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Constructing Non-Computable Julia Sets
[article]

2006
*
arXiv
*
pre-print

As a consequence, we

arXiv:math/0604371v1
fatcat:mrybqz4tivhjjjpsuylxjlfdgu
*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*. ...##
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Constructing non-computable Julia sets

2007
*
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing - STOC '07
*

We show how to

doi:10.1145/1250790.1250893
dblp:conf/stoc/BravermanY07
fatcat:s5xwtsfxbfhhvmxyxquexhljzi
*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. ...##
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Constructing Locally Connected Non-Computable Julia Sets

2009
*
Communications in Mathematical Physics
*

We

doi:10.1007/s00220-009-0858-5
fatcat:ce4ramq3azbchi367dmaf7vd3i
*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*. ...##
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Research Synergy of Multi-Stage Mathematical and Informational Tasks on Fractal Geometry

2017
*
US-China Education Review. A
*

The

doi:10.17265/2161-623x/2017.08.003
fatcat:rjzqbt4gvfd2tgpc7dqkliv4gm
*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. ...##
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Almost every real quadratic polynomial has a poly-time computable Julia set
[article]

2017
*
arXiv
*
pre-print

We prove that Collet-Eckmann rational maps have poly-time

arXiv:1702.05768v2
fatcat:yggkxuwztjefjdfkxwt5ozlycq
*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. ...##
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On computability of Julia sets: answers to questions of Milnor and Shub
[article]

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

arXiv:math/0604175v1
fatcat:tijjyywdafhn3gwb7kenw7tjpa
*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. ...##
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Computability and Complexity of Julia Sets

2014
*
IEICE Proceeding Series
*

In this review paper, we introduce a recently developed

doi:10.15248/proc.2.2
fatcat:ewbeetz6bffopnmxduqxfh26qq
*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. ...##
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Computability and complexity of Julia sets: a review

2014
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Nonlinear Theory and Its Applications IEICE
*

In this review paper, we introduce a recently developed

doi:10.1587/nolta.5.410
fatcat:rjexh5kffngsbjbfs62w2hwwti
*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. ...##
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Non-Computable Impressions of Computable External Rays of Quadratic Polynomials

2014
*
Communications in Mathematical Physics
*

We discuss

doi:10.1007/s00220-014-2218-3
fatcat:ambirwf3pfbfdmobmi4wxxkyf4
*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] . ...##
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RELATIVE SUPERIOR MANDELBROT AND JULIA SETS FOR INTEGER AND NON-INTEGER VALUES

2012
*
International Journal of Research in Engineering and Technology
*

Superior

doi:10.15623/ijret.2012.0102011
fatcat:vquhqotdonadhkognufenq5aoy
*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 ...##
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Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays
[article]

2020
*
arXiv
*
pre-print

The

arXiv:1910.11848v3
fatcat:3okqlmuh2ncihjq2ha2vvk3ikm
*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. ...##
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Computability of Julia Sets

2008
*
Moscow Mathematical Journal
*

In this paper we settle most of the open questions on algorithmic

doi:10.17323/1609-4514-2008-8-2-185-231
fatcat:x6ymucv2m5dbxdqdak5v3snppi
*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. ...##
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Computable geometric complex analysis and complex dynamics
[article]

2017
*
arXiv
*
pre-print

As applications, we review the state of the art regarding

arXiv:1703.06459v1
fatcat:qxmigwp56fcmrpha4ubkvunxii
*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*. ...##
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Polymake.jl: A new interface to polymake
[article]

2020
*
arXiv
*
pre-print

We present the

arXiv:2003.11381v1
fatcat:syzpg6f2zreslmwfj6z226ap7q
*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*> ? ...##
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A New Approach in Fractals Models

2012
*
International Journal of Computer Applications
*

In colloquial usage, a fractal is a shape that is recursively

doi:10.5120/7517-9070
fatcat:wjodm7ijfrebdklejftgf332we
*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*. ...
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