scholarly journals Set estimation under biconvexity restrictions

2020 ◽  
Vol 24 ◽  
pp. 770-788
Author(s):  
Alejandro Cholaquidis ◽  
Antonio Cuevas
Keyword(s):  
Convex Hull ◽  
Convex Sets ◽  
Optimization Theory ◽  
Minimal Set ◽  
Set Estimation ◽  
Convex Case ◽  
Natural Notion ◽  
Inclination Angles ◽  
Definition Of ◽  
Straight Lines

A set in the Euclidean plane is said to be biconvex if, for some angle θ ∈ [0, π∕2), all its sections along straight lines with inclination angles θ and θ + π∕2 are convex sets (i.e., empty sets or segments). Biconvexity is a natural notion with some useful applications in optimization theory. It has also be independently used, under the name of “rectilinear convexity”, in computational geometry. We are concerned here with the problem of asymptotically reconstructing (or estimating) a biconvex set S from a random sample of points drawn on S. By analogy with the classical convex case, one would like to define the “biconvex hull” of the sample points as a natural estimator for S. However, as previously pointed out by several authors, the notion of “hull” for a given set A (understood as the “minimal” set including A and having the required property) has no obvious, useful translation to the biconvex case. This is in sharp contrast with the well-known elementary definition of convex hull. Thus, we have selected the most commonly accepted notion of “biconvex hull” (often called “rectilinear convex hull”): we first provide additional motivations for this definition, proving some useful relations with other convexity-related notions. Then, we prove some results concerning the consistent approximation of a biconvex set S and the corresponding biconvex hull. An analogous result is also provided for the boundaries. A method to approximate, from a sample of points on S, the biconvexity angle θ is also given.

scholarly journals Decomposable Convexities in Graphs and Hypergraphs

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Francesco M. Malvestuto
Keyword(s):  
Convex Hull ◽  
Convex Sets ◽  
Nonempty Subset ◽  
Convex Set ◽  
Convex Geometry ◽  
Convexity Space ◽  
Vertex Set ◽  
Convex Geometries ◽  
Acyclic Hypergraph ◽  
Maximal Clusters

Given a connected hypergraph with vertex set V, a convexity space on is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called convex sets. The convex hull of a subset X of V is the smallest convex set containing X. By a cluster of we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on is decomposable if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs).

Ball characterizations in spaces of constant curvature

2018 ◽  
Vol 55 (4) ◽  
pp. 421-478
Author(s):  
Jesus Jerónimo-Castro ◽  
Endre Makai, Jr.
Keyword(s):  
Convex Sets ◽  
Central Symmetry ◽  
Connected Components ◽  
Plane Convex ◽  
Converse Implication ◽  
Plane Convex Body ◽  
Dual Theorem ◽  
Centrally Symmetric ◽  
Whole Space ◽  
Straight Lines

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical, Euclidean and hyperbolic spaces, under some regularity assumptions. Suppose that in any of these spaces there is a pair of closed convex sets of class C+2 with interior points, different from the whole space, and the intersections of any congruent copies of these sets are centrally symmetric (provided they have non-empty interiors). Then our sets are congruent balls. Under the same hypotheses, but if we require only central symmetry of small intersections, then our sets are either congruent balls, or paraballs, or have as connected components of their boundaries congruent hyperspheres (and the converse implication also holds). Under the same hypotheses, if we require central symmetry of all compact intersections, then either our sets are congruent balls or paraballs, or have as connected components of their boundaries congruent hyperspheres, and either d ≥ 3, or d = 2 and one of the sets is bounded by one hypercycle, or both sets are congruent parallel domains of straight lines, or there are no more compact intersections than those bounded by two finite hypercycle arcs (and the converse implication also holds). We also prove a dual theorem. If in any of these spaces there is a pair of smooth closed convex sets, such that both of them have supporting spheres at any of their boundary points Sd for Sd of radius less than π/2- and the closed convex hulls of any congruent copies of these sets are centrally symmetric, then our sets are congruent balls.

Recognizing straight lines on raster display with a type of convex hull

2001 ◽  
Author(s):  
Shuping Chen ◽  
Yongfa Xia ◽  
Mingshun Wang ◽  
Xin Mao
Keyword(s):  
Convex Hull ◽  
Straight Lines

On the definition of post-newtonian approximations

1992 ◽  
Vol 438 (1903) ◽  
pp. 341-360 ◽  
Keyword(s):  
General Relativity ◽  
Newtonian Limit ◽  
Minimal Set ◽  
Asymptotically Flat ◽  
Precise Formulation ◽  
The Third ◽  
Newtonian Approximation ◽  
Definition Of

A definition of post-newtonian approximations is presented where the whole formalism is derived from a minimal set of axioms. This establishes a link between the existing precise formulation of the newtonian limit of general relativity and the post-newtonian equations which are used in practical calculations. The breakdown of higher post-newtonian approximations is examined within this framework. It is shown that the choice of harmonic gauge leads to equations which do not admit asymptotically flat solutions at the second post-newtonian level if one starts with a generic newtonian solution. The most simple choice of gauge gives equations which are solvable at the 2PN level but which in general have no solutions in the case of the third post-newtonian approximation.

