Related papers: The tile assembly model is intrinsically universal
Let $X$ be any rational surface. We construct a tilting bundle $T$ on $X$. Moreover, we can choose $T$ in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on $X$ is…
A p-divisible group over a complete local domain determines a Galois representation on the Tate module of its generic fibre. We determine the image of this representation for the universal deformation in mixed characteristic of a…
The evolution of multicellular organisms from monocellular ancestors represents one of the greatest advances of the history of life. The assembly of such multicellular organisms requires signalling and response between cells: over millions…
We prove that an effective temperature naturally emerges from the algorithmic structure of a regular universal Turing machine (UTM), without introducing any external physical parameter. In particular, the redundancy growth of the machine's…
Experiments have reached a monumental capacity for designing and synthesizing microscopic particles for self-assembly, making it possible to precisely control particle concentrations, shapes, and interactions. However, more physical insight…
Self-assembly is a phenomenon observed in nature at all scales where autonomous entities build complex structures, without external influences nor centralised master plan. Modelling such entities and programming correct interactions among…
We introduce a new type of generalized Turing machines (GTMs), which are intended as a tool for the mathematician who studies computability in Analysis. In a single tape cell a GTM can store a symbol, a real number, a continuous real…
We use computational modeling to investigate the assembly thermodynamics of a particle-based model for geometrically frustrated assembly, in which the local packing geometry of subunits is incompatible with uniform, strain-free large-scale…
We completely characterize the condition when a tile structure provides an unextendible product basis (UPB), and construct UPBs of different large sizes in $\mathbb{C}^m\otimes\mathbb{C}^n$ for any $n\geq m\geq 3$. This solves an open…
Unified multimodal models (UMMs) aim to jointly perform multimodal understanding and generation within a single framework. We present TUNA, a native UMM that builds a unified continuous visual representation by cascading a VAE encoder with…
In the age of artificial intelligence and biotechnology, a unified understanding of technology and biology is critically needed but still lacking. A cornerstone of such unification is evolvable design. I present a formalism, called goal…
In this paper we first show that every non-zero $\tau$-rigid $A$-module induces at least one stratifying system in the module category of $A$. Moreover, we show that each of these stratifying systems can be seen as a signed…
Microstructural geometry plays a critical role in the response of heterogeneous materials. Consequently, methods for generating microstructural samples are increasingly crucial to advanced numerical analyses. We extend Sonon et al.'s…
A visceral structure on M is given by a definable base for a uniform topology on its universe in which all basic open sets are infinite and any infinite definable subset X of M has non-empty interior. This context includes o-minimal ordered…
Consider a self-similar space X. A typical situation is that X looks like several copies of itself glued to several copies of another space Y, and Y looks like several copies of itself glued to several copies of X, or the same kind of thing…
Complex nanophotonic structures hold the potential to deliver exquisitely tailored optical responses for a range of applications. Metal-insulator-metal (MIM) metasurfaces arranged in supercells, for instance, can be tailored by geometry and…
We present a generic construction of finite realisations of amalgamation patterns. An amalgamation pattern is specified by a finite collection of finite template structures together with a collection of partial isomorphisms between them. A…
Self-assembly processes allow us to design and create complex nanostructures using molecules as building blocks and surfaces as scaffolds. This autonomous driven construction is possible due to a complex thermodynamic balance of…
We use $\tau$-tilting theory to give a description of the wall and chamber structure of a finite dimensional algebra. We also study $\mathfrak{D}$-generic paths in the wall and chamber structure of an algebra $A$ and show that every maximal…
Winfree (1998) showed that discrete Sierpinski triangles can self-assemble in the Tile Assembly Model. A striking molecular realization of this self-assembly, using DNA tiles a few nanometers long and verifying the results by atomic-force…