Reflex Angles Research Articles

Pseudo-integrable billiards and arithmetic dynamics

We introduce a new class of billiard systems in the plane, withboundaries formed by finitely many arcs of confocal conics such thatthey contain some reflex angles. Fundamental dynamical, topological,geometric, and arithmetic properties of such billiards are studied.The novelty, caused by reflex angles on boundary, induces invariantleaves of higher genera and dynamical behavior different fromLiouville--Arnold's Theorem. Its analog is derived from the MaierTheorem on measured foliations. The billiard flow generates ameasurable foliation defined by a closed 1-form $w$. Using theclosed form, a transformation of the given billiard table to arectangular cylinder is constructed and a trajectory equivalencebetween corresponding billiards has been established. A local versionof Poncelet Theorem is formulated and necessary algebro-geometricconditions for periodicity are presented. It is proved that thedynamics depends on arithmetic of rotation numbers, but not ongeometry of a given confocal pencil of conics.

Upper bounds for the perimeter of plane convex bodies

We show that the maximum total perimeter of k plane convex bodies with disjoint interiors lying inside a given convex body C is equal to $\operatorname{per}\, (C)+2(k-1)\operatorname{diam}\, (C)$ , in the case when C is a square or an arbitrary triangle. A weaker bound is obtained for general plane convex bodies. As a consequence, we establish a bound on the perimeter of a polygon with at most k reflex angles lying inside a given plane convex body.

Green functions and twist correlators for N branes at angles

We compute the Green functions and correlator functions for N twist fields for branes at angles on T2 and we show that there are N−2 different configurations labeled by an integer M which is roughly associated with the number of reflex angles of the configuration. In order to perform this computation we use an SL(2,R) invariant formulation and geometric constraints instead of Pochammer contours. In particular the M=1 or M=N−1 amplitude can be expressed without using transcendental functions. We determine the amplitudes normalization from N→N−1 reduction without using the factorization into the untwisted sector. Both the amplitudes normalization and the OPE of two twist fields are unique (up to one constant) when the ϵ↔1−ϵ symmetry is imposed. For consistency we find also an infinite number of relations among Lauricella hypergeometric functions.

Shooting Permanent Rays among Disjoint Polygons in the Plane

We present a data structure for ray shooting and insertion in the free space between disjoint polygonal obstacles with a total of $n$ vertices in the plane, where each ray starts at the boundary of some obstacle. The portion of each query ray between the starting point and the first obstacle hit is inserted permanently as a new obstacle. Our data structure uses $O(n\log n)$ space and preprocessing time, and it supports $m$ successive ray shooting and insertion queries in $O((n+m)\log^2 n + m\log m)$ total time in the real RAM model of computation. We present two applications: (1) Our data structure supports efficient implementation of auto-partitions in the plane, that is, binary space partitions where each partition is done along the supporting line of an input segment. If $n$ input line segments are fragmented into $m$ pieces by an auto-partition, then it can now be implemented in $O(n\log^2n+m\log m)$ time. This improves the expected runtime of Patersen and Yao's classical randomized auto-partition algorithm for $n$ disjoint line segments in the plane to $O(n\log^2 n)$. (2) If we are given disjoint polygonal obstacles with a total of $n$ vertices in the plane, a permutation of the reflex vertices, and a half-line at each reflex vertex that partitions the reflex angle into two convex angles, then the convex partitioning algorithm draws a ray emanating from each reflex vertex in the prescribed order in the given direction until it hits another obstacle, a previous ray, or infinity. The previously best implementation (with a semidynamic ray shooting data structure) requires $O(n^{3/2-\varepsilon/2})$ time using $O(n^{1+\varepsilon})$ space for any $\varepsilon>0$. Our data structure improves the runtime to $O(n\log^2 n)$.

Corner multifractality for reflex angles and conformal invariance at 2D Anderson metal–insulator transition with spin–orbit scattering

We investigate boundary multifractality of critical wave functions at the Anderson metal–insulator transition in two-dimensional disordered non-interacting electron systems with spin–orbit scattering. We show numerically that multifractal exponents at a corner with an opening angle θ = 3 π / 2 are directly related to those near a straight boundary in the way dictated by conformal symmetry. This result extends our previous numerical results on corner multifractality obtained for θ < π to θ > π , and gives further supporting evidence for conformal invariance at criticality. We also propose a refinement of the validity of the symmetry relation of A.D. Mirlin et al. [Phys. Rev. Lett. 97 (2006) 046803] for corners.

Competition for space between encrusting excavating Caribbean sponges and other coral reef organisms

