Showing papers on "Regular polygon published in 2012"

Sharp Holder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications

Abstract: We prove a new estimate on manifolds with a lower Ricci bound which asserts that the geometry of balls centered on a minimizing geodesic can change in at most a Holder continuous way along the geodesic. We give examples that show that the Holder exponent, along with essentially all the other consequences that follow from this estimate, are sharp. Among the applications is that the regular set is convex for any non- collapsed limit of Einstein metrics. In the general case of a potentially collapsed limit of manifolds with just a lower Ricci curvature bound we show that the regular set is weakly convex and a.e. convex. We also show two conjectures of Cheeger-Colding. One of these asserts that the isometry group of any, even collapsed, limit of manifolds with a uniform lower Ricci curvature bound is a Lie group. The other asserts that the dimension of any limit space is the same everywhere.

A Convex Approach to Minimal Partitions

Abstract: We describe a convex relaxation for a family of problems of minimal perimeter partitions. The minimization of the relaxed problem can be tackled numerically: we describe an algorithm and show some results. In most cases, our relaxed problem finds a correct numerical approximation of the optimal solution: we give some arguments to explain why it should be so and also discuss some situations where it fails.

On the Linear Convergence of the Alternating Direction Method of Multipliers

Abstract: We analyze the convergence rate of the alternating direction method of multipliers (ADMM) for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints. Previous analysis of the ADMM typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks. Moreover, there has been no rate of convergence analysis for the ADMM without strong convexity in the objective function. In this paper we establish the global linear convergence of the ADMM for minimizing the sum of any number of convex separable functions. This result settles a key question regarding the convergence of the ADMM when the number of blocks is more than two or if the strong convexity is absent. It also implies the linear convergence of the ADMM for several contemporary applications including LASSO, Group LASSO and Sparse Group LASSO without any strong convexity assumption. Our proof is based on estimating the distance from a dual feasible solution to the optimal dual solution set by the norm of a certain proximal residual, and by requiring the dual stepsize to be sufficiently small.

Error estimates for generalized barycentric interpolation

Abstract: We prove the optimal convergence estimate for first-order interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson approach uses Voronoi diagrams on the vertices of the polygon to define the functions, and the Harmonic approach defines the functions as the solution of a PDE. We show that given certain conditions on the geometry of the polygon, each of these constructions can obtain the optimal convergence estimate. In particular, we show that the well-known maximum interior angle condition required for interpolants over triangles is still required for Wachspress functions but not for Sibson functions.

Polygonal homographic orbits of the curved $n$-body problem

Abstract: In the 2-dimensional n-body problem, n ≥ 3, in spaces of constant curvature, κ 6= 0, we study polygonal homographic solutions. We first provide necessary and sufficient conditions for the existence of these orbits and then consider the case of regular polygons. We further use this criterion to show that, for any n ≥ 3, the regular ngon is a polygonal homographic orbit if and only if all masses are equal. Then we prove the existence of relative equilibria of non-equal masses on the sphere of curvature κ > 0 for n = 3 in the case of scalene triangles. Such triangular relative equilibria occur only along fixed geodesics and are generated from fixed points of the sphere. Finally, through a classification of the isosceles case, we prove that not any three masses can form a triangular relative equilibrium.

Moving Out the Edges of a Lattice Polygon

Abstract: We review previous work of (mainly) Koelman, Haase and Schicho, and Poonen and Rodriguez-Villegas on the dual operations of (i) taking the interior hull and (ii) moving out the edges of a two-dimensional lattice polygon. We show how the latter operation naturally gives rise to an algorithm for enumerating lattice polygons by their genus. We then report on an implementation of this algorithm, by means of which we produce the list of all lattice polygons (up to equivalence) whose genus is contained in {1,…,30}. In particular, we obtain the number of inequivalent lattice polygons for each of these genera. As a byproduct, we prove that the minimal possible genus for a lattice 15-gon is 45.

Arnold diffusion in nearly integrable Hamiltonian systems

Abstract: In this paper, Arnold diffusion is proved to be generic phenomenon in nearly integrable convex Hamiltonian systems with three degrees of freedom: $$ H(x,y)=h(y)+\epsilon P(x,y), \qquad x\in\mathbb{T}^3,\ y\in\mathbb{R}^3. $$ Under typical perturbation $\epsilon P$, the system admits "connecting" orbit that passes through any two prescribed small balls in the same energy level $H^{-1}(E)$ provided $E$ is bigger than the minimum of the average action, namely, $E>\min\alpha$.

