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Patterned self-assembly tile set synthesis (PATS) aims at finding a minimum tile set to uniquely self-assemble a given rectangular color pattern. For $k \ge 1$, $k$-PATS is a variant of PATS that restricts input patterns to those with at…
Self-similarity refers to the image prior widely used in image restoration algorithms that small but similar patterns tend to occur at different locations and scales. However, recent advanced deep convolutional neural network based methods…
The formation of quasi-spherical cages from protein building blocks is a remarkable self-assembly process in many natural systems, where a small number of elementary building blocks are assembled to build a highly symmetric icosahedral…
This short contribution presents a method for generating $N$-point spherical configurations with low mesh ratios. The method extends Caspar-Klug icosahedral point-grids to non-icosahedral nets through the use of planar barycentric…
Semidefinite programming (SDP) is a fundamental class of convex optimization problems with diverse applications in mathematics, engineering, machine learning, and related disciplines. This paper investigates the application of the…
Multi-objective optimization problems require simultaneously optimizing two or more objective functions. Many studies have reported that the solution set of an M-objective optimization problem often forms an (M-1)-dimensional topological…
This paper addresses the question of whether a single tile with nearest neighbor matching rules can force a tiling in which the tiles fall into a large number of isohedral classes. A single tile is exhibited that can fill the Euclidean…
We give an explicit geometric way to build mixed-integer programming (MIP) formulations for unions of polyhedra. The construction is simply described in terms of spanning hyperplanes in an r-dimensional linear space. The resulting MIP…
The field of algorithmic self-assembly is concerned with the design and analysis of self-assembly systems from a computational perspective, that is, from the perspective of mathematical problems whose study may give insight into the natural…
A novel algorithm that produces a quad layout based on imposed set of singularities is proposed. In this paper, we either use singularities that appear naturally, e.g., by minimizing Ginzburg-Landau energy, or use as an input user-defined…
We study the two-dimensional hierarchical rectangle packing problem, motivated by applications in analog integrated circuit layout, facility layout, and logistics. Unlike classical strip or bin packing, the dimensions of the container are…
In the Escherization problem, given a closed figure in a plane, the objective is to find a closed figure that is as close as possible to the input figure and tiles the plane. Koizumi and Sugihara's formulation reduces this problem to an…
Self-assembled monolayers of microparticles encoding Archimedean and non-regular tessellations promise unprecedented structure-property relationships for a wide spectrum of applications in fields ranging from optoelectronics to surface…
Optimally convergent (with respect to the regularity) quadratic finite element method for two dimensional obstacle problem on simplicial meshes is studied in (Brezzi, Hager, Raviart, Numer. Math, 28:431--443, 1977). There was no analogue of…
This paper views the honeycomb conjecture and the Kepler problem essentially as extreme value problems and solves them by partitioning 2-space and 3-space into building blocks and determining those blocks that have the universal extreme…
Segmentation algorithms based on an energy minimisation framework often depend on a scale parameter which balances a fit to data and a regularising term. Irregular pyramids are defined as a stack of graphs successively reduced. Within this…
We develop an all-hex meshing strategy for the interstitial space in beds of densely packed spheres that is tailored to turbulent flow simulations based on the spectral element method (SEM). The SEM achieves resolution through elevated…
This paper addresses two problems needed to support four-dimensional ($3d + t$) spacetime numerical simulations. The first contribution is a general algorithm for producing conforming spacetime meshes of moving geometries. Here, the surface…
Template matching is a basic method in image analysis to extract useful information from images. In this paper, we suggest a new method for pattern matching. Our method transform the template image from two dimensional image into one…
Plane arrangements are a useful tool for surface and volume modelling. However, their main drawback is poor scalability. We introduce two key novelties that enable the construction of plane arrangements for complex objects and entire…