scholarly journals Disjoint Compatibility Graph of Non-Crossing Matchings of Points in Convex Position

10.37236/4403 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Oswin Aichholzer ◽  
Andrei Asinowski ◽  
Tillmann Miltzow
Keyword(s):  
Perfect Matchings ◽  
Connected Components ◽  
Isomorphism Classes ◽  
Medium Components ◽  
Convex Position ◽  
Compatibility Graph ◽  
Straight Line

Let $X_{2k}$ be a set of $2k$ labeled points in convex position in the plane. We consider geometric non-intersecting straight-line perfect matchings of $X_{2k}$. Two such matchings, $M$ and $M'$, are disjoint compatible if they do not have common edges, and no edge of $M$ crosses an edge of $M'$. Denote by $\rm{DCM}_k$ the graph whose vertices correspond to such matchings, and two vertices are adjacent if and only if the corresponding matchings are disjoint compatible. We show that for each $k \geq 9$, the connected components of $\rm{DCM}_k$ form exactly three isomorphism classes - namely, there is a certain number of isomorphic small components, a certain number of isomorphic medium components, and one big component. The number and the structure of small and medium components is determined precisely.

scholarly journals Commuting elements in central products of special unitary groups

2012 ◽  
Vol 56 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Alejandro Adem ◽  
F. R. Cohen ◽  
José Manuel Gómez
Keyword(s):  
Prime Number ◽  
Moduli Space ◽  
Unitary Group ◽  
Unitary Groups ◽  
Connected Components ◽  
Flat Connections ◽  
Isomorphism Classes ◽  
Central Product ◽  
Central Products ◽  
Path Connected

AbstractWe study the space of commuting elements in the central product Gm,p of m copies of the special unitary group SU(p), where p is a prime number. In particular, a computation for the number of path-connected components of these spaces is given and the geometry of the moduli space Rep(ℤn, Gm,p) of isomorphism classes of flat connections on principal Gm,p-bundles over the n-torus is completely described for all values of n, m and p.

scholarly journals Edge-Removal and Non-Crossing Configurations in Geometric Graphs

2010 ◽  
Vol Vol. 12 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Oswin Aichholzer ◽  
Sergio Cabello ◽  
Ruy Fabila-Monroy ◽  
David Flores-Peñaloza ◽  
Thomas Hackl ◽  
...  
Keyword(s):  
Extremal Problem ◽  
General Position ◽  
Geometric Graph ◽  
Perfect Matchings ◽  
Geometric Graphs ◽  
Line Segments ◽  
Straight Line ◽  
Edge Removal ◽  
International Audience ◽  
Point Set

Graphs and Algorithms International audience A geometric graph is a graph G = (V, E) drawn in the plane, such that V is a point set in general position and E is a set of straight-line segments whose endpoints belong to V. We study the following extremal problem for geometric graphs: How many arbitrary edges can be removed from a complete geometric graph with n vertices such that the remaining graph still contains a certain non-crossing subgraph. The non-crossing subgraphs that we consider are perfect matchings, subtrees of a given size, and triangulations. In each case, we obtain tight bounds on the maximum number of removable edges.

On Degenerations of Modules With Nondirecting Indecomposable Summands

1996 ◽  
Vol 48 (5) ◽  
pp. 1091-1120 ◽  
Author(s):  
A. Skowroński ◽  
G. Zwara
Keyword(s):  
Natural Number ◽  
Partial Order ◽  
General Linear Group ◽  
Partial Orders ◽  
Connected Components ◽  
Closed Field ◽  
Affine Variety ◽  
Indecomposable Modules ◽  
Isomorphism Classes ◽  
Finite Dimensional

AbstractLet A be a finite dimensional associative K-algebra with an identity over an algebraically closed field K, d a natural number, and modA(d) the affine variety of d-dimensional A-modules. The general linear group Gld(K) acts on modA(d) by conjugation, and the orbits correspond to the isomorphism classes of d-dimensional modules. For M and N in modA(d), N is called a degeneration of M, if TV belongs to the closure of the orbit of M. This defines a partial order ≤deg on modA(d). There has been a work [1], [10], [11], [21] connecting ≤deg with other partial orders ≤ext and ≤deg on modA(d) defined in terms of extensions and homomorphisms. In particular, it is known that these partial orders coincide in the case A is representation-finite and its Auslander-Reiten quiver is directed. We study degenerations of modules from the additive categories given by connected components of the Auslander-Reiten quiver of A having oriented cycles. We show that the partial orders ≤ext, ≤deg and < coincide on modules from the additive categories of quasi-tubes [24], and describe minimal degenerations of such modules. Moreover, we show that M ≤degN does not imply M ≤ext N for some indecomposable modules M and N lying in coils in the sense of [4].

scholarly journals Drawing Hamiltonian Cycles with no Large Angles

10.37236/2356 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Adrian Dumitrescu ◽  
János Pach ◽  
Géza Tóth
Keyword(s):  
Hamiltonian Cycle ◽  
Hamiltonian Cycles ◽  
Point Sets ◽  
Convex Position ◽  
Straight Line ◽  
Open Region ◽  
Point Set ◽  
Rectifiable Curves ◽  
Element Set

Let $n \geq 4$ be even. It is shown that every set $S$ of $n$ points in the plane can be connected by a (possibly self-intersecting) spanning tour (Hamiltonian cycle) consisting of $n$ straight-line edges such that the angle between any two consecutive edges is at most $2\pi/3$. For $n=4$ and $6$, this statement is tight. It is also shown that every even-element point set $S$ can be partitioned  into at most two subsets, $S_1$ and $S_2$, each admitting a spanning tour with no angle larger than $\pi/2$. Fekete and Woeginger conjectured that for sufficiently large even $n$, every $n$-element set admits such a spanning tour. We confirm this conjecture for point sets in convex position. A much stronger result holds for large point sets randomly and uniformly selected from an open region bounded by finitely many rectifiable curves: for any $\epsilon>0$, these sets almost surely admit a spanning tour with no angle larger than $\epsilon$.

