Shading of regions on vector display devises

For a set S of n lines labeled from 1 to n, we say that S supports an n-vertex planar graph G if for every labeling from 1 to n of its vertices, G has a straight-line crossing-free drawing with each vertex drawn as a point on its associated line. It is known from previous work [4] that no set of n parallel lines supports all n-vertex planar graphs. We show that intersecting lines, even if they intersect at a common point, are more “powerful” than a set of parallel lines. In particular, we prove that every such set of lines supports outerpaths, lobsters, and squids, none of which are supported by any set of parallel lines. On the negative side, we prove that no set of n lines that intersect in a common point supports all n-vertex planar graphs. Finally, we show that there exists a set of n lines in general position that does not support all n-vertex planar graphs.

A new upper bound for the VC-dimension of visibility regions

In this paper we are proving the following fact. Let P be an arbitrary simple polygon, and let S be an arbitrary set of 15 points inside P. Then there exists a subset T of S that is not visually discernible, that is, T ≠ vis(v) ∩ S holds for the visibility regions vis(v) of all points v in P. In other words, the VC-dimension $d$ of visibility regions in a simple polygon cannot exceed 14. Since Valtr [v-ggwps-98] proved in 1998 that d ∈ [6,23] holds, no progress has been made on this bound. Our reduction immediately implies a smaller upper bound to the number of guards needed to cover P by e-net theorems.

Given an arbitrary simple polygon, we construct a polytopal complex analogous to the associahedron based on its convex diagonalizations. This polytopal complex is shown to be contractible, and a geometric realization is provided based on the theory of secondary polytopes. We then reformulate a combinatorial deformation theory in terms of visibility and presents some open problems.

Catadioptric sensors refer to the combination of lens-based devices and reflective surfaces. These systems are useful because they may have a field of view which is greater than hemispherical, providing the ability to simultaneously view in any direction. Configurations which have a unique effective viewpoint are of primary interest, among these is the case where the reflective surface is a parabolic mirror and the camera is such that it induces an orthographic projection and which we call paracatadioptric. We present an algorithm for the calibration of such a device using only the images of lines in space. In fact, we show that we may obtain all of the intrinsic parameters from the images of only three lines and that this is possible without any metric information. We propose a closed-form solution for focal length, image center, and aspect ratio for skewless cameras and a polynomial root solution in the presence of skew. We also give a method for determining the orientation of a plane containing two sets of parallel lines from one uncalibrated view. Such an orientation recovery enables a rectification which is impossible to achieve in the case of a single uncalibrated view taken by a conventional camera. We study the performance of the algorithm in simulated setups and compare results on real images with an approach based on the image of the mirror's bounding circle.

Rotational splitting Δω(n,l,m) of the eigenfrequencies of a star rotating with angular velocity Ω(r, θ) about a unique axis can be represented as a weighted integral of Ω over r and θ, (r, θ, φ) being spherical polar coordinates about the axis of rotation. For high-frequency acoustic modes, Δω/m collapses essentially to a function of ω — ω /(l+1/2) and M = m/(l + 1/2) alone, and the weighting kernel K(r,θ) becomes asymptotically degenerate, each factor being of essentially Abel type. Therefore, formally, the splitting integral can be inverted, once a procedure has been found for extending Δω over the domain of (ω, M) such that the turning points (rt,ωt), given by (c(rt)/rt, sinωt) = (ω, M) where c is sound speed, span the star. Obtaining that representation is the most difficult stage of the inversion. We report on a procedure that treats the inverted two-dimensional Abel integral as a repeated double integral, representing the data successively along a set of parallel lines M =constant. The method is illustrated by an inversion of artificial data which is compared with the angular velocity from which those data were computed.

The structure of a strange attractor

The desire to increase the amount of information that can be encoded onto a single photon has driven research into many areas of optics. One such area is optical orbital angular momentum (OAM) [1]. These beams have helical phasefronts and carry an orbital angular momentum of mbar per photon, where the integer m is unbounded, giving a large state space in which to encode information. We recently developed a telescope system comprising two bespoke refractive optical elements to transform OAM states into transverse momentum states [2]. This is achieved by mapping the azimuthal position of the input plane to the lateral position in the output [3]. A mapping of this type transforms a set of concentric rings at the input plane into a set of parallel lines in the output plane. A lens can then separate the resulting transverse momentum states into specified lateral positions, allowing for the efficient measurement of multiple OAM states simultaneously. Separating OAM states in this way presents an opportunity for this larger alphabet to improve the data capacity of a free space link and has potential application in both the classical and quantum regimes. We will present our latest design, increasing the bandwidth of measurable states to over 50 OAM modes. In such a system we study the crosstalk introduced by a thin phase turbulence, showing that turbulence similarly degrades the purity of all the modes within this range.

