The distribution of the sequence {nξ}(n = 0, 1, 2, …)

1965 ◽  
Vol 61 (3) ◽  
pp. 665-670 ◽  
Author(s):  
John H. Halton
Keyword(s):  
Unit Interval ◽  
Simple Generalization ◽  
Unit Hypercube

Introduction and statement of results. We shall describe how, for successive integers N, the points {nξ}, with n = 0, 1, …,N – 1, are distributed in the closed unit interval U = [0, 1]; by showing how successive points {Nξ,} modify the partition of U produced by the previous points. The simple generalization to the k-dimensional sequence {nξ} = ({nξ(1)},{nξ(2)}, …,{nξ(k)}), in the unit hypercube Uk, is also made.

Bounding the stochastic performance of continuum structure functions. II

1987 ◽  
Vol 24 (03) ◽  
pp. 609-618 ◽  
Author(s):  
Laurence A. Baxter ◽  
Chul Kim
Keyword(s):  
Structure Function ◽  
Random Variables ◽  
Structure Functions ◽  
Unit Interval ◽  
Modular Decomposition ◽  
Associated Random Variables ◽  
Unit Hypercube ◽  
Continuum Structure

A continuum structure function γ is a non-decreasing mapping from the unit hypercube to the unit interval. Block and Savits (1984) use the sets and to determine bounds on the distribution of γ (X) when X is a vector of associated random variables. It is shown that, if γ admits of a modular decomposition, improved bounds may be obtained.

Continuum structures. II

1986 ◽  
Vol 99 (2) ◽  
pp. 331-338 ◽  
Author(s):  
Laurence A. Baxter
Keyword(s):  
Structure Function ◽  
Unit Interval ◽  
Continuum Structures ◽  
Unit Hypercube ◽  
Continuum Structure ◽  
The Subject ◽  
Nondecreasing Mapping

AbstractA continuum structure function is a nondecreasing mapping from the unit hypercube to the unit interval. This paper continues the author's work on the subject, extending Griffith's definitions of coherency to such functions and studying the analytic properties of a continuum structure function based on Natvig's ‘second suggestion’.

Bounding the stochastic performance of continuum structure functions. II

1987 ◽  
Vol 24 (3) ◽  
pp. 609-618 ◽  
Author(s):  
Laurence A. Baxter ◽  
Chul Kim
Keyword(s):  
Structure Function ◽  
Random Variables ◽  
Structure Functions ◽  
Unit Interval ◽  
Modular Decomposition ◽  
Associated Random Variables ◽  
Unit Hypercube ◽  
Continuum Structure

A continuum structure function γ is a non-decreasing mapping from the unit hypercube to the unit interval. Block and Savits (1984) use the sets and to determine bounds on the distribution of γ (X) when X is a vector of associated random variables. It is shown that, if γ admits of a modular decomposition, improved bounds may be obtained.

Structure functions with finite minimal vector sets

1989 ◽  
Vol 26 (1) ◽  
pp. 196-201 ◽  
Author(s):  
Laurence A. Baxter ◽  
Seung Min Lee
Keyword(s):  
Structure Function ◽  
Random Variables ◽  
Structure Functions ◽  
Unit Interval ◽  
Independent Random Variables ◽  
Unit Hypercube ◽  
Continuum Structure ◽  
Minimal Vectors

A continuum structure function (CSF) y is a non-decreasing mapping from the unit hypercube to the unit interval. Define whereas γ (γ) < α for all y < x}, the set of minimal vectors to level α. This paper examines CSFs for which each Pα is finite. It is shown that if γ is such a CSF and X is a vector of independent random variables, the distribution of γ (X) is readily calculated. Further, if γ is an arbitrary right-continuous CSF, the distribution of γ (X) may be approximated arbitrarily closely by that of γ′(X) where γ′ is a right-continuous CSF for which each minimal vector set is finite.

Bounding the stochastic performance of continuum structure functions. I

1986 ◽  
Vol 23 (3) ◽  
pp. 660-669 ◽  
Author(s):  
Laurence A. Baxter ◽  
Chul Kim
Keyword(s):  
Structure Function ◽  
Random Variables ◽  
Structure Functions ◽  
Unit Interval ◽  
Minimal Path ◽  
Modular Decomposition ◽  
Associated Random Variables ◽  
Unit Hypercube ◽  
Continuum Structure

A continuum structure function γ is a non-decreasing mapping from the unit hypercube to the unit interval. Minimal path (cut) sets of upper (lower) simple continuum structure functions are introduced and are used to determine bounds on the distribution of γ (Χ) when X is a vector of associated random variables and when γ is right (left)-continuous. It is shown that, if γ admits of a modular decomposition, improved bounds may be obtained.