History of Air Maps and Charts

1998 ◽  
Vol 51 (2) ◽  
pp. 203-215 ◽  
Author(s):  
Philip Steele
Keyword(s):  
Nineteenth Century ◽  
Ordnance Survey ◽  
History Of ◽  
Minor Significance ◽  
Aerial Navigation ◽  
The Difference ◽  
Definition Of ◽  
Straight Lines ◽  
The Hill

There is no generally accepted definition of the difference between a map and a chart. A widespread feeling probably exists favouring the old saying that maps are to look at and charts to work on. It is true that the term ‘aeronautical chart’ gained a general currency over alternative terms as contact flying gave way to aerial navigation. But, in this paper, the terms ‘map’ and ‘chart’ will be used as seems appropriate to each occasion, without attempt to conform to any particular definition.We can get an idea of what was available to the earliest aviators by looking at an Ordnance Survey reprint of one of their nineteenth century maps (Fig. 1). They are printed in one colour only, black on white. By far the predominant feature is the hill shading. Quite gentle hills are hachured with a heaviness which tends to obscure both natural features like rivers, lakes and woodlands and man-made constructions such as towns and villages, roads, canals and railways. Hills are, of course, very important features to those on the ground, since they limit the extent to which other features can be seen. To the soldier, the significance of high ground is self-evident, and it was principally for the ordnance requirements of soldiers that these maps had been developed. But when men began to view the ground from the air, the perspective changed. Hills appeared flattened out and, provided that you knew the height of the tallest in the area and were sure none would impede your take-off or landing, were of minor significance. Lakes and woods, though, were spread out before you in their distinctive shapes, while railway lines and canals presented bold straight lines and curves, and rivers their unique courses, to your view. The need was for new kinds of maps which would give due prominence to such features.

The distribution of the convex hull of a Gaussian sample

1980 ◽  
Vol 17 (03) ◽  
pp. 686-695 ◽  
Author(s):  
William F. Eddy
Keyword(s):  
Weak Convergence ◽  
Convex Hull ◽  
Poisson Point Process ◽  
Point Processes ◽  
Convex Sets ◽  
Continuous Mapping ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Limit Process ◽  
Gaussian Sample

The distribution of the convex hull of a random sample ofd-dimensional variables is described by embedding the collection of convex sets into the space of continuous functions on the unit sphere. Weak convergence of the normalized convex hull of a circular Gaussian sample to a process with extreme-value marginal distributions is demonstrated. The proof shows that an underlying sequence of point processes converges to a Poisson point process and then applies the continuous mapping theorem. Several properties of the limit process are determined.

scholarly journals On a conjecture concerning helly circle graphs

2003 ◽  
Vol 23 (1) ◽  
pp. 221-229 ◽  
Author(s):  
Guillermo Durán ◽  
Agustín Gravano ◽  
Marina Groshaus ◽  
Fábio Protti ◽  
Jayme L. Szwarcfiter
Keyword(s):  
Systems Of Equations ◽  
Intersection Graphs ◽  
Circle Graph ◽  
Circle Graphs ◽  
Analytic Tool ◽  
Two Systems ◽  
Cartesian Plane ◽  
Definition Of ◽  
Straight Lines

We say that G is an e-circle graph if there is a bijection between its vertices and straight lines on the cartesian plane such that two vertices are adjacent in G if and only if the corresponding lines intersect inside the circle of radius one. This definition suggests a method for deciding whether a given graph G is an e-circle graph, by constructing a convenient system S of equations and inequations which represents the structure of G, in such a way that G is an e-circle graph if and only if S has a solution. In fact, e-circle graphs are exactly the circle graphs (intersection graphs of chords in a circle), and thus this method provides an analytic way for recognizing circle graphs. A graph G is a Helly circle graph if G is a circle graph and there exists a model of G by chords such that every three pairwise intersecting chords intersect at the same point. A conjecture by Durán (2000) states that G is a Helly circle graph if and only if G is a circle graph and contains no induced diamonds (a diamond is a graph formed by four vertices and five edges). Many unsuccessful efforts - mainly based on combinatorial and geometrical approaches - have been done in order to validate this conjecture. In this work, we utilize the ideas behind the definition of e-circle graphs and restate this conjecture in terms of an equivalence between two systems of equations and inequations, providing a new, analytic tool to deal with it.

Reliability and Functionality of Repairable Systems Using a Minimal Set of Metrics: Design and Maintenance of a Smart Charging Microgrid

2013 ◽  
Author(s):  
Vijitashwa Pandey ◽  
Annette G. Skowronska ◽  
Zissimos P. Mourelatos ◽  
David Gorsich ◽  
Matthew Castanier
Keyword(s):  
Decision Making ◽  
Spare Parts ◽  
Careful Consideration ◽  
Repairable Systems ◽  
Minimal Set ◽  
Us Army ◽  
Military Installation ◽  
Smart Charging ◽  
Definition Of ◽  
Multiple Metrics

The definition of reliability may not be readily applicable for repairable systems. Our recent work has shown that multiple metrics are needed to fully account for the performance of a repairable system under uncertainty. Optimal tradeoffs among a minimal set of metrics can be used in the design and maintenance of these systems. A minimal set of metrics provides the most information about the system with the smallest number of metrics using a set of desirable properties. Critical installations such as a remote microgrid powering a military installation require a careful consideration of cost and repair strategies. This is because of logistical challenges in performing repairs and supplying necessary spare parts, particularly in unsafe locations. This paper shows how a minimal set of metrics enhances decision making in such a scenario. It enables optimal tradeoffs between critical attributes in decision making, while guaranteeing that all important performance measures are satisfied. As a result, cost targets and inventory planning can be achieved in an optimal way. We demonstrate the value of the proposed approach using a US Army smart-charging microgrid installation.

scholarly journals Typical Convex Sets of Convex Sets

1987 ◽  
Vol 43 (3) ◽  
pp. 287-290 ◽  
Author(s):  
T. Schwarz ◽  
T. Zamfirescu
Keyword(s):  
Convex Sets ◽  
Compact Convex ◽  
Natural Notion

There exists a natural notion of convexity in the space of all compact convex sets in D. Thus, we may consider the space of all bounded closed convex families of compact convex sets. We present here a strange generic extremal behaviour of the elements of this space.

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