Random space filling and moments of coverage in geometrical probability

1978 ◽  
Vol 15 (2) ◽  
pp. 340-355 ◽  
Author(s):  
Andrew F. Siegel
Keyword(s):  
Coverage Probability ◽  
Arc Length ◽  
Random Set ◽  
Space Filling ◽  
Geometrical Probability ◽  
Almost Everywhere ◽  
Coverage Probabilities ◽  
Special Cases ◽  
Fixed Set ◽  
General Conditions

The moments of the random proportion of a fixed set that is covered by a random set (moments of coverage) are shown to converge under very general conditions to the probability that the fixed set is almost everywhere covered by the random set. Moments and coverage probabilities are calculated for several cases of random arcs of random sizes on the circle. When comparing arc length distributions having the same expectation, it is conjectured that if one concentrates more mass near that expectation, the corresponding coverage probability will be smaller. Support for this conjecture is provided in special cases.

Random space filling and moments of coverage in geometrical probability

1978 ◽  
Vol 15 (02) ◽  
pp. 340-355 ◽  
Author(s):  
Andrew F. Siegel
Keyword(s):  
Coverage Probability ◽  
Arc Length ◽  
Random Set ◽  
Space Filling ◽  
Geometrical Probability ◽  
Almost Everywhere ◽  
Coverage Probabilities ◽  
Special Cases ◽  
Fixed Set ◽  
General Conditions

The moments of the random proportion of a fixed set that is covered by a random set (moments of coverage) are shown to converge under very general conditions to the probability that the fixed set is almost everywhere covered by the random set. Moments and coverage probabilities are calculated for several cases of random arcs of random sizes on the circle. When comparing arc length distributions having the same expectation, it is conjectured that if one concentrates more mass near that expectation, the corresponding coverage probability will be smaller. Support for this conjecture is provided in special cases.

scholarly journals GENERALIZED W3-STRINGS FROM FREE FIELDS

1995 ◽  
Vol 10 (06) ◽  
pp. 515-524 ◽  
Author(s):  
J. M. FIGUEROA-O'FARRILL ◽  
C. M. HULL ◽  
L. PALACIOS ◽  
E. RAMOS
Keyword(s):  
Bosonic Strings ◽  
Free Field ◽  
Space Time ◽  
General Construction ◽  
Special Cases ◽  
Free Fields ◽  
Rule Out ◽  
General Conditions ◽  
String Theories

The conventional quantization of w3-strings gives theories which are equivalent to special cases of bosonic strings. We explore whether a more general quantization can lead to new generalized W3-string theories by seeking to construct quantum BRST charges directly without requiring the existence of a quantum W3-algebra. We study W3-like strings with a direct space-time interpretation — that is, with matter given by explicit free field realizations. Special emphasis is placed on the attempt to construct a quantum W-string associated with the magic realizations of the classical w3-algebra. We give the general conditions for the existence of W3-like strings, and comment on how the known results fit into our general construction. Our results are negative: we find no new consistent string theories, and in particular rule out the possibility of critical strings based on the magic realizations.

Bounds for coverage probabilities with applications to sequential coverage problems

1974 ◽  
Vol 11 (02) ◽  
pp. 281-293 ◽  
Author(s):  
Peter J. Cooke
Keyword(s):  
Asymptotic Behaviour ◽  
Stopping Rules ◽  
Geometrical Probability ◽  
Coverage Probabilities ◽  
Coverage Problems

This paper discusses general bounds for coverage probabilities and moments of stopping rules for sequential coverage problems in geometrical probability. An approach to the study of the asymptotic behaviour of these moments is also presented.

scholarly journals Coverage versus Confidence

2021 ◽  
Vol 23 ◽  
Author(s):  
Peyton Cook
Keyword(s):  
Confidence Interval ◽  
Confidence Intervals ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Coverage Probability ◽  
Negative Binomial ◽  
Asymptotic Confidence Interval ◽  
Population Proportion ◽  
Coverage Probabilities ◽  
The Mean

