Related papers: On computational complexity of Siegel Julia sets
We discuss a classical complexity of finite-dimensional unitary transformations, which can been seen as a computable approximation of classical descriptional complexity of a unitary transformation acting on a set of qubits.
Computational complexity is a new quantum information concept that may play an important role in holography and in understanding the physics of the black hole interior. We consider quantum computational complexity for $n$ qubits using…
We construct "large" Cantor sets whose complements resemble the unit disk arbitrarily well from the point of view of the squeezing function, and we construct "large" Cantor sets whose complements do not resemble the unit disk from the point…
For quadratic polynomials with an indifferent fixed point with bounded type rotation number (they have a Siegel disk), much of what is known of their Julia set comes from the study of a quasiconformal model. The model is build from a…
Genetic Programming (GP) has found various applications. Understanding this type of algorithm from a theoretical point of view is a challenging task. The first results on the computational complexity of GP have been obtained for problems…
We prove that a nonempty, proper subset $S$ of the complex plane can be approximated in a strong sense by polynomial filled Julia sets if and only if $S$ is bounded and $\hat{\mathbb{C}} \setminus \textrm{int}(S)$ is connected. The proof…
Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings…
The dynamics of all quadratic Newton maps of rational functions are completely described. The Julia set of such a map is found to be either a Jordan curve or totally disconnected. It is proved that no Newton map with degree at least three…
In studying the complexity of iterative processes it is usually assumed that the arithmetic operations of addition, multiplication, and division can be performed in certain constant times. This assumption is invalid if the precision…
We prove that there is no algorithm that can determine whether or not a finitely presented group has a non-trivial finite quotient; indeed, this remains undecidable among the fundamental groups of compact, non-positively curved square…
Computational problems are classified into computable and uncomputable problems. If there exists an effective procedure (algorithm) to compute a problem then the problem is computable otherwise it is uncomputable. Turing machines can…
Many combinatorial optimization problems are often considered intractable to solve exactly or by approximation. An example of such problem is maximum clique which -- under standard assumptions in complexity theory -- cannot be solved in…
Sofic shifts are symbolic dynamical systems defined by the set of bi-infinite sequences on an edge-labeled directed graph, called a presentation. We study the computational complexity of an array of natural decision problems about…
Special exotic class of dynamical systems~ -- the implicit maps~ -- is considered. Such maps, particularly, can appear as a result of using of implicit and semi-implicit iterative numerical methods. In the present work we propose the…
After a brief discussion of the computational complexity of Clifford algebras, we present a new basis for even Clifford algebra Cl(2m) that simplifies greatly the actual calculations and, without resorting to the conventional matrix…
In this paper we explore by means of the method of Lagrangian descriptors the Julia sets arising from complex maps, and we analyze their underlying dynamics. In particular, we take a look at two classical examples: the quadratic mapping…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
We prove several new rigidity results for polynomial automorphisms of $\mathbb C^2$ with positive entropy. A first result is that a complex slice of the (forward or backward) Julia set is never a smooth, or even rectifiable, curve. We also…
As it follows from G\"odel's incompleteness theorems, any consistent formal system of axioms and rules of inference should imply a true unprovable statement. Actually, this fundamental principle can be efficiently applicable in…
In general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of…