Showing papers on "Regular polygon published in 2017"

Shape and Symmetry Determine Two-Dimensional Melting Transitions of Hard Regular Polygons

Abstract: Not much is known about what shapes of molecules lead to an $x$-atic phase, an exotic phase of two-dimensional matter that shares properties of both fluids and solids. A new set of simulations investigates the behavior of a range of polygon-shaped particles and shows how body symmetry influences these phase transitions.

Origamizer: A Practical Algorithm for Folding Any Polyhedron

Abstract: It was established at SoCG'99 that every polyhedral complex can be folded from a sufficiently large square of paper, but the known algorithms are extremely impractical, wasting most of the material and making folds through many layers of paper. At a deeper level, these foldings get the topology wrong, introducing many gaps (boundaries) in the surface, which results in flimsy foldings in practice. We develop a new algorithm designed specifically for the practical folding of real paper into complicated polyhedral models. We prove that the algorithm correctly folds any oriented polyhedral manifold, plus an arbitrarily small amount of additional structure on one side of the surface (so for closed manifolds, inside the model). This algorithm is the first to attain the watertight property: for a specified cutting of the manifold into a topological disk with boundary, the folding maps the boundary of the paper to within epsilon of the specified boundary of the surface (in Frechet distance). Our foldings also have the geometric feature that every convex face is folded seamlessly, i.e., as one unfolded convex polygon of the piece of paper. This work provides the theoretical underpinnings for Origamizer, freely available software written by the second author, which has enabled practical folding of many complex polyhedral models such as the Stanford bunny.

Minimal inequalities for an infinite relaxation of integer programs

Abstract: We show that maximal $S$-free convex sets are polyhedra when $S$ is the set of integral points in some rational polyhedron of $\mathbb{R}^n$. This result extends a theorem of Lov\'asz characterizing maximal lattice-free convex sets. Our theorem has implications in integer programming. In particular, we show that maximal $S$-free convex sets are in one-to-one correspondence with minimal inequalities.

Asymptotic for a second-order evolution equation with convex potential and vanishing damping term

Abstract: In this short note, we recover by a different method the new result due to Attouch, Peyrouqet and Redont concerning the weak convergence as $t\rightarrow+\infty$ of solutions $x(t)$ to the second order differential equation \[ x^{\prime\prime}(t)+\frac{K}{t}x^{\prime}(t)+ abla\Phi(x(t))=0, \] where $K>3$ and $\Phi$ is a smooth convex function defined on an Hilbert Space $\mathcal{H}.$ Moreover, we improve slightly their result on the rate of convergence of $\Phi(x(t))-\min\Phi.$

Bridging the Gap: Many-Objective Optimization and Informed Decision-Making

Abstract: The field of many-objective optimization has grown out of infancy and a number of contemporary algorithms can deliver well converged and diverse sets of solutions close to the Pareto optimal front. Concurrently, the studies in cognitive science have highlighted the pitfalls of imprecise decision-making in presence of a large number of alternatives. Thus, for effective decision-making, it is important to devise methods to identify a handful (7 ± 2) of solutions from a potentially large set of tradeoff solutions. Existing measures such as reflex/bend angle, expected marginal utility (EMU), maximum convex bulge/distance from hyperplane, hypervolume contribution, and local curvature are inadequate for the purpose as: 1) they may not create complete ordering of the solutions; 2) they cannot deal with large number of objectives and/or solutions; and 3) they typically do not provide any insight on the nature of selected solutions (internal, peripheral, and extremal). In this letter, we introduce a scheme to identify solutions of interest based on recursive use of the EMU measure. The nature of the solutions (internal or peripheral) is then characterized using reference directions generated via systematic sampling and the top ${K}$ solutions with the largest relative EMU measure are presented to the decision maker. The performance of the approach is illustrated using a number of benchmarks and engineering problems. In our opinion, the development of such methods is necessary to bridge the gap between theoretical development and real-world adoption of many-objective optimization algorithms.

