A half-normal distribution scheme for generating functions

Wallner, Michael

doi:10.1016/j.ejc.2020.103138

Cited by 12 publications

(4 citation statements)

References 26 publications

(58 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Instead of characterizing all possible limit laws for which we have few applications, we address the following question, based on the OEIS examples we collected: under which conditions will the limit law of the coefficients be either Rayleigh or half-normal (two of the most common non-normal laws in lattice paths, random trees, random mappings, etc.)? For more instances and techniques for these two laws, see [79,220] and the references therein. It turns out that these are very special laws from our framework and very strong restrictions are needed.…”

Section: Rayleigh and Half-normal Limit Laws (mentioning

confidence: 99%

An asymptotic distribution theory for Eulerian recurrences with applications

Hwang

Chern

Duh

2020

Advances in Applied Mathematics

View full text Add to dashboard Cite

We study linear recurrences of Eulerian type of the formwith P 0 (v) given, where α(v), β(v) and γ(v) are in most cases polynomials of low degrees. We characterize the various limit laws of the coefficients of P n (v) for large n using the method of moments and analytic combinatorial tools under varying α(v), β(v) and γ(v), and apply our results to more than two hundred of concrete examples when β(v) = 0 and more than three hundred when β(v) = 0 that we collected from the literature and from Sloane's OEIS database. The limit laws and the convergence rates we worked out are almost all new and include normal, half-normal, Rayleigh, beta, Poisson, negative binomial, Mittag-Leffler, Bernoulli, etc., showing the surprising richness and diversity of such a simple framework, as well as the power of the approaches used.

show abstract

Section: Rayleigh and Half-normal Limit Laws (mentioning

confidence: 99%

An asymptotic distribution theory for Eulerian recurrences with applications

Hwang

Chern

Duh

2020

Advances in Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…5,28 For a half-normal distribution the mean and standard deviation are proportional to each other, and also to the standard deviation of the combined PDF valid over the whole viscuit domain. 29 Therefore the η − (t) and η + (t) individually should also scale as t 1/2 , assuming gaussian statistics. Hence the difference between η − (t) and η + (t) for large t is also a measure of the breadth of the η(t) distribution of each side in the large t limit.…”

Section: Simulation Detailsmentioning

confidence: 99%

Viscuit and the fluctuation theorem investigation of shear viscosity by molecular dynamics simulations: The information and the noise

Heyes

Dini

Smith

2021

The Journal of Chemical Physics

View full text Add to dashboard Cite

The shear viscosity, η, of model liquids and solids is investigated within the framework of the viscuit and Fluctuation Theory (FT) probability distribution function (PDF) theories, following on from Heyes et al. [J. Chem. Phys. 152, 194504 (2020).] using equilibrium molecular dynamics (MD) simulations on Lennard-Jones and Weeks-Chandler-Andersen model systems. The viscosity can be obtained in equilibrium MD simulation from the first moment of the viscuit PDF which is shown for finite simulation lengths to give a less noisy plateau region than the Green-Kubo method. Two other formulas for the shear viscosity in terms of the viscuit and PDF analysis are also derived. A separation of the time-dependent average negative and positive viscuits extrapolated from the noise dominated region to zero time provide another route to η. The third method involves the relative number of positive and negative viscuits, and their PDF standard deviations on the two sides for an equilibrium system. For the FT and finite shear rates, accurate analytic expressions for the relative number of positive to negative block average shear stresses is derived assuming a shifted gaussian PDF, which is shown to agree well with Non-equilibrium Molecular Dynamics simulations. A similar treatment of the positive and negative block average contributions to the viscosity is also shown to match the simulation data very well.

show abstract

“…In the next subsections we will use this equation by marking certain parameters in order to deduce information on their distribution. For more information on this concept see e.g., [8,20]. We start with the number of elements on level 0.…”

Section: Parameters Of Relaxed Binary Treesmentioning

confidence: 99%

A bijection of plane increasing trees with relaxed binary trees of right height at most one

Wallner

2019

Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

Plane increasing trees are rooted labeled trees embedded into the plane such that the sequence of labels is increasing on any branch starting at the root. Relaxed binary trees are a subclass of unlabeled directed acyclic graphs. We construct a bijection between these two combinatorial objects and study the therefrom arising connections of certain parameters. Furthermore, we show central limit theorems for two statistics on leaves. We end the study by considering more than 20 subclasses and their bijective counterparts. Many of these subclasses are enumerated by known counting sequences, and thus enrich their combinatorial interpretation.

show abstract

A half-normal distribution scheme for generating functions

Cited by 12 publications

References 26 publications

An asymptotic distribution theory for Eulerian recurrences with applications

An asymptotic distribution theory for Eulerian recurrences with applications

Viscuit and the fluctuation theorem investigation of shear viscosity by molecular dynamics simulations: The information and the noise

A bijection of plane increasing trees with relaxed binary trees of right height at most one

Contact Info

Product

Resources

About