Related papers: Strict Self-Assembly of Fractals using Multiple Ha…
This paper presents an efficient parallel direct algorithm with near-optimal complexity for the compact fourth and sixth-order approximation of the three-dimensional Helmholtz equations [1] with the problem coefficient depending on only one…
We study several variants of the classical Sierpinski Carpet (SC) fractal. The main examples we call infinite magic carpets (IMC), obtained by taking an infinite blowup of a discrete graph approximation to SC and identifying edges using…
We study Minkowski contents and fractal curvatures of arbitrary self-similar tilings (constructed on a feasible open set of an IFS) and the general relations to the corresponding functionals for self-similar sets. In particular, we…
We present a systematic construction of finite element exact sequences with a commuting diagram for the de Rham complex in one-, two- and three-space dimensions. We apply the construction in two-space dimensions to rediscover two families…
In the abstract Tile Assembly Model (aTAM), the phenomenon of cooperation occurs when the attachment of a new tile to a growing assembly requires it to bind to more than one tile already in the assembly. Often referred to as…
A generalized theory of the self-limiting assembly of twisted bundles of filaments and columns is presented. Bundles and fibers form in a broad variety of supramolecular systems, from biological to synthetic materials. A widely-invoked…
Fractal structures emerge from statistical and hierarchical processes in urban development or network evolution. In a class of efficient and robust geographical networks, we derive the size distribution of layered areas, and estimate the…
Dynamic tetrahedral simulation pipelines rebuild topology-dependent solver state after every fracture, refinement, or merge event - discarding structural continuity that survives each edit and spending global work on what are often local…
The Abelian Sandpile Model, seen as a deterministic lattice automaton, on two-dimensional periodic graphs generates complex regular patterns displaying (fractal) self-similarity. In particular, on a variety of lattices and initial…
We examine the dimensions of various inf-sup stable mixed finite element spaces on tetrahedral meshes in 3D with exact divergence constraints. More precisely, we compare the standard Scott-Vogelius elements of higher polynomial degree and…
The abelian sandpile serves as a model to study self-organized criticality, a phenomenon occurring in biological, physical and social processes. The identity of the abelian group is a fractal composed of self-similar patches, and its limit…
We advance the program of connections between final coalgebras as sources of circularity in mathematics and fractal sets of real numbers. In particular, we are interested in the Sierpinski carpet, taking it as a fractal subset of the unit…
The ability to control forces between sub-micron-scale building blocks offers considerable potential for designing new materials through self-assembly. A typical paradigm is to first identify a particular (crystal) structure that has some…
We consider the tile self-assembly model and how tile complexity can be eliminated by permitting the temperature of the self-assembly system to be adjusted throughout the assembly process. To do this, we propose novel techniques for…
Our main goal in this long survey article is to provide an overview of the theory of complex fractal dimensions and of the associated geometric or fractal zeta functions, first in the case of fractal strings (one-dimensional drums with…
Exponential growth describes an extremely rapid process ubiquitous across mathematics and diverse physical, biological, and technological systems. Here, we introduce a class of fractal-inspired lattices that combine long-range periodic…
Tendon-driven underactuated hands excel in adaptive grasping but often suffer from kinematic unpredictability and highly non-linear force transmission. This ambiguity limits their ability to perform precise free-motion shaping and deliver…
Majumder, Reif and Sahu have presented a stochastic model of reversible, error-permitting, two-dimensional tile self-assembly, and showed that restricted classes of tile assembly systems achieved equilibrium in (expected) polynomial time.…
We consider a trace theorem for self-similar Dirichlet forms on self-similar sets to self-similar subsets. In particular, we characterize the trace of the domains of Dirichlet forms on the Sierpinski gaskets and the Sierpinski carpets to…
We demonstrate the realization of large, fully loaded, arbitrarily-shaped three-dimensional arrays of single atoms. Using holographic methods and real-time, atom-by-atom, plane-by-plane assembly, we engineer atomic structures with up to 72…