Modeling crack discontinuities without element-partitioning in the extended finite element method

Abstract: In this paper, we model crack discontinuities in two-dimensional linear elastic continua using the extended nite element method without the need to partition an enriched element into a collection of triangles or quadrilaterals. For crack modeling in the X-FEM, the standard finite element approximation is enriched with a discontinuous function and the near-tip crack functions. Each element that is fully cut by the crack is decomposed into two simple (convex or nonconvex) polygons, whereas the element that contains the crack tip is treated as a nonconvex polygon. On using Euler's homogeneous function theorem and Stokes's theorem to numerically integrate homogeneous functions on convex and nonconvex polygons, the exact contributions to the sti ness matrix from discontinuous enriched basis functions are computed. For contributions to the sti ness matrix from weakly singular integrals (due to enrichment with asymptotic crack-tip functions), we only require a one-dimensional quadrature rule along the edges of a polygon. Hence, neither element-partitioning on either side of the crack discontinuity nor use of any cubature rule within an enriched element are needed. Structured fi nite element meshes consisting of rectangular elements, as well as unstructured triangular meshes, are used. We demonstrate the exibility of the approach and its excellent accuracy in stress intensity factor computations for two-dimensional crack problems.

Mixed powerdomains for probability and nondeterminism

Abstract: We consider mixed powerdomains combining ordinary nondeterminism and probabilistic nondeterminism. We characterise them as free algebras for suitable (in)equation-al theories; we establish functional representation theorems; and we show equivalencies between state transformers and appropriately healthy predicate transformers. The extended nonnegative reals serve as `truth-values'. As usual with powerdomains, everything comes in three flavours: lower, upper, and order-convex. The powerdomains are suitable convex sets of subprobability valuations, corresponding to resolving nondeterministic choice before probabilistic choice. Algebraically this corresponds to the probabilistic choice operator distributing over the nondeterministic choice operator. (An alternative approach to combining the two forms of nondeterminism would be to resolve probabilistic choice first, arriving at a domain-theoretic version of random sets. However, as we also show, the algebraic approach then runs into difficulties.) Rather than working directly with valuations, we take a domain-theoretic functional-analytic approach, employing domain-theoretic abstract convex sets called Kegelspitzen; these are equivalent to the abstract probabilistic algebras of Graham and Jones, but are more convenient to work with. So we define power Kegelspitzen, and consider free algebras, functional representations, and predicate transformers. To do so we make use of previous work on domain-theoretic cones (d-cones), with the bridge between the two of them being provided by a free d-cone construction on Kegelspitzen.

Anosov subgroups: Dynamical and geometric characterizations

Abstract: We study infinite covolume discrete subgroups of higher rank semisimple Lie groups, motivated by understanding basic properties of Anosov subgroups from various viewpoints (geometric, coarse geometric and dynamical). The class of Anosov subgroups constitutes a natural generalization of convex cocompact subgroups of rank one Lie groups to higher rank. Our main goal is to give several new equivalent characterizations for this important class of discrete subgroups. Our characterizations capture "rank one behavior" of Anosov subgroups and are direct generalizations of rank one equivalents to convex cocompactness. Along the way, we considerably simplify the original definition, avoiding the geodesic flow. We also show that the Anosov condition can be relaxed further by requiring only non-uniform unbounded expansion along the (quasi)geodesics in the group.

A dual scaled boundary finite element formulation over arbitrary faceted star convex polyhedra

Abstract: A novel technique to formulate arbritrary faceted polyhedral elements in three-dimensions is presented The formulation is applicable for arbitrary faceted polyhedra, provided that a scaling requirement is satisfied and the polyhedron facets are planar A triangulation process can be applied to non-planar facets to generate an admissible geometry The formulation adopts two separate scaled boundary coordinate systems with respect to: (i) a scaling centre located within a polyhedron and; (ii) a scaling centre on a polyhedron’s facets The polyhedron geometry is scaled with respect to both the scaling centres Polygonal shape functions are derived using the scaled boundary finite element method on the polyhedron facets The stiffness matrix of a polyhedron is obtained semi-analytically Numerical integration is required only for the line elements that discretise the polyhedron boundaries The new formulation passes the patch test Application of the new formulation in computational solid mechanics is demonstrated using a few numerical benchmarks

