A Geometric Derivation of the Irwin-Hall Distribution

Marengo, James E.; Farnsworth, David L.; Stefanic, Lucas

doi:10.1155/2017/3571419

Cited by 24 publications

(9 citation statements)

References 28 publications

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“…We note that the distribution described in Eq. ( 26) represents a generalized form of the Irwin-Hall distribution [40,41]. Therefore, it is possible to compute the volume (F L ) exactly (see Appendix D for details).…”

Section: A Local Contact Breaking Distributionsmentioning

confidence: 99%

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First contact breaking distributions in strained disordered crystals

Maharana¹,

Nampoothiri²,

Ramola³

2022

Preprint

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We derive exact probability distributions for the strain ( ) at which the first plastic drop event occurs in uniformly strained disordered crystals, with quenched disorder introduced through polydispersity in particle sizes. We identify the first plastic drop in the system with the first contact breaking event and characterize these events numerically as well as theoretically. Our theoretical results are corroborated with numerical simulations of quasistatic volumetric strain applied to disordered near-crystalline configurations of athermal soft particles. We develop a general technique to determine the distribution of strains at which the first contact breaking events occur, through an exact mapping between the cumulative distribution of such events and the volume of a convex polytope whose dimension is determined by the number of defects N d in the system. An exact numerical computation of this polytope volume for systems with small numbers of defects displays a remarkable match with the distribution of strains generated through direct numerical simulations. Finally, we derive the distribution of strains at which the first plastic failure occurs, assuming that individual contact breaking events are uncorrelated, which accurately reproduces distributions obtained from direct numerical simulations.

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Section: A Local Contact Breaking Distributionsmentioning

confidence: 99%

“…This variable has the following exact distribution[40,41] P (T ) = 1 (N − 1)! N i=1 γ i T N −1 − N s=1 (−1) s−1 N dg s (T ) dT ,(D3)…”

mentioning

confidence: 99%

First contact breaking distributions in strained disordered crystals

Maharana¹,

Nampoothiri²,

Ramola³

2022

Preprint

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“…For a geometric derivation of the above formula, see [28]. On the other hand, the characteristic function of S l−1 is given by…”

Section: Appendix Bmentioning

confidence: 99%

CLT for non-Hermitian random band matrices with variance profiles

Jana¹

2019

Preprint

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We show that the fluctuations of the linear eigenvalue statistics of a non-Hermitian random band matrix of increasing bandwidth bn with a continuous variance profile wν (x) converges to a N (0, σ 2 f (ν)), where ν = limn→∞(2bn/n) ∈ [0, 1] and f is the test function. When ν ∈ (0, 1], we obtain an explicit formula for σ 2 f (ν), which depends on f , and variance profile wν . When ν = 1, the formula is consistent with Rider, and Silverstein ( 2006) [33]. We also independently compute an explicit formula for σ 2 f (0) i.e., when the bandwidth bn grows slower compared to n. In addition, we show that σ 2 f (ν) → σ 2 f (0) as ν ↓ 0.

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“…Before discussing the numerical instability that is present in the distribution function of the sum of uniform(0, θ) random variables, we must first derive that function. It can be shown that if X i iid ∼ Unif(0, 1) and we take a random sample of size n from this population, then Y = n i=1 X i ∼ Irwin-Hall(n) (Marengo et al 2017), where Y has the following density and distribution functions:…”

Section: Acknowledgmentsmentioning

confidence: 99%

Not just normal: Exploring power with Shiny apps

Stratton¹,

Green²,

Hoegh³

2021

Technology Innovations in Statistics Education

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Statistical power is an important topic taught in most graduate-level and undergraduatelevel mathematical statistics courses, but it is often difficult to understand conceptually. Visualizing the power curve, sampling distributions, and how they interact can help students more easily conceptualize power, but the creation of such visuals can be difficult and time-consuming. Interactive web applications provide a way for students to dynamically visualize power, and many web applications for understanding power exist. Most existing applications assume samples are drawn from a Normal population, concern only the sample mean, and/or were created for introductory classes. We developed a new web application suitable for undergraduate-level and graduate-level mathematical statistics courses that allows users to visualize the complex relationships underlying power for multiple different estimators and population distributions (available at https://shiny.stt.msu.edu/jg/powerapp/); source code is provided for instructors who wish to modify the application. Our experience implementing this application across two different semesters is also discussed, and example activities are provided in the appendix.

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A Geometric Derivation of the Irwin-Hall Distribution

Cited by 24 publications

References 28 publications

First contact breaking distributions in strained disordered crystals

First contact breaking distributions in strained disordered crystals

CLT for non-Hermitian random band matrices with variance profiles

Not just normal: Exploring power with Shiny apps

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