scholarly journals Modeling the Dirichlet distribution using multiplicative functions

2020 ◽  
Vol 25 (2) ◽  
Author(s):  
Gintautas Bareikis ◽  
Algirdas Mačiulis
Keyword(s):  
Distribution Function ◽  
Limit Distribution ◽  
Multiplicative Function ◽  
Remainder Term ◽  
Dirichlet Distribution ◽  
Distribution Functions ◽  
Multiplicative Functions ◽  
Cesàro Mean ◽  
Cesaro Mean ◽  
The One

For q,m,n,d ∈ N and some multiplicative function f > 0, we denote by T3(n) the sum of f(d) over the ordered triples (q,m,d) with qmd = n. We prove that Cesaro mean of distribution functions defined by means of T3 uniformly converges to the one-parameter Dirichlet distribution function. The parameter of the limit distribution depends on the values of f on primes. The remainder term is estimated as well. 

A Frequency-function form of the central limit theorem

1953 ◽  
Vol 49 (3) ◽  
pp. 462-472 ◽  
Author(s):  
Walter L. Smith
Keyword(s):  
Distribution Function ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Distribution Functions ◽  
Frequency Function ◽  
The Central Limit Theorem ◽  
Gaussian Distribution Function ◽  
Limiting Behaviour ◽  
The One

The central limit theorem in the calculus of probability has been extensively studied in recent years. In its simplest form the theorem states that if X1, X2,… is a sequence of independent, identically distributed random variables of mean zero, then under general conditions the distribution function of Zm = (X1 + … + Xn)/√ n converges as n → ∞ to the normal or Gaussian distribution function. This form of the theorem in terms of distribution functions is the one required in statistical work, since it enables statements to be made about the limiting behaviour of prob {a ≤ Zn ≤ b}.

scholarly journals Hazard rates and subexponential distributions

2006 ◽  
Vol 80 (94) ◽  
pp. 29-46 ◽  
Author(s):  
A. Baltrunas ◽  
E. Omey ◽  
Van Gulck
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Rate Of Convergence ◽  
Remainder Term ◽  
Hazard Rate ◽  
Sufficient Conditions ◽  
Distribution Functions ◽  
Hazard Rates ◽  
Subexponential Distributions

A distribution function F on the nonnegative halfline is called subexponential if limx??(1?F*n(x))/(1?F(x)) = n for all n>_ 2. We obtain new sufficient conditions for subexponential distributions and related classes of distribution functions. Our results are formulated in terms of the hazard rate. We also analyze the rate of convergence in the definition and discuss the asymptotic behaviour of the remainder term Rn(x) = 1?F*n(x)?n(1?F(x)). We use the results in studying subordinated distributions and we conclude the paper with some multivariate extensions of our results.

scholarly journals Non-uniform Estimates in the Approximation by the Irwin Law

2016 ◽  
Vol 55 (1) ◽  
pp. 112-118
Author(s):  
Kazimieras Padvelskis ◽  
Ruslan Prigodin
Keyword(s):  
Distribution Function ◽  
Remainder Term ◽  
Cumulative Distribution Function ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Type Condition ◽  
Uniform Estimates ◽  
Cumulative Distribution Functions ◽  
Cumulant Method ◽  
The Difference

We consider an approximation of a cumulative distribution function F(x) by the cumulative distributionfunction G(x) of the Irwin law. In this case, a function F(x) can be cumulative distribution functions of sums (products) ofindependent (dependent) random variables. Remainder term of the approximation is estimated by the cumulant method.The cumulant method is used by introducing special cumulants, satisfying the V. Statulevičius type condition. The mainresult is a nonuniform bound for the difference |F(x)-G(x)| in terms of special cumulants of the symmetric cumulativedistribution function F(x).

scholarly journals THE DISTRIBUTION FUNCTION FOR IMPURITY STATES IN SEMICONDUCTORS

1966 ◽  
Vol 44 (10) ◽  
pp. 2231-2240 ◽  
Author(s):  
Robert Barrie ◽  
C. Y. Cheung
Keyword(s):  
Distribution Function ◽  
Distribution Functions ◽  
Temperature Dependent ◽  
Impurity States ◽  
Impurity Site ◽  
Dirac Distribution ◽  
Double Time ◽  
The One ◽  
Bound Electron ◽  
Creation Operators