Synthesis, physico-chemical and DNA interaction studies of homo- and hetero-trinuclear complexes

The electrostatic potential in an ionic crystal is presented in terms of the zeroth-order MacDonald function by means of decomposition of the crystal lattice into sets of parallel lines with the periodic charge distribution with the zero net charge within a period. The method is applied to the calculation of the Madelung constants of a series of standard cubic lattices, doped fullerides and idealized Y-Ba-Cu-O crystals.

Determining object shape from local velocity measurements

With a still developing and largely unpoliced code of ethics, the Internet forces users to renegotiate continually the boundaries and registers of publicity and privacy, and thus, it is the technic that constellates the ideal conditions for gossip, quite apart from its speed and efficiency in circulating tittle-tattle. Proceeding from observations about how the Internet challenges definitional certitudes about the practice and performance of gossip, this article considers the queer performances of Stephen Karam's darkly comic play Speech & Debate, in which three high-school misfits meet and bond over shared sexual secrets that begin to circulate online. Karam's representation suggests how the new platforms or stages for gossip that develop out of networking sites and technologies may create new counter-publics, capable, in their turn, of occasioning new forms of counter-intimacy. These platforms offer not only new potentialities for intertextual citation and public discourse but also mechanisms for self-promotion and, in this way, raise questions about the distinction between confession and gossip and between intimacy and publicity: two sets of parallel lines that have begun to bend toward one another.

In chromatic variants of the art gallery problem, simple polygons are guarded with point guards that are assigned one of k colors each. We say these guards cover the polygon. Here we consider the conflict-free chromatic art gallery problem, first studied by Bartschi and Suri (Algorithmica 2013): A covering of the polygon is conflict-free if each point of the polygon is seen by some guard whose color appears exactly once among the guards visible to that point. We are interested in the smallest number k(n) of colors that ensure such a covering for every n-vertex polygon. It is known that k(n) is O(log n) for orthogonal and for monotone polygons, and O(log2 n) for arbitrary simple polygons. Our main contribution in this paper is an improvement of the upper bound on k(n) to O(log n) for simple polygons. The bound is achieved through a partitioning of the polygon into weak visibility subpolygons, which is known as a window partition. In a weak visibility polygon, there is a boundary edge e such that each point of the polygon is seen by some point on e. We show for the first time for this special class of polygons an upper bound of O(log n). For the subpolygons of the window partition we prove a novel concept of independence that allows to reuse colors in independent subpolygons. Combining these results leads to the upper bound of O(log n) for arbitrary simple polygons.

The most generally used method for estimating the basin-wide sequence and scaling of first and last occurrences, based on their occurrence in local sections, is Shaw’s graphic correlation method. The key step in this method is the determination of the line of correlation (LOC), which represents the best estimate of the correlation between two local sections, or between a local section and a composite standard. In general, available techniques for fitting the LOC for multiple sections are tedious, subjective, or computationally expensive. A new method employing genetic algorithms can dramatically reduce the effort involved in determining the LOC and produces stable biostratigraphic correlations and composite range charts objectively and efficiently. Genetic algorithms are an artificial intelligence technique that excels in locating the optimum solution from a large number of alternative choices. In the case of the LOC, the alternative choices are the number of line segments comprising the complete line and the positions of each segment’s beginning and end points. For a given number of segments, a wide range of alternative LOCs can be rapidly evaluated and a potential optimum fit determined. It is also possible to estimate the point when no further refinement of the fit by adding line segments is necessary. Genetic algorithms can also be applied to other methods for quantitative biostratigraphy.

This paper outlines the following generalization of the classical maximal-empty-rectangle (MER) problem: given n arbitrarily-oriented non-intersecting line segments of finite length on a rectangular floor, locate an empty isothetic rectangle of maximum area. Thus, the earlier restriction on isotheticity of the obstacles is relaxed. Based on the wellknown technique of matrix searching, a novel algorithm of time complexity O(nlog2n) and space complexity O(n), is proposed. Next, the technique is extended to handle the following two related open problems: locating the largest isothetic MER (i) inside an arbitrary simple polygon and (ii) amidst a set of arbitrary polygonal obstacles.