Related papers: Patterns formed by coins
It is shown that any primitive integral Apollonian circle packing captures a fraction of the prime numbers. Basically the method consists in applying the circle method, considering the curvatures produced by a well-chosen family of binary…
Computational mechanics is a method for discovering, describing and quantifying patterns, using tools from statistical physics. It constructs optimal, minimal models of stochastic processes and their underlying causal structures. These…
A sorting network is a shortest path from 12..n to n..21 in the Cayley graph of the symmetric group S(n) generated by nearest-neighbor swaps. A pattern is a sequence of swaps that forms an initial segment of some sorting network. We prove…
Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. Previous efforts for exact algorithms have been unable to avoid structural problems that appear for…
A Bl-packing is a (branched) circle packing that ``properly covers'' the unit disc. We establish some fundamental properties of such packings. We give necessary and sufficient conditions for their existence, prove their uniqueness, and show…
We consider the problem of finding all enclosing rectangles of minimum area that can contain a given set of rectangles without overlap. Our rectangle packer chooses the x-coordinates of all the rectangles before any of the y-coordinates. We…
Clustering can be defined as the process of assembling objects into a number of groups whose elements are similar to each other in some manner. As a technique that is used in many domains, such as face clustering, plant categorization,…
Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and…
Enumeration of tilings is the mathematical study concerning the total number of coverings of regions by similar pieces without gaps or overlaps. Enumeration of tilings has become a vibrant subfield of combinatorics with connections and…
We consider a modification of Winkler's "dots and coins" problem, where we constrain the dots to lie on a square lattice in the plane. We construct packings of "coins" (closed unit disks) using motif patterns.
In a circle (an S^1) with circumference 1 assume m objects distributed pseudo-randomly. In the universal covering R^1 assume the objects replicated accordingly, and take an interval L>1. In this interval, make the normalized histogram of…
In this article, the notion of a mathematical model in science is attempted to be enlightened from several points of view. In particular, it is shown that mathematical models are introduced differently and used differently in different…
We introduce and study the model of simply generated non-crossing partitions, which are, roughly speaking, chosen at random according to a sequence of weights. This framework encompasses the particular case of uniform non-crossing…
Assembling parts into an object is a combinatorial problem that arises in a variety of contexts in the real world and involves numerous applications in science and engineering. Previous related work tackles limited cases with identical unit…
Theoretical background is provided towards the mathematical foundation of the minimum enclosing ball problem. This problem concerns the determination of the unique spherical surface of smallest radius enclosing a given bounded set in the…
This paper proves a deformation circle pattern theorem, which gives a complete description of those circle patterns with interstices in terms of the combinatorial type, the exterior intersections angles and the conformal structures of…
Universal cycles are generalizations of de Bruijn cycles and Gray codes that were introduced originally by Chung, Diaconis, and Graham in 1992. They have been developed by many authors since, for various combinatorial objects such as…
Circuit topology refers to the arrangement of interactions between objects belonging to a linearly ordered object set. Linearly ordered set of objects are common in nature and occur in a wide range of applications in economics, computer…
Circle packings with specified patterns of tangencies form a discrete counterpart of analytic functions. In this paper we study univalent packings (with a combinatorial closed disk as tangent graph) which are embedded in (or fill) a…
The Circle Pattern Theorem characterizes the existence and rigidity of circle patterns with prescribed intersection angles on simplicial triangulations of closed surfaces. In this paper we extend the theorem to quasi-simplicial…