. In particular, we present an algorithm for constructing quadratics whose Julia sets are uncomputable.  ...  We also show that a filled Julia set of a polynomial is always computable.  ...  We thank John Milnor for posing the question on computability of filled Julia sets to us.  ... 
doi:10.17323/1609-4514-2008-8-2-185-231 fatcat:x6ymucv2m5dbxdqdak5v3snppi

Computable geometric complex analysis and complex dynamics [article]

Cristobal Rojas, Michael Yampolsky
2017 arXiv   pre-print
As applications, we review the state of the art regarding computability and complexity of Julia sets, their invariant measures and external rays impressions.  ...  We discuss computability and computational complexity of conformal mappings and their boundary extensions.  ...  Computability of Julia sets.  ... 
arXiv:1703.06459v1 fatcat:qxmigwp56fcmrpha4ubkvunxii

Polymake.jl: A new interface to polymake [article]

Marek Kaluba, Benjamin Lorenz, Sascha Timme
2020 arXiv   pre-print
We present the Julia interface Polymake.jl to polymake, a software for research in polyhedral geometry.  ...  We describe the technical design and how the integration into Julia makes it possible to combine polymake with state-of-the-art numerical software.  ...  ::cmp> {{0 1 2 3 4}} julia> @convert_to Array{Set} K5.MAX_CLIQUES pm::Array<pm::Set<long, pm::operations::cmp> > {0 1 2 3 4} julia> ?  ... 
arXiv:2003.11381v1 fatcat:syzpg6f2zreslmwfj6z226ap7q

A New Approach in Fractals Models

Deepak Negi, Ashish Negi, Priti Dimri
2012 International Journal of Computer Applications  
In colloquial usage, a fractal is a shape that is recursively constructed or self-similar, that is, a shape that appears similar at all scales of magnification and is therefore often referred to as "infinitely  ...  Relative Superior Julia set of Q is boundary of Julia set RSK . Escape Criterion for Cubic' s [17] 1/ 1 1/ 1 max{| |,(2 / ) ,(2 / ) }.  ...  Definition 2.5 [3] function () c Qz is called Julia set.  ... 
doi:10.5120/7517-9070 fatcat:wjodm7ijfrebdklejftgf332we
« Previous Showing results 1 — 15 out of 27,343 results