Inequalities with applications to percolation and reliability

1985 ◽  
Vol 22 (03) ◽  
pp. 556-569 ◽  
Author(s):  
J. Van Den Berg ◽  
H. Kesten
Keyword(s):  
Lower Bound ◽  
Probability Measure ◽  
Cluster Size ◽  
Discrete Analog ◽  
Reliability Theory ◽  
Probability Measures ◽  
Bond Percolation ◽  
Critical Probability ◽  
New Better Than Used ◽  
Better Than

A probability measure μ on ℝn + is defined to be strongly new better than used (SNBU) if for all increasing subsets . For n = 1 this is equivalent to being new better than used (NBU distributions play an important role in reliability theory). We derive an inequality concerning products of NBU probability measures, which has as a consequence that if μ 1, μ 2, ···, μn are NBU probability measures on ℝ+, then the product-measure μ = μ × μ 2 × ··· × μn on ℝn + is SNBU. A discrete analog (i.e., with N instead of ℝ+) also holds. Applications are given to reliability and percolation. The latter are based on a new inequality for Bernoulli sequences, going in the opposite direction to the FKG–Harris inequality. The main application (3.15) gives a lower bound for the tail of the cluster size distribution for bond-percolation at the critical probability. Further applications are simplified proofs of some known results in percolation. A more general inequality (which contains the above as well as the FKG-Harris inequality) is conjectured, and connections with an inequality of Hammersley [12] and others ([17], [19] and [7]) are indicated.

Inequalities with applications to percolation and reliability

1985 ◽  
Vol 22 (3) ◽  
pp. 556-569 ◽  
Author(s):  
J. Van Den Berg ◽  
H. Kesten
Keyword(s):  
Lower Bound ◽  
Probability Measure ◽  
Cluster Size ◽  
Discrete Analog ◽  
Reliability Theory ◽  
Probability Measures ◽  
Bond Percolation ◽  
Critical Probability ◽  
New Better Than Used ◽  
Better Than

A probability measure μ on ℝn+ is defined to be strongly new better than used (SNBU) if for all increasing subsets . For n = 1 this is equivalent to being new better than used (NBU distributions play an important role in reliability theory). We derive an inequality concerning products of NBU probability measures, which has as a consequence that if μ1, μ2, ···, μn are NBU probability measures on ℝ+, then the product-measure μ = μ × μ2 × ··· × μn on ℝn+ is SNBU. A discrete analog (i.e., with N instead of ℝ+) also holds.Applications are given to reliability and percolation. The latter are based on a new inequality for Bernoulli sequences, going in the opposite direction to the FKG–Harris inequality. The main application (3.15) gives a lower bound for the tail of the cluster size distribution for bond-percolation at the critical probability. Further applications are simplified proofs of some known results in percolation. A more general inequality (which contains the above as well as the FKG-Harris inequality) is conjectured, and connections with an inequality of Hammersley [12] and others ([17], [19] and [7]) are indicated.

Characterization of Some First Passage Times Using Log-Concavity and Log-Convexity as Aging Notions

1987 ◽  
Vol 1 (3) ◽  
pp. 279-291 ◽  
Author(s):  
Moshe Shaked ◽  
J. George Shanthikumar
Keyword(s):  
Branching Processes ◽  
Stochastic Ordering ◽  
Reliability Theory ◽  
First Passage Times ◽  
First Passage ◽  
Likelihood Ratio Ordering ◽  
New Better Than Used ◽  
Better Than ◽  
Passage Times

An interpretation of log-concavity and log-convexity as aging notions is given in this paper. It imitates a stochastic ordering characterization of the NBU (new better than used) and the NWU (new worse than used) notions but stochastic ordering is now replaced by the likelihood ratio ordering. The new characterization of log-concavity and log-convexity sheds new light on these properties and enables one to obtain intuitively simple proofs of the log-convexity and log-concavity of some first passage times of interest in branching processes and in reliability theory.

Bond percolation on honeycomb and triangular lattices

1981 ◽  
Vol 13 (2) ◽  
pp. 298-313 ◽  
Author(s):  
John C. Wierman
Keyword(s):  
Lower Bounds ◽  
Triangular Lattice ◽  
Cluster Size ◽  
Honeycomb Lattice ◽  
Bond Percolation ◽  
Critical Probability ◽  
Lattice Graph ◽  
The Mean ◽  
Triangular Lattices ◽  
Crossing Probabilities

The two common critical probabilities for a lattice graph L are the cluster size critical probability pH(L) and the mean cluster size critical probability pT(L). The values for the honeycomb lattice H and the triangular lattice T are proved to be pH(H) = pT(H) = 1–2 sin (π/18) and PH(T) = pT(T) = 2 sin (π/18). The proof uses the duality relationship and the star-triangle relationship between the two lattices, to find lower bounds for sponge-crossing probabilities.