Variational Geometric Approach to Generalized Differential and Conjugate Calculi in Convex Analysis

Abstract: This paper develops a geometric approach of variational analysis for the case of convex objects considered in locally convex topological spaces and also in Banach space settings. Besides deriving in this way new results of convex calculus, we present an overview of some known achievements with their unified and simplified proofs based on the developed geometric variational schemes.

Uniqueness of convex ancient solutions to mean curvature flow in $\mathbb{R}^3$

Abstract: A well-known question of Perelman concerns the classification of noncompact ancient solutions to the Ricci flow in dimension $3$ which have positive sectional curvature and are $\kappa$-noncollapsed. In this paper, we solve the analogous problem for mean curvature flow in $\mathbb{R}^3$, and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in $\mathbb{R}^3$ which is strictly convex and noncollapsed.

Dynamic fracture analysis of the soil-structure interaction system using the scaled boundary finite element method

Abstract: Dynamic fracture analysis of the soil-structure interaction system by using the scaled boundary finite element method is presented in this paper. The polygon scaled boundary finite elements, which have some salient features to model any star convex polygons, are employed for modelling the near-field bounded domains. A procedure for coupling the bounded domains with an improved continued-fraction-based high-order transmitting boundary is established. The formulations of the soil-structure interaction system are coupled via the interaction force vector at the interface. The dynamic stress intensity factors and T-stress are extracted according to the definition of the generalized stress intensity factors. The dynamic stress intensity factors of the coupled system are evaluated accurately and efficiently. Two numerical examples are demonstrated to validate the developed method.

Convex Extensions in Combinatorial Optimization and Their Applications

Abstract: This research focuses on problems of combinatorial optimization necessary for mapping combinatorial sets into arithmetic Euclidean space. The analysis shows that there is a class of vertex-located sets that coincide with the set of vertices within their convex hull. The author has proved the theorems on the existence of convex, strongly convex, and differentiable extensions for functions defined on vertex-located sets. An equivalent problem of mathematical programming with convex objective function and functional constraints has been formulated. The author has studied the properties of convex function extremes on vertex-located sets. The research contains the examples of vertex-located combinatorial sets and algorithms for constructing convex, strongly convex, and differentiable extensions for functions defined on these sets. The conditions have been formulated that are sufficient for a minimum value of functions, as well as lower bounds of functions have been defined on the permutation set. The results obtained can be well used for developing new methods of combinatorial optimization.

On the Finite Convergence of the Douglas--Rachford Algorithm for Solving (Not Necessarily Convex) Feasibility Problems in Euclidean Spaces

Abstract: Solving feasibility problems is a central task in mathematics and the applied sciences. One particularly successful method is the Douglas--Rachford algorithm. In this paper, we provide many new conditions sufficient for finite convergence. Numerous examples illustrate our results.

Computing mixed volume and all mixed cells in quermassintegral time

Abstract: The mixed volume counts the roots of generic sparse polynomial systems. Mixed cells are used to provide starting systems for homotopy algorithms that can find all those roots and track no unnecessary path. Up to now, algorithms for that task were of enumerative type, with no general non-exponential complexity bound. A geometric algorithm is introduced in this paper. Its complexity is bounded in the average and probability-one settings in terms of some geometric invariants: quermassintegrals associated with the tuple of convex hulls of the support of each polynomial. Besides the complexity bounds, numerical results are reported. Those are consistent with an output-sensitive running time for each benchmark family where data are available. For some of those families, an asymptotic running time gain over the best code available at this time was noticed.

