Related papers: Small tile sets that compute while solving mazes
We present a construction of a family of non-periodic tilings using elementary tools such as modular arithmetic and vector geometry. These tilings exhibit a distinct type of structural regularity, which we term modulo-staggered rotational…
We show that every tiling of a convex set in the Euclidean plane $\mathbb{R}^2$ by equilateral triangles of mutually different sizes contains arbitrarily small tiles. The proof is purely elementary up to the discussion of one family of…
Self-assembly materials are traditionally designed so that molecular or meso-scale components form a single kind of large structure. Here, we propose a scheme to create "multifarious assembly mixtures", which self-assemble many different…
We establish a relationship between the Small-World behavior found in complex networks and a family of Random Walks trajectories using, as a linking bridge, a maze iconography. Simple methods to generate mazes using Random Walks are…
Natural and man-made transport webs are frequently dominated by dense sets of nested cycles. The architecture of these networks, as defined by the topology and edge weights, determines how efficiently the networks perform their function.…
In this paper, we study abstract dynamical systems with discrete phase spaces. One example of such a system is induced by the $3 x{+}1$-map on the set of all natural numbers, also known as the Collatz map. Our main focus is on dynamical…
Monte Carlo simulation studies are at the core of the modern applied, computational, and theoretical statistical literature. Simulation is a broadly applicable research tool, used to collect data on the relative performance of methods or…
An effort has been made to show mathematicians some new ideas applied to image analysis. Gray images are presented as tilings. Based on topological properties of the tiling, a number of gray convex hulls: maximal, minimal, and oriented ones…
Given a finite collection of two-dimensional tile types, the field of study concerned with covering the plane with tiles of these types exclusively has a long history, having enjoyed great prominence in the last six to seven decades. Much…
Natural frequencies and normal modes are basic properties of a structure which play important roles in analyses of its vibrational characteristics. As their computation reduces to solving eigenvalue problems, it is a natural arena for…
The recently introduced tile codes are a promising alternative to surface codes, combining two-dimensional locality with higher encoding efficiency. While surface codes are well understood in terms of their logical operators and boundary…
Minimal balanced collections are a generalization of partitions of a finite set of n elements and have important applications in cooperative game theory and discrete mathematics. However, their number is not known beyond n = 4. In this…
The algorithmic self-assembly of shapes has been considered in several models of self-assembly. For the problem of \emph{shape construction}, we consider an extended version of the Two-Handed Tile Assembly Model (2HAM), which contains…
The world of 2D materials is rapidly expanding with new discoveries of stackable and twistable layered systems composed of lattices of different symmetries, orbital character, and structural motifs. Often, however, it is not clear a priori…
This paper studies random lozenge tilings of general non-convex polygonal regions. We show that the pairwise interaction of the non-convexities leads asymptotically to new kernels and thus to new statistics for the tiling fluctuations. The…
Structural DNA nanotechnology has advanced to the extent that extremely complex structures can be designed. Much of this advancement has been due to the development of automated DNA design and simulation tools. Typically, the tools (e.g.…
Inspired by the challenge of scaling up existing silicon quantum hardware, we investigate compilation strategies for sparsely-connected 2d qubit arrangements and propose a spin-qubit architecture with minimal compilation overhead. Our…
Wang tile based representation of a heterogeneous material facilitates fast synthesis of non-periodic microstructure realizations. In this paper, we apply the tiling approach in numerical homogenization to determine the Representative…
General matrix multiplication (GEMM) is the computational backbone of modern AI workloads, and its efficiency is critically dependent on effective tiling strategies. Conventional approaches employ symmetric tile buffering, where the…
An approach that combines Self-Organizing maps, hierarchical clustering and network components is presented, aimed at comparing protein conformational ensembles obtained from multiple Molecular Dynamic simulations. As a first result the…