This article is intended to help students understand the concept of a coverage probability involving confidence intervals. Mathematica is used as a language for describing an algorithm to compute the coverage probability for a simple confidence interval based on the binomial distribution. Then, higher-level functions are used to compute probabilities of expressions in order to obtain coverage probabilities. Several examples are presented: two confidence intervals for a population proportion based on the binomial distribution, an asymptotic confidence interval for the mean of the Poisson distribution, and an asymptotic confidence interval for a population proportion based on the negative binomial distribution.

scholarly journals Triangles of Baumslag–Solitar Groups

2012 ◽  
Vol 64 (2) ◽  
pp. 241-253 ◽  
Author(s):  
Daniel Allcock
Keyword(s):  
Finite Groups ◽  
Derived Length ◽  
Special Cases ◽  
Finite Quotients ◽  
General Conditions

Abstract Our main result is that many triangles of Baumslag–Solitar groups collapse to finite groups, generalizing a famous example of Hirsch and other examples due to several authors. A triangle of Baumslag–Solitar groups means a group with three generators, cyclically ordered, with each generator conjugating some power of the previous one to another power. There are six parameters, occurring in pairs, and we show that the triangle fails to be developable whenever one of the parameters divides its partner, except for a few special cases. Furthermore, under fairly general conditions, the group turns out to be finite and solvable of derived length ≤ 3. We obtain a lot of information about finite quotients, even when we cannot determine developability.

10. Visitors: entry for temporary purposes

2018 ◽  
Author(s):  
Gina Clayton ◽  
Georgina Firth ◽  
Caroline Sawyer ◽  
Rowena Moffatt ◽  
Helena Wray
Keyword(s):  
Medical Treatment ◽  
Family Members ◽  
Short Term ◽  
Special Cases ◽  
The Law ◽  
The Uk ◽  
General Conditions

Course-focused and comprehensive, the Textbook on series provides an accessible overview of the key areas on the law curriculum. This chapter discusses the law relating to individuals coming to the UK as visitors for short-term or finite purposes such as tourism, business visits, sporting and entertainment engagements, or for private medical treatment. There is a discussion of the withdrawal, reinstatement, and restriction of rights of appeal for those visiting family members in the UK, and the application of Article 8 ECHR to these situations. The revised visitor rules in Appendix V are described in detail. The chapter also discusses the special cases of marriage visitors, carers and transit visitors, and general conditions such as prohibited activities and the need for maintenance and accommodation.

Refined Bonferroni prediction bands for autoregressive models

2019 ◽  
Vol 65 (2) ◽  
pp. 155-172
Author(s):  
Anna Staszewska-Bystrova
Keyword(s):  
Monte Carlo ◽  
Coverage Probability ◽  
Higher Order ◽  
Autoregressive Processes ◽  
Monte Carlo Experiments ◽  
Coverage Probabilities ◽  
Bonferroni's Inequality ◽  
Large Width ◽  
Joint Prediction ◽  
Prediction Bands

Joint prediction bands are often constructed using Bonferroni’s inequality. The drawback of such bands may be their large width and excessive coverage probability. The paper proposes two refinements to the basic Bonferroni method of constructing bootstrap prediction bands. These are based on higher order inequalities and optimization of the width of the band. The procedures are applied to the problem of predicting univariate autoregressive processes. Their properties are studied by means of Monte Carlo experiments. It is shown that the proposed methods lead, in many scenarios, to obtaining relatively narrow prediction bands with desired coverage probabilities.

scholarly journals A class of Newton maps with Julia sets of Lebesgue measure zero

2021 ◽  
Author(s):  
Mareike Wolff
Keyword(s):  
Lebesgue Measure ◽  
Newton’S Method ◽  
Newton's Method ◽  
Measure Zero ◽  
Julia Set ◽  
Julia Sets ◽  
Lebesgue Measure Zero ◽  
Almost Everywhere ◽  
General Conditions

AbstractLet $$g(z)=\int _0^zp(t)\exp (q(t))\,dt+c$$ g ( z ) = ∫ 0 z p ( t ) exp ( q ( t ) ) d t + c where p, q are polynomials and $$c\in {\mathbb {C}}$$ c ∈ C , and let f be the function from Newton’s method for g. We show that under suitable assumptions on the zeros of $$g''$$ g ′ ′ the Julia set of f has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that $$f^n(z)$$ f n ( z ) converges to zeros of g almost everywhere in $${\mathbb {C}}$$ C if this is the case for each zero of $$g''$$ g ′ ′ that is not a zero of g or $$g'$$ g ′ . In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.

