Related papers: Computability of Julia sets
A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not…
We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We…
For the solvable polynomial algebras introduced and studied by Kandri-Rody and Weispfenning [J. Symbolic Comput., 9(1990)], a constructive characterization is given in terms of Gr\"obner bases for ideals of free algebras, thereby solvable…
In this paper, we consider iterative propositional calculi, which are finite sets of propositional formulas together with the rules of modus ponens and weak substitution (when formula being substituted must be already inferred). We…
Computations over the rational numbers often suffer from intermediate coefficient swell. One solution to this problem is to apply the given algorithm modulo a number of primes and then lift the modular results to the rationals. This method…
We prove that the Julia set $J(f)$ of at most finitely renormalizable unicritical polynomial $f:z\mapsto z^d+c$ with all periodic points repelling is locally connected. (For $d=2$ it was proved by Yoccoz around 1990.) It follows from a…
We discuss some claims that certain UCOMP devices can perform hypercomputation (compute Turing-uncomputable functions) or perform super-Turing computation (solve NP-complete problems in polynomial time). We discover that all these claims…
For any polynomial diffeomorphism $f$ of $\mathbb{C}^2$ with positive entropy, neither the Julia set of $f$ nor of its inverse $f^{-1}$ is $C^1$ smooth as a manifold-with-boundary.
Any Jordan curve in the complex plane can be approximated arbitrarily well in the Hausdorff topology by Julia sets of polynomials. Finite collections of disjoint Jordan domains can be approximated by the basins of attraction of rational…
A compact set has computable type if any homeomorphic copy of the set which is semicomputable is actually computable. Miller proved that finite-dimensional spheres have computable type, Iljazovi\'c and other authors established the property…
It is well known that many problems in interval computation are intractable, which restricts our attempts to solve large problems in reasonable time. This does not mean, however, that all problems are computationally hard. Identifying…
We study polynomials with coefficients in a field L as dynamical systems where L is any algebraically closed and complete ultrametric field with dense valuation group and characteristic zero residual field. We give a complete description of…
We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split…
We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $\Pi_{2^{m}}^*$ represent the set of polynomials computable with…
How to handle division in systems that compute with logical formulas involving what would otherwise be polynomial constraints over the real numbers is a surprisingly difficult question. This paper argues that existing approaches from both…
In this work we define an universal arithmetical algorithm, by means of the standard quantum mechanical formalism, called universal qm-arithmetical algorithm. By universal qm-arithmetical algorithm any decidable arithmetical formula…
The paper studies constructions of irreducible polynomials over finite fields using polynomial composition method.
We consider the dynamics of rational semigroups (semigroups of rational maps) on the Riemann sphere. We provide proof that a random backward iteration algorithm to draw the pictures of the Julia sets, previously proven to work in the…
We prove a number of elementary facts about computability in partial combinatory algebras (pca's). We disprove a suggestion made by Kreisel about using Friedberg numberings to construct extensional pca's. We then discuss separability and…
Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called \emph{dendritic}. By results of Kiwi, any dendritic polynomial is semi-conjugate to a topological polynomial whose topological Julia set…