scholarly journals Refined Inversion Statistics on Permutations

10.37236/1993 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Joshua Sack ◽  
Henning Úlfarsson
Keyword(s):  
Distribution Function ◽  
Distribution Functions ◽  
Fixed Position ◽  
Fixed Integer ◽  
Left Boundary ◽  
Eulerian Polynomials ◽  
Identity Permutation ◽  
Special Cases ◽  
Dot Product

We introduce and study new refinements of inversion statistics for permutations, such as $k$-step inversions, (the number of inversions with fixed position differences) and non-inversion sums (the sum of the differences of positions of the non-inversions of a permutation). We also provide a distribution function for non-inversion sums, a distribution function for $k$-step inversions that relates to the Eulerian polynomials, and special cases of distribution functions for other statistics we introduce, such as $(\le\!\!k)$-step inversions and $(k_1,k_2)$-step inversions (that fix the value separation as well as the position).  We connect our refinements to other work, such as inversion tops that are $0$ modulo a fixed integer $d$, left boundary sums of paths, and marked meshed patterns.   Finally, we use non-inversion sums to show that for every number $n>34$, there is a permutation such that the dot product of that permutation and the identity permutation (of the same length) is $n$.

scholarly journals Measures of concordance determined byD4-invariant copulas

2004 ◽  
Vol 2004 (70) ◽  
pp. 3867-3875 ◽  
Author(s):  
H. H. Edwards ◽  
P. Mikusiński ◽  
M. D. Taylor
Keyword(s):  
Distribution Function ◽  
Mass Distribution ◽  
Random Vector ◽  
Joint Distribution ◽  
Distribution Functions ◽  
Joint Distribution Function ◽  
Unit Square ◽  
Special Cases ◽  
Measures Of Concordance ◽  
Measure Of Association

A continuous random vector(X,Y)uniquely determines a copulaC:[0,1]2→[0,1]such that when the distribution functions ofXandYare properly composed intoC, the joint distribution function of(X,Y)results. A copula is said to beD4-invariant if its mass distribution is invariant with respect to the symmetries of the unit square. AD4-invariant copula leads naturally to a family of measures of concordance having a particular form, and all copulas generating this family areD4-invariant. The construction examined here includes Spearman’s rho and Gini’s measure of association as special cases.

A study of the mass ratio dependence of the mixed species collision integral

1982 ◽  
Vol 28 (2) ◽  
pp. 233-254 ◽  
Author(s):  
A. J. M. Garrett
Keyword(s):  
Distribution Function ◽  
Spherical Harmonic ◽  
Particle Distribution ◽  
Collision Integral ◽  
Distribution Functions ◽  
Particle Distribution Function ◽  
Isotropic Case ◽  
Mass Ratio ◽  
Test Species ◽  
Special Cases

This paper is concerned with the Boltzmann collision integral for the one-particle distribution function of a test species of particle undergoing elastic collisions with particles of a second species which is in thermal equilibrium. A previous paper studied this expression as a function of the mass ratio for the two species of particle when the test particle distribution function was isotropic in velocity space; this work generalizes that analysis to anisotropic distribution functions by expanding the distribution function in tensorial spherical harmonics. First the limit of zero mass ratio is considered: this simplifies the calculation dramatically. There is no contribution to the collision integral from the zeroth-order spherical harmonic in this limit. Then the main calculation shows how to find the terms arising from the existence of a finite mass ratio as an ascending power series in this quantity, and evaluates for each spherical harmonic the next term, linear in mass ratio. This is checked for two special cases: that of an isotropic distribution function, when the expression reduces to Davydov's form, and that arising from a cross-section inversely proportional to the collision velocity, when a comparison with the exact solution of the associated eigen problem can be made. As in the isotropic case, an exact representation of the collision integral as an expansion in mass ratio must include some terms non-analytic in this quantity and vanishing more quickly than any positive power: it is shown how these arise in the present formalism. The formulae derived here have applications to the transport theory of electrons and light ions in a predominantly neutral gas as governed by the Boltzmann equation.