Building complex networks with Platonic solids.

Abstract: We propose a unified model to build planar graphs with diverse topological characteristics which are of relevance in real applications. Here convex regular polyhedra (Platonic solids) are used as the building blocks for the construction of a variety of complex planar networks. These networks are obtained by merging polyhedra face by face on a tree-structure leading to planar graphs. We investigate two different constructions: (1) a fully deterministic construction where a self-similar fractal structure is built by using a single kind of polyhedron which is iteratively attached to every face and (2) a stochastic construction where at each step a polyhedron is attached to a randomly chosen face. These networks are scale-free, small-world, clustered, and sparse, sharing several characteristics of real-world complex networks. We derive analytical expressions for the degree distribution, the clustering coefficient, and the mean degree of nearest neighbors showing that these networks have power-law degree distributions with tunable exponents associated with the building polyhedron, and they possess a hierarchical organization that is determined by planarity.

Polyarc discrete element for efficiently simulating arbitrarily shaped 2D particles

Abstract: SUMMARY A new two-dimensional discrete element type, termed the ‘polyarc’ element is presented in this paper. Compared to other discrete element types, the new element is capable of representing any two-dimensional convex particle shape with arbitrary angularity and elongation using a small number of shape parameters. Contact resolution between polyarc elements, which is the most computation-extensive task in DEM simulation only involves simple closed-form solutions. Two undesirable contact scenarios common for polygon elements can be avoided by the polyarc element, so the contact resolution algorithm for polyarc elements is simpler than that for polygon elements. The extra flexibility in particle shape representation induces little or no additional computational cost. The key algorithmic aspects of the new element, including the particle shape representation scheme, the quick neighbor search algorithm, the contact resolution algorithm, and the contact law are presented. The recommended contact law for the polyarc model was formulated on the basis of an evaluation of various contact law schemes for polygon type discrete elements. The capability and efficiency of the new element type were demonstrated through an investigation of strength anisotropy of a virtual sand consisting of a random mix of angular and smooth elongated particles subjected to biaxial compression tests. Copyright © 2011 John Wiley & Sons, Ltd.

Solving an optimization packing problem of circles and non-convex polygons with rotations into a multiply connected region

Abstract: This paper deals with the packing problem of circles and non-convex polygons, which can be both translated and rotated into a strip with prohibited regions. Using the Φ-function technique, a mathematical model of the problem is constructed and its characteristics are investigated. Based on the characteristics, a solution approach to the problem is offered. The approach includes the following methods: an optimization method by groups of variables to construct starting points, a modification of the Zoutendijk feasible direction method to search for local minima and a special non-exhaustive search of local minima to find an approximation to a global minimum. A number of numerical results are given. The numerical results are compared with the best known ones.

Drawing Graphs in the Plane with a Prescribed Outer Face and Polynomial Area

Abstract: We study the classic graph drawing problem of drawing a planar graph using straight-line edges with a prescribed convex polygon as the outer face. Unlike previous algorithms for this problem, which may produce drawings with exponential area, our method produces drawings with polynomial area. In addition, we allow for collinear points on the boundary, provided such vertices do not create overlapping edges. Thus, we solve an open problem of Duncan et al., which, when combined with their work, implies that we can produce a planar straight-line drawing of a combinatorially-embedded genus-g graph with the graph's canonical polygonal schema drawn as a convex polygonal external face.

Planar polygon extraction and merging from depth images

Abstract: There has been considerable interest recently in building 3D maps of environments using inexpensive depth cameras like the Microsoft Kinect sensor. We exploit the fact that typical indoor scenes have an abundance of planar features by modeling environments as sets of plane polygons. To this end, we build upon the Fast Sampling Plane Filtering (FSPF) algorithm that extracts points belonging to local neighborhoods of planes from depth images, even in the presence of clutter. We introduce an algorithm that uses the FSPF-generated plane filtered point clouds to generate convex polygons from individual observed depth images. We then contribute an approach of merging these detected polygons across successive frames while accounting for a complete history of observed plane filtered points without explicitly maintaining a list of all observed points. The FSPF and polygon merging algorithms run in real time at full camera frame rates with low CPU requirements: in a real world indoor environment scene, the FSPF and polygon merging algorithms take 2.5 ms on average to process a single 640 × 480 depth image. We provide experimental results demonstrating the computational efficiency of the algorithm and the accuracy of the detected plane polygons by comparing with ground truth.