On Concentration for (Regularized) Empirical Risk Minimization

Abstract: Rates of convergence for empirical risk minimizers have been well studied in the literature. In this paper, we aim to provide a complementary set of results, in particular by showing that after normalization, the risk of the empirical minimizer concentrates on a single point. Such results have been established by Chatterjee (The Annals of Statistics, 42(6):2340–2381 2014) for constrained estimators in the normal sequence model. We first generalize and sharpen this result to regularized least squares with convex penalties, making use of a “direct” argument based on Borell’s theorem. We then study generalizations to other loss functions, including the negative log-likelihood for exponential families combined with a strictly convex regularization penalty. The results in this general setting are based on more “indirect” arguments as well as on concentration inequalities for maxima of empirical processes.

Dynamical convexity and elliptic periodic orbits for Reeb flows

Abstract: A long-standing conjecture in Hamiltonian Dynamics states that the Reeb flow of any convex hypersurface in $$\mathbb {R}^{2n}$$ carries an elliptic closed orbit. Two important contributions toward its proof were given by Ekeland (An index theory for periodic solutions of convex Hamiltonian systems, 1986) and Dell’Antonio–D’Onofrio–Ekeland (Periodic solutions of elliptic type for strongly nonlinear Hamiltonian systems, 1995) proving this for convex hypersurfaces satisfying suitable pinching conditions and for antipodal invariant convex hypersurfaces respectively. In this work we present a generalization of these results using contact homology and a notion of dynamical convexity first introduced by Hofer–Wysocki–Zehnder for tight contact forms on $$S^3$$ . Applications include geodesic flows under pinching conditions, magnetic flows and toric contact manifolds.

On aggregation for heavy-tailed classes

Abstract: We introduce an alternative to the notion of ‘fast rate’ in Learning Theory, which coincides with the optimal error rate when the given class happens to be convex and regular in some sense. While it is well known that such a rate cannot always be attained by a learning procedure (i.e., a procedure that selects a function in the given class), we introduce an aggregation procedure that attains that rate under rather minimal assumptions—for example, that the \(L_q\) and \(L_2\) norms are equivalent on the linear span of the class for some \(q>2\), and the target random variable is square-integrable. The key components in the proof include a two-sided isomorphic estimator on distances between class members, which is based on the median-of-means; and an almost isometric lower bound of the form \(N^{-1}\sum _{i=1}^N f^2(X_i) \ge (1-\zeta )\mathbb {E}f^2\) which holds uniformly in the class. Both results only require that the \(L_q\) and \(L_2\) norms are equivalent on the linear span of the class for some \(q>2\).

Convex Central Configurations of the 4-Body Problem with Two Pairs of Equal Adjacent Masses

Abstract: We study the convex central configurations of the 4-body problem assuming that they have two pairs of equal masses located at two adjacent vertices of a convex quadrilateral. Under these assumptions we prove that the isosceles trapezoid is the unique central configuration.

Random-to-Random Nodal Distance Distributions in Finite Wireless Networks

Abstract: Most performance metrics in wireless networks, such as outage probability, link capacity, etc., are functions of the distances between communicating/interfering nodes. A probabilistic distance-based model is definitely needed in quantifying these metrics, which eventually involves the nodal distance distribution (NDD) in a finite network intrinsically depending on the network coverage and nodal spatial distribution. Recently, the NDD from a reference node to a uniformly distributed node has been extended to the networks in the shape of arbitrary polygons. In contrast, the NDD between two uniformly distributed nodes (Ran2Ran) is still confined to the networks in certain specific shapes, including disks, triangles, rectangles, rhombuses, trapezoids, and regular polygons, which greatly limits its applicable network scenarios. By extending a tool in integral geometry, called Kinematic Measure, and using decomposition and recursion methods, this paper shows a systematic, algorithmic approach to Ran2Ran NDDs, which can handle arbitrarily-shaped networks, including convex, concave, disjoint, and tiered networks. Besides validating our approach through extensive simulations and comparisons with the known results if applicable, we also demonstrate its potentials in handling nonuniform nodal distributions, and in modeling two wireless networks of particular interest in the current literature, where the existing approaches are inapplicable.

Extended formulations for convex hulls of graphs of bilinear functions.

