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Constructing Non-Computable Julia Sets
[article]
2006
arXiv
pre-print
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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
2007
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing - STOC '07
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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
2009
Communications in Mathematical Physics
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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
2017
US-China Education Review. A
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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]
2017
arXiv
pre-print
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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]
2006
arXiv
pre-print
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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
2014
IEICE Proceeding Series
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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
2014
Nonlinear Theory and Its Applications IEICE
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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
2014
Communications in Mathematical Physics
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2014 pre-print arXiv:1402.0440v1 fatcat:7iye4544pjdl7af3syx72onsnq
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
2012
International Journal of Research in Engineering and Technology
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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]
2020
arXiv
pre-print
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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
2008
Moscow Mathematical Journal
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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]
2017
arXiv
pre-print
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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]
2020
arXiv
pre-print
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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
2012
International Journal of Computer Applications
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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
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