scholarly journals MEASURABLE PERFECT MATCHINGS FOR ACYCLIC LOCALLY COUNTABLE BOREL GRAPHS

2017 ◽  
Vol 82 (1) ◽  
pp. 258-271
Author(s):  
CLINTON T. CONLEY ◽  
BENJAMIN D. MILLER
Keyword(s):  
Analogous Result ◽  
Perfect Matchings ◽  
Equivalence Relations ◽  
Connected Components ◽  
Locally Finite ◽  
Finite Case ◽  
Locally Countable ◽  
Countable Case ◽  
Selection Of

AbstractWe characterize the structural impediments to the existence of Borel perfect matchings for acyclic locally countable Borel graphs admitting a Borel selection of finitely many ends from their connected components. In particular, this yields the existence of Borel matchings for such graphs of degree at least three. As a corollary, it follows that acyclic locally countable Borel graphs of degree at least three generating μ-hyperfinite equivalence relations admit μ-measurable matchings. We establish the analogous result for Baire measurable matchings in the locally finite case, and provide a counterexample in the locally countable case.

Microdensitometer—computer correlation analysis of ultrastructural periodicity

1978 ◽  
Vol 36 (2) ◽  
pp. 32-33
Author(s):  
D.R. Ensor ◽  
C.G. Jensen ◽  
J.A. Fillery ◽  
R.J.K. Baker
Keyword(s):  
Correlation Analysis ◽  
Background Noise ◽  
Computer Control ◽  
Optical Diffraction ◽  
Screw Thread ◽  
Straight Line ◽  
Computer Storage ◽  
Correlation Process ◽  
Electron Microscope Image ◽  
Fixed Beam

Because periodicity is a major indicator of structural organisation numerous methods have been devised to demonstrate periodicity masked by background “noise” in the electron microscope image (e.g. photographic image reinforcement, Markham et al, 1964; optical diffraction techniques, Horne, 1977; McIntosh,1974). Computer correlation analysis of a densitometer tracing provides another means of minimising "noise". The correlation process uncovers periodic information by cancelling random elements. The technique is easily executed, the results are readily interpreted and the computer removes tedium, lends accuracy and assists in impartiality.A scanning densitometer was adapted to allow computer control of the scan and to give direct computer storage of the data. A photographic transparency of the image to be scanned is mounted on a stage coupled directly to an accurate screw thread driven by a stepping motor. The stage is moved so that the fixed beam of the densitometer (which is directed normal to the transparency) traces a straight line along the structure of interest in the image.

Freeze fracture imaging and computer simulation of liposome structure: Membrane geometry and defects

1983 ◽  
Vol 41 ◽  
pp. 646-647
Author(s):  
Joseph A. Zasadzinski
Keyword(s):  
Liquid Crystalline ◽  
Freeze Fracture ◽  
Continuum Theory ◽  
Closed Surfaces ◽  
Straight Line ◽  
Low Weight ◽  
Concentric Spheres ◽  
Membrane Geometry ◽  
Dupin Cyclides ◽  
Limiting Case

At low weight fractions, many surfactant and biological amphiphiles form dispersions of lamellar liquid crystalline liposomes in water. Amphiphile molecules tend to align themselves in parallel bilayers which are free to bend. Bilayers must form closed surfaces to separate hydrophobic and hydrophilic domains completely. Continuum theory of liquid crystals requires that the constant spacing of bilayer surfaces be maintained except at singularities of no more than line extent. Maxwell demonstrated that only two types of closed surfaces can satisfy this constraint: concentric spheres and Dupin cyclides. Dupin cyclides (Figure 1) are parallel closed surfaces which have a conjugate ellipse (r1) and hyperbola (r2) as singularities in the bilayer spacing. Any straight line drawn from a point on the ellipse to a point on the hyperbola is normal to every surface it intersects (broken lines in Figure 1). A simple example, and limiting case, is a family of concentric tori (Figure 1b).To distinguish between the allowable arrangements, freeze fracture TEM micrographs of representative biological (L-α phosphotidylcholine: L-α PC) and surfactant (sodium heptylnonyl benzenesulfonate: SHBS)liposomes are compared to mathematically derived sections of Dupin cyclides and concentric spheres.

Human Enamel Replication

1976 ◽  
Vol 34 ◽  
pp. 180-181
Author(s):  
Norman L. Dockum ◽  
John G. Dockum
Keyword(s):  
Organic Matrix ◽  
Human Teeth ◽  
Human Enamel ◽  
Straight Line ◽  
Ultrastructural Characteristics

Ultrastructural characteristics of fractured human enamel and acid-etched enamel were compared using acetate replicas shadowed with platinum and palladium. Shadowed replications of acid-etched surfaces were also obtained by the same method.Enamel from human teeth has a rod structure within which there are crystals of hydroxyapatite contained within a structureless organic matrix composed of keratin. The rods which run at right angles from the dentino-enamel junction are considered to run in a straight line perpendicular to the perimeter of the enamel, however, in many areas these enamel rods overlap, interlacing and intertwining with one another.

The Connection Between Communication and Self-Esteem: A "Straight" Line?

1994 ◽  
Vol 39 (2) ◽  
pp. 214-215
Author(s):  
Volker Thomas
Keyword(s):  
Self Esteem ◽  
Straight Line

How to Draw a Straight Line

1877 ◽  
Vol 4 (86supp) ◽  
pp. 1364-1365
Keyword(s):  
Straight Line
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