Related papers: The hat polykite as an Iterated Function System
In this paper we present a systematic study of continuous local iterated function systems. We prove local iterated function systems admit compact attractors and, under a contractivity assumption, construct their code space and present an…
The aim of this paper is to summarize some recently obtained relations between the Ablowitz-Ladik hierarchy (ALH) and other integrable equations. It has been shown that solutions of finite subsystems of the ALH can be used to derive a wide…
This paper discusses, certain algebraic, analytic, and topological results on partial iterated function systems($IFS_p$'s). Also, the article proves the Collage theorem for partial iterated function systems. Further, it provides a method to…
Complex continuous or mixed joint distributions (e.g., P(Y | z_1, z_2, ..., z_N)) generally lack closed-form solutions, often necessitating approximations such as MCMC. This paper proposes Indeterminate Probability Theory (IPT), which makes…
A combinatorial substitution is a map over tilings which allows to define sets of tilings with a strong hierarchical structure. In this paper, we show that such sets of tilings are sofic, that is, can be enforced by finitely many local…
The Ablowitz-Ladik hierarchy (ALH) is considered in the framework of the inverse scattering approach. After establishing the structure of solutions of the auxiliary linear problems, the ALH, which has been originally introduced as an…
Chain-of-Thought (CoT) reasoning is known to improve Large Language Models both empirically and in terms of theoretical approximation power. However, our understanding of the inner workings and conditions of apparition of CoT capabilities…
We present here an elementary construction of an aperiodic tile set. Although there already exist dozens of examples of aperiodic tile sets we believe this construction introduces an approach that is different enough to be interesting and…
This paper considers the egodicity properties in iterated function systems. First, we will introduce chain mixing and chain transitive iterated function systems then some results and examples are presented to compare with these notions in…
This article presents an overview of the theory of integrable systems with symmetries, focusing on toric systems, semitoric systems, and their classifications via decorated polygons. We discuss certain one-parameter families of integrable…
We study tilings of the plane that combine strong properties of different nature: combinatorial and algorithmic. We prove existence of a tile set that accepts only quasiperiodic and non-recursive tilings. Our construction is based on the…
We consider linear iterated function systems (IFS) with a constant contraction ratio in the plane for which the "overlap set" $\Ok$ is finite, and which are "invertible" on the attractor $A$, the sense that there is a continuous surjection…
In the present paper, as we did previously in [7], we investigate the relations between the geometric properties of tilings and the algebraic properties of associated relational structures. Our study is motivated by the existence of…
Prolog's ability to return multiple answers on backtracking provides an elegant mechanism to derive reversible encodings of combinatorial objects as Natural Numbers i.e. {\em ranking} and {\em unranking} functions. Starting from a…
Classical results on aperiodic tilings are rather complicated and not widely understood. Below, an alternative approach is discussed in hope to provide additional intuition not apparent in classical works.
In this paper an iterated function system on the space of distribution functions is built. The inverse problem is introduced and studied by convex optimization problems. Some applications of this method to approximation of distribution…
A new family of decagonal quasiperiodic tilings are constructed by the use of generalized point substitution processes, which is a new substitution formalism developed by the author [N. Fujita, Acta Cryst. A 65, 342 (2009)]. These tilings…
A new mathematical notation is proposed for the iteration of functions. It facilitates the application of the iteration of functions in mathematical and logical expressions, definitions of sets, and formulations of algorithms. Illustrations…
In this paper, we propose an Integer Linear Model whose solutions are the aperiodic rhythms tiling with a given rhythm A. We show how this model can be used to efficiently check the necessity of the Coven-Meyerowitz's $(T2)$ condition and…
Functors with an instance of the Traversable type class can be thought of as data structures which permit a traversal of their elements. This has been made precise by the correspondence between traversable functors and finitary containers…