Bounding the stochastic performance of continuum structure functions. I

1986 ◽  
Vol 23 (03) ◽  
pp. 660-669 ◽  
Author(s):  
Laurence A. Baxter ◽  
Chul Kim
Keyword(s):  
Structure Function ◽  
Random Variables ◽  
Structure Functions ◽  
Unit Interval ◽  
Minimal Path ◽  
Modular Decomposition ◽  
Associated Random Variables ◽  
Unit Hypercube ◽  
Continuum Structure

A continuum structure function γ is a non-decreasing mapping from the unit hypercube to the unit interval. Minimal path (cut) sets of upper (lower) simple continuum structure functions are introduced and are used to determine bounds on the distribution of γ (Χ) when X is a vector of associated random variables and when γ is right (left)-continuous. It is shown that, if γ admits of a modular decomposition, improved bounds may be obtained.

Reliability importance for continuum structure functions

1987 ◽  
Vol 24 (3) ◽  
pp. 779-785 ◽  
Author(s):  
Chul Kim ◽  
Laurence A. Baxter
Keyword(s):  
Structure Function ◽  
Structure Functions ◽  
Unit Interval ◽  
System Level ◽  
Unit Hypercube ◽  
Continuum Structure ◽  
Definition Of

A continuum structure function is a non-decreasing mapping from the unit hypercube to the unit interval. A definition of the reliability importance, ℛi(α) say, of component i at system level α(0 < α ≦ 1) is proposed. Some properties of this function are deduced, in particular conditions under which and conditions under which ℛi(α) is positive (0 < α < 1).

Reliability importance for continuum structure functions

1987 ◽  
Vol 24 (03) ◽  
pp. 779-785
Author(s):  
Chul Kim ◽  
Laurence A. Baxter
Keyword(s):  
Structure Function ◽  
Structure Functions ◽  
Unit Interval ◽  
System Level ◽  
Unit Hypercube ◽  
Continuum Structure ◽  
Definition Of

A continuum structure function is a non-decreasing mapping from the unit hypercube to the unit interval. A definition of the reliability importance, ℛi(α) say, of component i at system level α(0 &lt; α ≦ 1) is proposed. Some properties of this function are deduced, in particular conditions under which and conditions under which ℛi(α) is positive (0 &lt; α &lt; 1).

Structure functions with finite minimal vector sets

1989 ◽  
Vol 26 (01) ◽  
pp. 196-201
Author(s):  
Laurence A. Baxter ◽  
Seung Min Lee
Keyword(s):  
Structure Function ◽  
Random Variables ◽  
Structure Functions ◽  
Unit Interval ◽  
Independent Random Variables ◽  
Unit Hypercube ◽  
Continuum Structure ◽  
Minimal Vectors

A continuum structure function (CSF) y is a non-decreasing mapping from the unit hypercube to the unit interval. Define whereas γ (γ) &lt; α for all y &lt; x}, the set of minimal vectors to level α. This paper examines CSFs for which each Pα is finite. It is shown that if γ is such a CSF and X is a vector of independent random variables, the distribution of γ (X) is readily calculated. Further, if γ is an arbitrary right-continuous CSF, the distribution of γ (X) may be approximated arbitrarily closely by that of γ′(X) where γ′ is a right-continuous CSF for which each minimal vector set is finite.

Further Properties of Reliability Importance for Continuum Structure Functions

1989 ◽  
Vol 3 (2) ◽  
pp. 237-246 ◽  
Author(s):  
Laurence A. Baxter ◽  
Seung Min Lee
Keyword(s):  
Continuous Function ◽  
Structure Function ◽  
Simple Algorithm ◽  
Structure Functions ◽  
Unit Interval ◽  
System Level ◽  
Unit Hypercube ◽  
Continuum Structure ◽  
Definition Of ◽  
Nondecreasing Mapping

A continuum structure function (CSF) is a nondecreasing mapping from the unit hypercube to the unit interval. The Kim-Baxter definition of the reliability importance of component i in a CSF at system level α, Ri(α), say, is reviewed. Conditions under which Ri(α) is positive, under which Ri(α) is a continuous function of α, and under which Ri(α) ≥ Rj(α) uniformly in α are presented. A simple algorithm for evaluating Ri(α) is described.

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