scholarly journals The entropic pressure of a lattice polygon

2013 ◽  
Vol 2013 (10) ◽  
pp. P10005 ◽  
Author(s):  
F Gassoumov ◽  
E J Janse van Rensburg
Keyword(s):  
Lattice Polygon

scholarly journals The lattice size of a lattice polygon

2015 ◽  
Vol 136 ◽  
pp. 64-95 ◽  
Author(s):  
Wouter Castryck ◽  
Filip Cools
Keyword(s):  
Lattice Polygon ◽  
Lattice Size

scholarly journals Moving Out the Edges of a Lattice Polygon

2011 ◽  
Vol 47 (3) ◽  
pp. 496-518 ◽  
Author(s):  
Wouter Castryck
Keyword(s):  
Lattice Polygon

scholarly journals On convex lattice polygons

1976 ◽  
Vol 15 (3) ◽  
pp. 395-399 ◽  
Author(s):  
P.R. Scott
Keyword(s):  
Boundary Points ◽  
Lattice Polygon ◽  
Convex Lattice ◽  
Convex Lattice Polygons ◽  
Interior Points ◽  
Lattice Polygons

Let π be a convex lattice polygon with b boundary points and c (≥ 1) interior points. We show that for any given c, the number b satisfies b ≤ 2c + 7, and identify the polygons for which equality holds.

scholarly journals Convex lattice polygons of minimum area

1990 ◽  
Vol 42 (3) ◽  
pp. 353-367 ◽  
Author(s):  
R.J. Simpson
Keyword(s):  
Integer Lattice ◽  
Minimum Area ◽  
Lattice Polygon ◽  
Convex Lattice ◽  
Convex Lattice Polygons ◽  
Lattice Polygons

A convex lattice polygon is a polygon whose vertices are points on the integer lattice and whose interior angles are strictly less than π radians. We define a(2n) to be the least possible area of a convex lattice polygon with 2n vertices. A method for constructing convex lattice polygons with area a(2n) is described, and values of a(2n) for low n are obtained.

scholarly journals A note on bounds on the minimum area of convex lattice polygons

1992 ◽  
Vol 45 (2) ◽  
pp. 237-240 ◽  
Author(s):  
Charles J. Colbourn ◽  
R.J. Simpson
Keyword(s):  
Positive Constant ◽  
Minimum Area ◽  
Lattice Polygon ◽  
Convex Lattice ◽  
Convex Lattice Polygons ◽  
Lattice Polygons

The minimum area a(v) of a v–sided convex lattice polygon is known to satisfy . We conjecture that a(v) = cv3 + o(v3), for c a constant; we prove that , and that for some positive constant c, .

scholarly journals Bounds on Multiple Self-avoiding Polygons

2018 ◽  
Vol 61 (3) ◽  
pp. 518-530 ◽  
Author(s):  
Kyungpyo Hong ◽  
Seungsang Oh
Keyword(s):  
Numerical Simulations ◽  
Upper Bounds ◽  
Square Lattice ◽  
Rectangular Grid ◽  
Lower And Upper Bounds ◽  
Lattice Polygon ◽  
Ring Polymers ◽  
Self Avoiding Walk ◽  
Multiple Ring

AbstractA self-avoiding polygon is a lattice polygon consisting of a closed self-avoiding walk on a square lattice. Surprisingly little is known rigorously about the enumeration of self-avoiding polygons, although there are numerous conjectures that are believed to be true and strongly supported by numerical simulations. As an analogous problemto this study, we considermultiple self-avoiding polygons in a confined region as a model for multiple ring polymers in physics. We find rigorous lower and upper bounds for the number pm×n of distinct multiple self-avoiding polygons in the m × n rectangular grid on the square lattice. For m = 2, p2×n = 2n−1 − 1. And for integers m, n ≥ 3,

scholarly journals The Projections of Convex Lattice Sets of Points in E2

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Yu Gu ◽  
Lin Si
Keyword(s):  
Discrete Version ◽  
Symmetric Convex ◽  
Projection Theorem ◽  
Lattice Polygon ◽  
Convex Lattice ◽  
Centrally Symmetric

Can one determine a centrally symmetric lattice polygon by its projections? In 2005, Gardner et al. proposed the above discrete version of Aleksandrov’s projection theorem. In this paper, we define a coordinate matrix for a centrally symmetric convex lattice set and suggest an algorithm to study this problem.

scholarly journals Graphs associated with triangulations of lattice polygons

1989 ◽  
Vol 47 (3) ◽  
pp. 391-398 ◽  
Author(s):  
Duane DeTemple ◽  
Jack M. Robertson
Keyword(s):  
Independent Sets ◽  
Factor Graph ◽  
Geometric Information ◽  
Edge Disjoint ◽  
Lattice Polygon ◽  
Simple Lattice ◽  
Edge Crossing ◽  
Partition Graph ◽  
Maximal Independent Sets ◽  
Lattice Polygons

AbstractTwo graphs, the edge crossing graph E and the triangle graph T are associated with a simple lattice polygon. The maximal independent sets of vertices of E and T are derived including a formula for the size of the fundamental triangles. Properties of E and T are derived including a formula for the size of the maximal independent sets in E and T. It is shown that T is a factor graph of edge-disjoint 4-cycles, which gives corresponding geometric information, and is a partition graph as recently defined by the authors and F. Harary.

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