scholarly journals Confounding adjustment performance of ordinal analysis methods in stroke studies

2019 ◽  
Author(s):  
T.P. Zonneveld ◽  
A. Aigner ◽  
R.H.H. Groenwold ◽  
A. Algra ◽  
P.J. Nederkoorn ◽  
...  
Keyword(s):  
Logistic Regression ◽  
Propensity Score ◽  
Coverage Probability ◽  
Ordinal Logistic Regression ◽  
Point Estimate ◽  
Confounding Variable ◽  
Application Range ◽  
Confounding Variables ◽  
Coverage Probabilities ◽  
Confounding Adjustment

AbstractBackgroundIn acute stroke studies, ordinal logistic regression (OLR) is often used to analyze outcome on the modified Rankin Scale (mRS), whereas the non-parametric Mann-Whitney measure of superiority (MWS) has also been suggested. It is unclear how these perform comparatively when confounding adjustment is warranted. Our aim is to quantify the performance of OLR and MWS in different confounding variable settings.MethodsWe set up a simulation study with three different scenarios; (1) dichotomous confounding variables, (2) continuous confounding variables, and (3) confounding variable settings mimicking a study on functional outcome after stroke. We compared adjusted ordinal logistic regression (aOLR) and stratified Mann-Whitney measure of superiority (sMWS), and also used propensity scores to stratify the MWS (psMWS). For comparability, OLR estimates were transformed to a MWS. We report bias, the percentage of runs that produced a point estimate deviating by more than 0.05 points (point estimate variation), and the coverage probability.ResultsIn scenario 1, there was no bias in both sMWS and aOLR, with similar point estimate variation and coverage probabilities. In scenario 2, sMWS resulted in more bias (0.04 versus 0.00), and higher point estimate variation (41.6% versus 3.3%), whereas coverage probabilities were similar. In scenario 3, there was no bias in both methods, point estimate variation was higher in the sMWS (6.7%) versus aOLR (1.1%), and coverage probabilities were 0.98 (sMWS) versus 0.95 (aOLR). With psMWS, bias remained 0.00, with less point estimate variation (1.5%) and a coverage probability of 0.95.ConclusionsThe bias of both adjustment methods was similar in our stroke simulation scenario, and the higher point estimate variation in the MWS improved with propensity score based stratification. The stratified MWS is a valid alternative for adjusted OLR only when the ratio of number of strata versus number of observations is relatively low, but propensity score based stratification extends the application range of the MWS.

scholarly journals Efficient algorithms for the reverse shortest path problem on trees under the hamming distance

2017 ◽  
Vol 27 (1) ◽  
pp. 46-60 ◽  
Author(s):  
Javad Tayyebi ◽  
Massoud Aman
Keyword(s):  
Shortest Path ◽  
Hamming Distance ◽  
Minimum Cost ◽  
Shortest Path Problem ◽  
Arc Length ◽  
Bound Constraints ◽  
Shortest Distance ◽  
Special Cases ◽  
Minimum Cost Flow Problem ◽  
Minimum Cut Problem

Given a network G(V,A,c) and a collection of origin-destination pairs with prescribed values, the reverse shortest path problem is to modify the arc length vector c as little as possible under some bound constraints such that the shortest distance between each origin-destination pair is upper bounded by the corresponding prescribed value. It is known that the reverse shortest path problem is NP-hard even on trees when the arc length modifications are measured by the weighted sum-type Hamming distance. In this paper, we consider two special cases of this problem which are polynomially solvable. The first is the case with uniform lengths. It is shown that this case transforms to a minimum cost flow problem on an auxiliary network. An efficient algorithm is also proposed for solving this case under the unit sum-type Hamming distance. The second case considered is the problem without bound constraints. It is shown that this case is reduced to a minimum cut problem on a tree-like network. Therefore, both cases studied can be solved in strongly polynomial time.

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