scholarly journals From Random Polygon to Ellipse: An Eigenanalysis

SIAM Review ◽  
2010 ◽  
Vol 52 (1) ◽  
pp. 151-170 ◽  
Author(s):  
Adam N. Elmachtoub ◽  
Charles F. Van Loan
Keyword(s):  
Random Polygon

A Statistical Study of Random Knotting Using the Vassiliev Invariants

1994 ◽  
Vol 03 (03) ◽  
pp. 321-353 ◽  
Author(s):  
Tetsuo Deguchi ◽  
Kyoichi Tsurusaki
Keyword(s):  
Statistical Study ◽  
Numerical Data ◽  
Vassiliev Invariants ◽  
Random Polygon ◽  
Type K ◽  
Random Polygons ◽  
Self Avoiding Walks ◽  
Asymptotic Scaling

Employing the Vassiliev invariants as tools for determining knot types of polygons in 3 dimensions, we evaluate numerically the knotting probability PK(N) of the Gaussian random polygon being equivalent to a knot type K. For prime knots and composite knots we plot the knotting probability PK(N) against the number N of polygonal nodes. Taking the analogy with the asymptotic scaling behaviors of self-avoiding walks, we propose a formula of fitting curves to the numerical data. The curves fit well the graphs of the knotting probability PK(N) versus N. This agreement suggests to us that the scaling formula for the knotting probability might also work for the random polygons other than the Gaussian random polygon.

ON RANDOM KNOTS

1994 ◽  
Vol 03 (03) ◽  
pp. 419-429 ◽  
Author(s):  
YUANAN DIAO ◽  
NICHOLAS PIPPENGER ◽  
DE WITT SUMNERS
Keyword(s):  
Gaussian Distribution ◽  
Piecewise Linear ◽  
Random Polygon ◽  
Random Polygons ◽  
Almost All ◽  
Random Knots

In this paper, we consider knotting of Gaussian random polygons in 3-space. A Gaussian random polygon is a piecewise linear circle with n edges in which the length of the edges follows a Gaussian distribution. We prove a continuum version of Kesten's Pattern Theorem for these polygons, and use this to prove that the probability that a Gaussian random polygon of n edges in 3-space is knotted tends to one exponentially rapidly as n tends to infinity. We study the properties of Gaussian random knots, and prove that the entanglement complexity of Gaussian random knots gets arbitrarily large as n tends to infinity. We also prove that almost all Gaussian random knots are chiral.

scholarly journals Limit theorems for generalized perimeters of random inscribed polygons. I

2020 ◽  
Vol 65 (4) ◽  
pp. 678-687
Author(s):  
Ekaterina N. Simarova ◽  
◽  
Keyword(s):  
Stochastic Geometry ◽  
Limit Theorems ◽  
Extreme Values ◽  
Limiting Behavior ◽  
Limit Distributions ◽  
Random Polygon ◽  
Generalized Perimeter ◽  
Definition Of ◽  
Inscribed Polygons

Lao and Mayer (2008) recently developed the theory of U-max-statistics, where instead of the usual averaging the values of the kernel over subsets, the maximum of the kernel is considered. Such statistics often appear in stochastic geometry. Their limit distributions are related to distributions of extreme values. This is the first article devoted to the study of the generalized perimeter (the sum of side powers) of an inscribed random polygon, and of U-max-statistics associated with it. It describes the limiting behavior for the extreme values of the generalized perimeter. This problem has not been studied in the literature so far. One obtains some limit theorems in the case when the parameter y, arising in the definition of the generalized perimeter does not exceed 1.

The Knotting of Equilateral Polygons in R3

1995 ◽  
Vol 04 (02) ◽  
pp. 189-196 ◽  
Author(s):  
YUANAN DIAO
Keyword(s):  
Positive Constant ◽  
Random Polygon ◽  
Random Polygons

It was proved in [4] that the knotting probability of a Gaussian random polygon goes to 1 as the length of the polygon goes to infinity. In this paper, we prove the same result for the equilateral random polygons in R3. More precisely, if EPn is an equilateral random polygon of n steps, then we have [Formula: see text] provided that n is large enough, where ∊ is some positive constant.

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