Principal Component Pursuit with Reduced Linear Measurements

Abstract: In this paper, we study the problem of decomposing a superposition of a low-rank matrix and a sparse matrix when a relatively few linear measurements are available. This problem arises in many data processing tasks such as aligning multiple images or rectifying regular texture, where the goal is to recover a low-rank matrix with a large fraction of corrupted entries in the presence of nonlinear domain transformation. We consider a natural convex heuristic to this problem which is a variant to the recently proposed Principal Component Pursuit. We prove that under suitable conditions, this convex program guarantees to recover the correct low-rank and sparse components despite reduced measurements. Our analysis covers both random and deterministic measurement models.

Compositions and convex combinations of asymptotically regular firmly nonexpansive mappings are also asymptotically regular

Abstract: Because of Minty’s classical correspondence between firmly nonexpansive mappings and maximally monotone operators, the notion of a firmly nonexpansive mapping has proven to be of basic importance in fixed point theory, monotone operator theory, and convex optimization. In this note, we show that if finitely many firmly nonexpansive mappings defined on a real Hilbert space are given and each of these mappings is asymptotically regular, which is equivalent to saying that they have or “almost have” fixed points, then the same is true for their composition. This significantly generalizes the result by Bauschke from 2003 for the case of projectors (nearest point mappings). The proof resides in a Hilbert product space and it relies upon the Brezis-Haraux range approximation result. By working in a suitably scaled Hilbert product space, we also establish the asymptotic regularity of convex combinations.

Fair partitions of polygons: An elementary introduction

Abstract: We introduce the question: Given a positive integer N, can any 2D convex polygonal region be partitioned into N convex pieces such that all pieces have the same area and the same perimeter? The answer to this question is easily ‘yes’ for N = 2. We give an elementary proof that the answer is ‘yes’ for N = 4 and generalize it to higher powers of 2.

Finite convex geometries of circles

Abstract: Let F be a finite set of circles in the plane. We point out that the usual convex closure restricted to F yields a convex geometry, that is, a combinatorial structure introduced by P. H Edelman in 1980 under the name "anti-exchange closure system". We prove that if the circles are collinear and they are arranged in a "concave way", then they determine a convex geometry of convex dimension at most 2, and each finite convex geometry of convex dimension at most 2 can be represented this way. The proof uses some recent results from Lattice Theory, and some of the auxiliary statements on lattices or convex geometries could be of separate interest. The paper is concluded with some open problems.

Curvature estimates in dimensions 2 and 3 for the level sets of p-harmonic functions in convex rings

Abstract: Sharp curvature estimates are given for the level sets of a class of p-harmonic functions in two and three dimensional convex rings.

Rectifiability of Self-contracted curves in the Euclidean space and applications

Abstract: It is hereby established that, in Euclidean spaces of finite dimension, bounded self-contracted curves have finite length. This extends the main result of Daniilidis, Ley, and Sabourau (J. Math. Pures Appl. 2010) concerning continuous planar self-contracted curves to any dimension, and dispenses entirely with the continuity requirement. The proof borrows heavily from a geometric idea of Manselli and Pucci (Geom. Dedicata 1991) employed for the study of regular enough curves, and can be seen as a nonsmooth adaptation of the latter, albeit a nontrivial one. Applications to continuous and discrete dynamical systems are discussed: continuous self-contracted curves appear as generalized solutions of nonsmooth convex foliation systems, recovering a hidden regularity after reparameterization, as consequence of our main result. In the discrete case, proximal sequences (obtained through implicit discretization of a gradient system) give rise to polygonal self-contracted curves. This yields a straightforward proof for the convergence of the exact proximal algorithm, under any choice of parameters.

