scholarly journals Variance estimates for random disc-polygons in smooth convex discs

2018 ◽  
Vol 55 (4) ◽  
pp. 1143-1157 ◽  
Author(s):  
Ferenc Fodor ◽  
Viktor Vígh
Keyword(s):  
Probability Model ◽  
Upper Bounds ◽  
Smooth Convex ◽  
Convex Disc ◽  
Variance Estimates

Abstract In this paper we prove asymptotic upper bounds on the variance of the number of vertices and the missed area of inscribed random disc-polygons in smooth convex discs whose boundary is C+2. We also consider a circumscribed variant of this probability model in which the convex disc is approximated by the intersection of random circles.

scholarly journals On Random Disc Polygons in Smooth Convex Discs

2014 ◽  
Vol 46 (4) ◽  
pp. 899-918 ◽  
Author(s):  
F. Fodor ◽  
P. Kevei ◽  
V. Vígh
Keyword(s):  
Smooth Convex ◽  
Convex Disc ◽  
Asymptotic Formulae ◽  
Planar Convex ◽  
Area And Perimeter ◽  
Spindle Convexity

In this paper we generalize some of the classical results of Rényi and Sulanke (1963), (1964) in the context of spindle convexity. A planar convex disc S is spindle convex if it is the intersection of congruent closed circular discs. The intersection of finitely many congruent closed circular discs is called a disc polygon. We prove asymptotic formulae for the expectation of the number of vertices, missed area, and perimeter difference of uniform random disc polygons contained in a sufficiently smooth spindle convex disc.

Statistical properties of hybrid composites. I. Recursion analysis

1983 ◽  
Vol 389 (1796) ◽  
pp. 67-100 ◽  
Keyword(s):  
Hybrid Composites ◽  
Probability Model ◽  
Load Sharing ◽  
Upper Bounds ◽  
Strength Distribution ◽  
Local Load ◽  
Sharing Rule ◽  
Critical Crack ◽  
Breaking Strain ◽  
Hybrid Effect

An adaptation of the chain-of-bundles probability model for unidirectional intraply hybrid composites consisting of two types of fibres is given. Local load sharing, which is sensitive to the different elastic moduli of the fibres, is assumed for the non-failed fibre segments in each bundle. A sequence of tight upper bounds is developed for the probability distribu­tion of strength for the hybrid. The upper bounds are based upon the occurrence of k or more adjacent broken fibre segments in a bundle; this event is necessary but not sufficient for bundle failure. This development allows for a description of a critical crack size k *, dependent upon the load on the hybrid, which is a characterization of the length of a crack that catastrophically propagates causing bundle failure with virtual certainty. The upper bound developed with k *, based upon the hybrid median strength, is essentially identical to the true probability distribution of hybrid strength. It is also shown that the strength distribution for the hybrid composite has a weakest link structure in terms of a charac­teristic distribution function that is highly dependent upon the local load sharing rule, the fibre properties, and the geometrical structure of the hybrid. Numerical results from the model show that typically there is a negative ‘hybrid effect’ for hybrid breaking strain, but there is a positive ‘hybrid effect’ for hybrid tensile strength.

scholarly journals Intrinsic volumes of inscribed random polytopes in smooth convex bodies

2010 ◽  
Vol 42 (3) ◽  
pp. 605-619 ◽  
Author(s):  
I. Bárány ◽  
F. Fodor ◽  
V. Vígh
Keyword(s):  
Convex Body ◽  
Convex Hull ◽  
Convex Bodies ◽  
Upper Bounds ◽  
Smooth Convex ◽  
Lower And Upper Bounds ◽  
Covering Theorem ◽  
Intrinsic Volumes ◽  
Large Numbers ◽  
Distribution Matching

Let K be a d-dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by Kn the convex hull of n points chosen randomly and independently from K according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the sth intrinsic volumes Vs(Kn) of Kn for s ∈ {1,…,d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of Kn. The essential tools are the economic cap covering theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.

scholarly journals Intrinsic volumes of inscribed random polytopes in smooth convex bodies

2010 ◽  
Vol 42 (03) ◽  
pp. 605-619
Author(s):  
I. Bárány ◽  
F. Fodor ◽  
V. Vígh
Keyword(s):  
Convex Body ◽  
Convex Hull ◽  
Convex Bodies ◽  
Upper Bounds ◽  
Smooth Convex ◽  
Lower And Upper Bounds ◽  
Covering Theorem ◽  
Intrinsic Volumes ◽  
Large Numbers ◽  
Distribution Matching

LetKbe ad-dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote byKnthe convex hull ofnpoints chosen randomly and independently fromKaccording to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of thesth intrinsic volumesVs(Kn) ofKnfors∈ {1,…,d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes ofKn. The essential tools are the economic cap covering theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.

scholarly journals An Analysis of Phase Transition in NK Landscapes

2002 ◽  
Vol 17 ◽  
pp. 309-332 ◽  
Author(s):  
Y. Gao ◽  
J. Culberson
Keyword(s):  
Phase Transition ◽  
Probability Model ◽  
Fixed Ratio ◽  
Upper Bounds ◽  
Ratio Model ◽  
Problem Size ◽  
Threshold Phenomena ◽  
Uniform Probability ◽  
Random Instances ◽  
Nk Landscapes

In this paper, we analyze the decision version of the NK landscape model from the perspective of threshold phenomena and phase transitions under two random distributions, the uniform probability model and the fixed ratio model. For the uniform probability model, we prove that the phase transition is easy in the sense that there is a polynomial algorithm that can solve a random instance of the problem with the probability asymptotic to 1 as the problem size tends to infinity. For the fixed ratio model, we establish several upper bounds for the solubility threshold, and prove that random instances with parameters above these upper bounds can be solved polynomially. This, together with our empirical study for random instances generated below and in the phase transition region, suggests that the phase transition of the fixed ratio model is also easy.

scholarly journals On Random Disc Polygons in Smooth Convex Discs

2014 ◽  
Vol 46 (04) ◽  
pp. 899-918 ◽  
Author(s):  
F. Fodor ◽  
P. Kevei ◽  
V. Vígh
Keyword(s):  
Smooth Convex ◽  
Convex Disc ◽  
Asymptotic Formulae ◽  
Planar Convex ◽  
Area And Perimeter ◽  
Spindle Convexity

In this paper we generalize some of the classical results of Rényi and Sulanke (1963), (1964) in the context of spindle convexity. A planar convex discSis spindle convex if it is the intersection of congruent closed circular discs. The intersection of finitely many congruent closed circular discs is called a disc polygon. We prove asymptotic formulae for the expectation of the number of vertices, missed area, and perimeter difference of uniform random disc polygons contained in a sufficiently smooth spindle convex disc.

Probability model for control parameters in the aircraft's information and control systems

2018 ◽  
Vol 2 ◽  
pp. 9-16
Author(s):  
A. Al-Ammouri ◽  
◽  
H.A. Al-Ammori ◽  
A.E. Klochan ◽  
A.M. Al-Akhmad ◽  
...  
Keyword(s):  
Control Systems ◽  
Probability Model ◽  
Control Parameters ◽  
And Control

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

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