Abstract: We consider the problem of characterizing the convex hull of the graph of a bilinear function $f$ on the $n$-dimensional unit cube $[0,1]^n$ Extended formulations for this convex hull are obtained by taking subsets of the facets of the Boolean Quadric Polytope (BQP) Extending existing results, we propose a systematic study of properties of $f$ that guarantee that certain classes of BQP facets are sufficient for an extended formulation We use a modification of Zuckerberg's geometric method for proving convex hull characterizations [Geometric proofs for convex hull defining formulations, Operations Research Letters \textbf{44} (2016), 625--629] to prove some initial results in this direction In particular, we provide small-sized extended formulations for bilinear functions whose corresponding graph is either a cycle with arbitrary edge weights or a clique or an almost clique with unit edge weights

Constructing Convex Inner Approximations of Steady-State Security Regions.

Abstract: We propose a scalable optimization framework for estimating convex inner approximations of the steady-state security sets. The framework is based on Brouwer fixed point theorem applied to a fixed-point form of the power flow equations. It establishes a certificate for the self-mapping of a polytope region constructed around a given feasible operating point. This certificate is based on the explicit bounds on the nonlinear terms that hold within the self-mapped polytope. The shape of the polytope is adapted to find the largest approximation of the steady-state security region. While the corresponding optimization problem is nonlinear and non-convex, every feasible solution found by local search defines a valid inner approximation. The number of variables scales linearly with the system size, and the general framework can naturally be applied to other nonlinear equations with affine dependence on inputs. Test cases, with the system sizes up to $1354$ buses, are used to illustrate the scalability of the approach. The results show that the approximated regions are not unreasonably conservative and that they cover substantial fractions of the true steady-state security regions for most medium-sized test cases.

Regular Sequences of Quasi-Nonexpansive Operators and Their Applications

Abstract: In this paper we present a systematic study of regular sequences of quasi-nonexpansive operators in Hilbert space. We are interested, in particular, in weakly, boundedly and linearly regular sequences of operators. We show that the type of the regularity is preserved under relaxations, convex combinations and products of operators. Moreover, in this connection, we show that weak, bounded and linear regularity lead to weak, strong and linear convergence, respectively, of various iterative methods. This applies, in particular, to block iterative and string averaging projection methods, which, in principle, are based on the above-mentioned algebraic operations applied to projections. Finally, we show an application of regular sequences of operators to variational inequality problems.

Tracking a mobile target by multi-robot circumnavigation using bearing measurements

Abstract: In this paper, we study a problem of target tracking and circumnavigation with a network of autonomous agents. We propose a distributed algorithm to estimate the position of the target and drive the agents to rotate around it while forming a regular polygon and keeping a desired distance. We formally show that the algorithm attains exponential convergence of the agents to the desired polygon if the target is stationary, and bounded convergence if the target is moving with bounded speed. Numerical simulations corroborate the theoretical results and demonstrate the resilience of the network to addition and removal of agents.

Optimal estimates for the perfect conductivity problem with inclusions close to the boundary

Abstract: When a convex perfectly conducting inclusion is closely spaced to the boundary of the matrix domain, a bigger convex domain containing the inclusion, the electric field can be arbitrary large. We establish both the pointwise upper bound and the lower bound of the gradient estimate for this perfect conductivity problem by using the energy method. These results give the optimal blow-up rates of electric field for conductors with arbitrary shape and in all dimensions. A particular case when a circular inclusion is close to the boundary of a circular matrix domain in dimension two is studied earlier by Ammari,Kang,Lee,Lee and Lim(2007). From the view of methodology, the technique we develop in this paper is significantly different from the previous one restricted to the circular case, which allows us further investigate the general elliptic equations with divergence form.

Exhaustive search of convex pentagons which tile the plane

Abstract: We present an exhaustive search of all families of convex pentagons which tile the plane. This research shows that there are no more than the already 15 known families. In particular, this implies that there is no convex polygon which allows only non-periodic tilings.