Tree rules in probabilistic transition system specifications with negative and quantitative premises

Abstract: Probabilistic transition system specifications (PTSSs) in the nt f /nt x format provide structural operational semantics for Segala-type systems that exhibit both probabilistic and nondeterministic behavior and guarantee that bisimilarity is a congruence. Similar to the nondeterministic case of the rule format tyft/tyxt, we show that the well-foundedness requirement is unnecessary in the probabilistic setting. To achieve this, we first define a generalized version of the nt f /nt x format in which quantitative premises and conclusions include nested convex combinations of distributions. Also this format guarantees that bisimilarity is a congruence. Then, for a given (possibly non-well-founded) PTSS in the new format, we construct an equivalent well-founded PTSS consisting of only rules of the simpler (well-founded) probabilistic ntree format. Furthermore, we develop a proof-theoretic notion for these PTSSs that coincides with the existing stratification-based meaning in case the PTSS is stratifiable. This continues the line of research lifting structural operational semantic results from the nondeterministic setting to systems with both probabilistic and nondeterministic behavior.

Two-Dimensional range diameter queries

Abstract: Given a set of n points in the plane, range diameter queries ask for the furthest pair of points in a given axis-parallel rectangular range. We provide evidence for the hardness of designing space-efficient data structures that support range diameter queries by giving a reduction from the set intersection problem. The difficulty of the latter problem is widely acknowledged and is conjectured to require nearly quadratic space in order to obtain constant query time, which is matched by known data structures for both problems, up to polylogarithmic factors. We strengthen the evidence by giving a lower bound for an important subproblem arising in solutions to the range diameter problem: computing the diameter of two convex polygons, that are separated by a vertical line and are preprocessed independently, requires almost linear time in the number of vertices of the smaller polygon, no matter how much space is used. We also show that range diameter queries can be answered much more efficiently for the case of points in convex position by describing a data structure of size O(nlogn) that supports queries in O(logn) time.

Named Area Generation

Abstract: Systems, methods, and computer program products for named area generation are disclosed. In some implementations, documents are processed to uncover pairs of text strings and geographical regions (e.g., a collection of simple convex polygons). For any string/polygon pair, each polygon defines a geographical region whose name is the associated string.

Successive radii and Minkowski addition

Abstract: In this paper we study the behavior of the so called successive inner and outer radii with respect to the Minkowski addition of convex bodies, generalizing the well-known cases of the diameter, minimal width, circumradius and inradius. We get all possible upper and lower bounds for the radii of the sum of two convex bodies in terms of the sum of the corresponding radii.

Minimum-Perimeter Intersecting Polygons

Abstract: Given a set $\mathcal{S}$ of segments in the plane, a polygon P is an intersecting polygon of $\mathcal{S}$ if every segment in $\mathcal{S}$ intersects the interior or the boundary of P. The problem MPIP of computing a minimum-perimeter intersecting polygon of a given set of n segments in the plane was first considered by Rappaport in 1995. This problem is not known to be polynomial, nor it is known to be NP-hard. Rappaport (Int. J. Comput. Geom. Appl. 5:243–265, 1995) gave an exponential-time exact algorithm for MPIP. Hassanzadeh and Rappaport (Proceedings of the 23rd International Workshop on Algorithms and Data Structures, LNCS, vol. 5664, pp. 363–374, 2009) gave a polynomial-time approximation algorithm with ratio $\frac{\pi}{2} \approx 1.57$. In this paper, we present two improved approximation algorithms for MPIP: a 1.28-approximation algorithm by linear programming, and a polynomial-time approximation scheme by discretization and enumeration. Our algorithms can be generalized for computing an approximate minimum-perimeter intersecting polygon of a set of convex polygons in the plane. From the other direction, we show that computing a minimum-perimeter intersecting polygon of a set of (not necessarily convex) simple polygons is NP-hard.

Modelling and design of polygon-shaped kinesin substrates for molecular communication

Abstract: One of the most prominent forms of information transmission between nano- or micro-scale devices is molecular communication, where molecules are used to transfer information inside a fluidic channel. The effects of channel shape on achievable information transmission rates is considered in this work. Specifically, regular convex polygons are studied. A mathematical framework for finding the optimal channel among this class of geometric shapes is derived. Using this framework it is shown that the optimal channel tends to be circular. This result is verified using computer simulations.

Perimeter of sublevel sets in infinite dimensional spaces

Abstract: We compare the perimeter measure with the Airault-Malliavin surface measure and we prove that all open convex subsets of abstract Wiener spaces have finite perimeter. By an explicit counter–example, we show that in general this is not true for compact convex domains.