Related papers: Translational tilings: structured or wild?
We address the challenging problem of Natural Language Comprehension beyond plain-text documents by introducing the TILT neural network architecture which simultaneously learns layout information, visual features, and textual semantics.…
Recent breakthroughs in large language models (LLM) have stirred up global attention, and the research has been accelerating non-stop since then. Philosophers and psychologists have also been researching the structure of language for…
Complex systems are characterized by specific time-dependent interactions among their many constituents. As a consequence they often manifest rich, non-trivial and unexpected behavior. Examples arise both in the physical and non-physical…
Machine translation (MT) plays an important role in benefiting linguists, sociologists, computer scientists, etc. by processing natural language to translate it into some other natural language. And this demand has grown exponentially over…
We define a moduli space of translation structures on the open topological disk with a basepoint and endow it with a locally-compact metrizable topology. We call this the immersive topology, because it is defined using the concept of…
Translation surfaces can be defined in an elementary way via polygons, and arise naturally in in the study of various basic dynamical systems. They can also be defined as Abelian differentials on Riemann surfaces, and have moduli spaces…
We investigate the dynamics of substitution subshifts and their associated tiling spaces. For a given subshift, the associated tiling spaces are all homeomorphic, but their dynamical properties may differ. We give criteria for such a tiling…
Statistical Topology emerged since topological aspects continue to gain importance in many areas of physics. It is most desirable to study topological invariants and their statistics in schematic models that facilitate the identification of…
Syntactic theory has traditionally adopted a constructivist approach, in which a set of atomic elements are manipulated by combinatory operations to yield derived, complex elements. Syntactic structure is thus seen as the result or discrete…
Is there a fixed dimension $n$ such that translational tiling of $\mathbb{Z}^n$ with a monotile is undecidable? Several recent results support a positive answer to this question. Greenfeld and Tao disprove the periodic tiling conjecture by…
In this review, we examine the problem of designing interpretable and explainable machine learning models. Interpretability and explainability lie at the core of many machine learning and statistical applications in medicine, economics,…
When $\mathbb{Z}^d$ is represented as a finite disjoint union of translated integer sublattices, the translated sublattices must possess some special properties. Such a representation is called a \emph{lattice tiling}. We develop a…
Tilt models offer intuitive and clean definitions of complex systems in which particles are influenced by global control commands. Despite a wide range of applications, there has been almost no theoretical investigation into the associated…
There are many dimensions of software complexity. In this article, we explore how structural complexity is measured and used to study and control evolving software systems. We also present the current research challenges and emerging trends…
A tessellation or tiling is a collection of sets, called tiles, that cover a plane without gaps and overlaps. The present note is an invitation to get to know the beauty and majesty of tessellations and triangulation of orientable surfaces.
To present a survey on known results from the theory of transposed Poisson algebras, as well as to establish new results on this subject, are the main aims of the present paper. Furthermore, a list of open questions for future research is…
Differentials on Riemann surfaces correspond to translation surfaces with conical singularities, and affine transformations acting on them preserve the orders of these singularities. This viewpoint allows the moduli spaces of differentials…
This paper presents a detailed symbolic approach to the study of self-similar tilings. It uses properties of addresses associated with graph-directed iterated function systems to establish conjugacy properties of tiling spaces. Tiles may be…
This paper focuses on the undecidability of translational tiling of $n$-dimensional space $\mathbb{Z}^n$ with a set of $k$ tiles. It is known that tiling $\mathbb{Z}^2$ with translated copies with a set of $8$ tiles is undecidable.…
Multidimensional combinatorial substitutions are rules that replace symbols by finite patterns of symbols in $\mathbb Z^d$. We focus on the case where the patterns are not necessarily rectangular, which requires a specific description of…