Bond percolation on honeycomb and triangular lattices

1981 ◽  
Vol 13 (02) ◽  
pp. 298-313 ◽  
Author(s):  
John C. Wierman
Keyword(s):  
Lower Bounds ◽  
Triangular Lattice ◽  
Cluster Size ◽  
Honeycomb Lattice ◽  
Bond Percolation ◽  
Critical Probability ◽  
Lattice Graph ◽  
The Mean ◽  
Triangular Lattices ◽  
Crossing Probabilities

The two common critical probabilities for a lattice graphLare the cluster size critical probabilitypH(L) and the mean cluster size critical probabilitypT(L). The values for the honeycomb latticeHand the triangular latticeTare proved to bepH(H) =pT(H) = 1–2 sin (π/18) andPH(T) =pT(T) = 2 sin (π/18). The proof uses the duality relationship and the star-triangle relationship between the two lattices, to find lower bounds for sponge-crossing probabilities.

scholarly journals Some remarks about the maximal perimeter of convex sets with respect to probability measures

2020 ◽  
pp. 2050037
Author(s):  
Galyna V. Livshyts
Keyword(s):  
Lower Bound ◽  
Probability Measure ◽  
Normal Distribution ◽  
Upper Bound ◽  
Convex Sets ◽  
Convex Set ◽  
Standard Normal Distribution ◽  
Probability Measures ◽  
Uniform Measure ◽  
Standard Normal

In this note, we study the maximal perimeter of a convex set in [Formula: see text] with respect to various classes of measures. Firstly, we show that for a probability measure [Formula: see text] on [Formula: see text], satisfying very mild assumptions, there exists a convex set of [Formula: see text]-perimeter at least [Formula: see text] This implies, in particular, that for any isotropic log-concave measure [Formula: see text], one may find a convex set of [Formula: see text]-perimeter of order [Formula: see text]. Secondly, we derive a general upper bound of [Formula: see text] on the maximal perimeter of a convex set with respect to any log-concave measure with density [Formula: see text] in an appropriate position. Our lower bound is attained for a class of distributions including the standard normal distribution. Our upper bound is attained, say, for a uniform measure on the cube. In addition, for isotropic log-concave measures, we prove an upper bound of order [Formula: see text] for the maximal [Formula: see text]-perimeter of a convex set.

scholarly journals A note on the idempotent measures on countable semigroups

1970 ◽  
Vol 11 (4) ◽  
pp. 417-420
Author(s):  
Tze-Chien Sun ◽  
N. A. Tserpes
Keyword(s):  
Probability Measure ◽  
Continuous Mapping ◽  
Compact Semigroup ◽  
Probability Measures ◽  
Left Zero ◽  
Product Topology ◽  
Locally Compact ◽  
Locally Compact Semigroup ◽  
Image Position ◽  
Completely Simple

In [6] we announced the following Conjecture: Let S be a locally compact semigroup and let μ be an idempotent regular probability measure on S with support F. Then(a) F is a closed completely simple subsemigroup.(b) F is isomorphic both algebraically and topologically to a paragroup ([2], p.46) X × G × Y where X and Y are locally compact left-zero and right-zero semi-groups respectively and G is a compact group. In X × G × Y the topology is the product topology and the multiplication of any two elements is defined by , x where [y, x′] is continuous mapping from Y × X → G.(c) The induced μ on X × G × Y can be decomposed as a product measure μX × μG× μY where μX and μY are two regular probability measures on X and Y respectively and μG is the normed Haar measure on G.

Bounded Elastic Moduli Multiplier Technique for Limit Loads: Part 2 - Case Studies

2021 ◽  
Author(s):  
Sathya Prasad Mangalaramanan
Keyword(s):  
Lower Bound ◽  
Case Studies ◽  
Power Law ◽  
Elastic Moduli ◽  
New Method ◽  
Stress Distributions ◽  
Limit Loads ◽  
Elastic Plastic ◽  
Thick Cylinder ◽  
Better Than

Abstract An accompanying paper provides the theoretical underpinnings of a new method to determine statically admissible stress distributions in a structure, called Bounded elastic moduli multiplier technique (BEMMT). It has been shown that, for textbook cases such as thick cylinder, beam, etc., the proposed method offers statically admissible stress distributions better than the power law and closer to elastic-plastic solutions. This paper offers several examples to demonstrate the robustness of this method. Upper and lower bound limit loads are calculated using iterative elastic analyses using both power law and BEMMT. These results are compared with the ones obtained from elastic-plastic FEA. Consistently BEMMT has outperformed power law when it comes to estimating lower bound limit loads.

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