MEPS Marine Ecology Progress Series Contact the journal Facebook Twitter RSS Mailing List Subscribe to our mailing list via Mailchimp HomeLatest VolumeAbout the JournalEditorsTheme Sections MEPS 312:113-121 (2006) - doi:10.3354/meps312113 Competition for space between encrusting excavating Caribbean sponges and other coral reef organisms Mateo López-Victoria1,2, Sven Zea2,*, Ernesto Weil3 1Instituto de Investigaciones Marinas y Costeras – INVEMAR, and 2Departamento de Biología and Centro de Estudios en Ciencias del Mar, CECIMAR, Universidad Nacional de Colombia, INVEMAR, Cerro Punta de Betín, A.A. 1016, Santa Marta, Colombia 3Islas Magueyez Marine Laboratory, Department of Marine Sciences, University of Puerto Rico, PO Box 908, Lajas 00667, Puerto Rico *Corresponding author. Email: szea@invemar.org.co ABSTRACT: Encrusting and excavating Caribbean sponges Cliona aprica, C. caribbaea and C. tenuis (Hadromerida: Clionaidae) aggressively undermine and displace live coral tissue. To establish modes and rates of lateral advance into neighboring organisms, 171 sponge individuals were studied for 13 mo. Sponge advancement into live coral, via excavation underneath boundary polyps, occurred only when the 2 confronted at an angle of ≥180°. Sponges stopped or retreated when coral upward and inward growth at the boundary closed the angle between the coral and sponge, placing polyps out of reach of excavating tissue. At a straight angle of confrontation, C. tenuis advanced into coral at higher rates than the other 2 sponge species. Its advance was more pronounced into Diploria strigosa than into Siderastrea siderea. Sponge undermining continuously opened space for transient settlement of turf algae and for accumulation of sediments. Corallivory at the sponge–coral boundary did not imply greater rates of sponge advance, but probably did contribute to coral bioerosion. Massive corals escaped sponge undermining by upward growth, their ability to do so depended partly on defensive mechanisms. Foliose and platy corals escaped by overtopping. Rates of sponge advance into substrata incrusted and overgrown by algae were in general lower than into live coral, while sponges lost space to some encrusting invertebrates. KEY WORDS: Competition · Excavating sponges · Cliona · Coral reefs · Caribbean · Corallivory · Turf algae · Reflex angle Full text in pdf format PreviousNextExport citation RSS - Facebook - Tweet - linkedIn Cited by Published in MEPS Vol. 312. Online publication date: April 24, 2006 Print ISSN: 0171-8630; Online ISSN: 1616-1599 Copyright © 2006 Inter-Research.

Minimum area polygons with two reflex angles enclosing k Points

We present in this paper, algorithms for finding the minimum area isothetic polygons having two reflex angles containing k points; (k<n) Our algorithms are based on line sweep paradigm. The idea is to enumerate all the polygons in each kind and find the minimum one. We also analyse the complexities of each of the algorithms. We find that out of the four orthogonal polygons with two reflex angles, three are orthoconvex and the algorithms presented for these polygons work in time, where n could be O(n-k)3 in the worst case. The algorithm for the non-orthoconvex polygon has a complexity of where n could be O((n-k)3 n) in the worst case.

Decomposing the boundary of a nonconvex polyhedron

We show that the boundary of a three-dimensional polyhedron withr reflex angles and arbitrary genus can be subdivided intoO(r) connected pieces, each of which lies on the boundary of its convex hull. A remarkable feature of this result is that the number of these convex-like pieces is independent of the number of vertices. Furthermore, it is linear inr, which contrasts with a quadratic worst-case lower bound in the number of convex pieces needed to decompose the polyhedron itself. The number of new vertices introduced in the process isO(n). The decomposition can be computed inO(n+rlogr) time.

Convex Partitions of Polyhedra: A Lower Bound and Worst-Case Optimal Algorithm

The problem of partitioning a polyhedron into a minimum number of convex pieces is known to be NP-hard. We establish here a quadratic lower bound on the complexity of this problem, and we describe an algorithm that produces a number of convex parts within a constant factor of optimal in the worst case. The algorithm is linear in the size of the polyhedron and cubic in the number of reflex angles. Since in most applications areas, the former quantity greatly exceeds the latter, the algorithm is viable in practice.

Non-Convex Pentahedra

Three straight lines are needed to enclose a finite region of a plane, a two-dimensional space. The polygon formed is a triangle. Different types of triangle are described by adjectives; scalene, isosceles, equilateral and acute angled, right angled, obtuse angled. All the triangles have a property in common: they are all convex. Four lines form a quadrilateral. The convex examples are well known. There are two further types: those which are “re-entrant” having an interior angle greater than 180 degrees, a reflex angle; those in which a pair of opposite sides cross within the quadrilateral, a “crossed” quadrilateral.

Non-Convex Pentahedra

Three straight lines are needed to enclose a finite region of a plane, a two-dimensional space. The polygon formed is a triangle. Different types of triangle are described by adjectives; scalene, isosceles, equilateral and acute angled, right angled, obtuse angled. All the triangles have a property in common: they are all convex.Four lines form a quadrilateral. The convex examples are well known. There are two further types: those which are “re-entrant” having an interior angle greater than 180 degrees, a reflex angle; those in which a pair of opposite sides cross within the quadrilateral, a “crossed” quadrilateral.

The Trigtractor—A Visual Aid for Teaching Trigonometry

The use of the word, function, in connection with the trigonometric ratios, implies related change. It is difficult, by means of a picture, or even by a series of still pictures, to develop in the pupil's mind this idea of functional change relationship. Failure to develop this concept adequately is the basic cause of very much of the difficulties which arise throughout a trigonometry course. To many, “sin A equals opposite side over hypotenuse, or ordinate over distance,” is a verbal memorization without a clear mental picture, and with the idea of functional dependence entirely lacking. To meet this difficulty in his trigonometry classes at the University of California, Dr. Merton Hill uses a large protractor with a movable radius strip carrying a movable ordinate strip at its outer extremity. The visual aid device pictured and described in this article, which for lack of a better term I call a “trigtractor,” is a further development of this protractor idea. It is a real time saver and simplifies matters much in: (1) teaching clear concepts of the trigonometric functions; (2) representing functions as a single line; (3) explaining changes in the functions from positive to negative and vice-versa from quadrant to quadrant; and (4) comparing functions of angles in one quadrant with those in another, especially in expressing functions of obtuse and reflex angles in terms of functions of acute angles.