Scaling

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Distribution Function ◽  
Dynamical Systems ◽  
Complex Systems ◽  
Power Law ◽  
Scaling Laws ◽  
Power Laws ◽  
Distribution Functions ◽  
Temporal Process ◽  
Approximate Power

Scaling appears practically everywhere in science; it basically quantifies how the properties or shapes of an object change with the scale of the object. Scaling laws are always associated with power laws. The scaling object can be a function, a structure, a physical law, or a distribution function that describes the statistics of a system or a temporal process. We focus on scaling laws that appear in the statistical description of stochastic complex systems, where scaling appears in the distribution functions of observable quantities of dynamical systems or processes. The distribution functions exhibit power laws, approximate power laws, or fat-tailed distributions. Understanding their origin and how power law exponents can be related to the particular nature of a system, is one of the aims of the book.We comment on fitting power laws.

scholarly journals Estimating a Distribution Function at the Boundary

2020 ◽  
Vol 49 (1) ◽  
pp. 1-23
Author(s):  
Shunpu Zhang ◽  
Zhong Li ◽  
Zhiying Zhang
Keyword(s):  
Distribution Function ◽  
Kernel Estimation ◽  
Kernel Estimator ◽  
Distribution Functions ◽  
Practical Application ◽  
Real World Applications ◽  
Distribution Kernel ◽  
Estimation Of Distribution ◽  
Simulation Results ◽  
Boundary Correction

Estimation of distribution functions has many real-world applications. We study kernel estimation of a distribution function when the density function has compact support. We show that, for densities taking value zero at the endpoints of the support, the kernel distribution estimator does not need boundary correction. Otherwise, boundary correction is necessary. In this paper, we propose a boundary distribution kernel estimator which is free of boundary problem and provides non-negative and non-decreasing distribution estimates between zero and one. Extensive simulation results show that boundary distribution kernel estimator provides better distribution estimates than the existing boundary correction methods. For practical application of the proposed methods, a data-dependent method for choosing the bandwidth is also proposed.

scholarly journals Contact and Chord Length Distribution Functions of the Poisson-Voronoi Tessellation in High Dimensions

2010 ◽  
Vol 42 (1) ◽  
pp. 48-68 ◽  
Author(s):  
L. Muche
Keyword(s):  
Voronoi Tessellation ◽  
Length Distribution ◽  
Distribution Functions ◽  
Chord Length ◽  
Dimensional Euclidean Space ◽  
Spherical Contact ◽  
Chord Length Distribution ◽  
Special Cases ◽  
Contact Distribution ◽  
Contact Distribution Function

In this paper we present formulae for contact distributions of a Voronoi tessellation generated by a homogeneous Poisson point process in the d-dimensional Euclidean space. Expressions are given for the probability density functions and moments of the linear and spherical contact distributions. They are double and simple integral formulae, which are tractable for numerical evaluation and for large d. The special cases d = 2 and d = 3 are investigated in detail, while, for d = 3, the moments of the spherical contact distribution function are expressed by standard functions. Also, the closely related chord length distribution functions are considered.

scholarly journals Transformation of circular random variables based on circular distribution functions

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5931-5947
Author(s):  
Hatami Mojtaba ◽  
Alamatsaz Hossein
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Real Data ◽  
Random Variable ◽  
Distribution Functions ◽  
Likelihood Method ◽  
Circular Distribution ◽  
Data Set ◽  
Trigonometric Moments ◽  
Von Mises

In this paper, we propose a new transformation of circular random variables based on circular distribution functions, which we shall call inverse distribution function (id f ) transformation. We show that M?bius transformation is a special case of our id f transformation. Very general results are provided for the properties of the proposed family of id f transformations, including their trigonometric moments, maximum entropy, random variate generation, finite mixture and modality properties. In particular, we shall focus our attention on a subfamily of the general family when id f transformation is based on the cardioid circular distribution function. Modality and shape properties are investigated for this subfamily. In addition, we obtain further statistical properties for the resulting distribution by applying the id f transformation to a random variable following a von Mises distribution. In fact, we shall introduce the Cardioid-von Mises (CvM) distribution and estimate its parameters by the maximum likelihood method. Finally, an application of CvM family and its inferential methods are illustrated using a real data set containing times of gun crimes in Pittsburgh, Pennsylvania.

A general shock model for a reliability system

2000 ◽  
Vol 37 (04) ◽  
pp. 925-935 ◽  
Author(s):  
Georgios Skoulakis
Keyword(s):  
Distribution Function ◽  
Point Process ◽  
Renewal Process ◽  
General System ◽  
System Structure ◽  
Random Numbers ◽  
System Failure ◽  
Shock Model ◽  
Stieltjes Transform ◽  
Special Cases

We study a reliability system subject to shocks generated by a renewal point process. When a shock occurs, components fail independently of each other with equal probabilities that are random numbers drawn from a distribution that may differ from shock to shock. We first consider the case of a parallel system and derive closed expressions for the Laplace-Stieltjes transform and the expectation of the time to system failure and for its density in the case that the distribution function of the renewal process possesses a density. We then treat a more general system structure, which has some very important special cases, such as k-out-of-n:F systems, and derive analogous formulae.

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