For a monovalent donor impurity in a semiconductor the number of electrons that can be bound to an impurity site is either zero or one. The one bound electron can have either direction of spin. This means that the usual Fermi–Dirac distribution function is not applied to such impurity states. A new derivation of the distribution function is presented in terms of annihilation and creation operators for the case of no interaction with the phonons. With the use of double-time temperature-dependent Green's functions, both the electron and the phonon distribution functions are derived when there is interaction between the bound electron and phonons.

scholarly journals On interpolation of operator, which is the sum of weighted Hardy-Littlewood and Cesaro mean operators

2019 ◽  
Vol 27 (1) ◽  
pp. 45
Author(s):  
B.I. Peleshenko
Keyword(s):  
Sufficient Conditions ◽  
Lorentz Spaces ◽  
Multiplicative Functions ◽  
Absolutely Continuous ◽  
Cesàro Mean ◽  
Cesaro Mean

It is proved that operators, which are the sum of weighted Hardy-Littlewood $\int\limits_0^1 f(xt) \psi(t) dt$ and Cesaro $\int\limits_0^1 f(\frac{x}{t}) t^{-n} \psi(t) dt$ mean operators, are limited on Lorentz spaces $\Lambda_{\varphi, a} (\mathbb{R})$, if the functions $f(x) \in \Lambda_{\varphi, a}(\mathbb{R})$ satisfy the condition $|f(-x)| = |f(x)|$, $x > 0$, for such non-increasing semi-multiplicative functions $\psi$, for which the next conditions are satisfied: $\frac{M_1}{\psi(t)} \leqslant \psi(\frac{1}{t}) \leqslant \frac{M_2}{\psi(t)}$, for all $0 < t \leqslant 1$; at some $0 < \varepsilon < \frac{1}{2}$, $0 < \delta < \frac{1}{2}$ functions $\psi(t) t^{1-\varepsilon}$, $\psi(\frac{1}{t}) t^{-\delta}$ do not decrease monotonically and functions $\psi(t) t$, $\psi (\frac{1}{t})$ are absolutely continuous. Also, there are proved sufficient conditions that the operators, which are the sum of weighted Hardy-Littlewood and Cesaro mean operators, when $\psi(t) = t^{-\alpha}$, where $\alpha \in (0, \frac{1}{2})$, on Lorentz spaces $\Lambda_{\varphi, a}(\mathbb{R})$, if the functions $f(x) \in \Lambda_{\varphi, a}(\mathbb{R})$ satisfy the condition $|f(-x)| = |f(x)|$, $x > 0$.

scholarly journals Structure Analysis of Amorphous and Liquid Magnesium Silicate

2021 ◽  
Vol 37 (1) ◽  
Author(s):  
Mai Thi Lan ◽  
Nguyen Thi Thao
Keyword(s):  
Molecular Dynamics ◽  
Boundary Condition ◽  
Distribution Function ◽  
Molecular Dynamics Simulation ◽  
Radial Distribution ◽  
Magnesium Silicate ◽  
Distribution Functions ◽  
Dynamics Simulation ◽  
Radial Distribution Functions ◽  
The One

We use the Oganov potentials and period boundary condition to perform molecular dynamics simulation of amorphous and liquid Mg2SiO4 systems under pressures 0 GPa and 40 GPa. We clarify structure of amorphous Mg2SiO4 at 0 and 40 GPa and compared with the one of Mg2SiO4 at liquid state. Especially, the origin of sub-peaks in radial distribution function of O-O, Si-Si and Mg-Mg pairs is explained clearly. The change of radial distribution functions, coordination number and the number of all types of bonds including the corner-, edge- and face-sharing bonds is also discussed in detail in this paper.

IRREVERSIBILITY IN RELATIVISTIC STATISTICAL MECHANICS

1994 ◽  
Vol 08 (29) ◽  
pp. 1847-1860 ◽  
Author(s):  
URI BEN-YA’ACOV
Keyword(s):  
Distribution Function ◽  
Statistical Mechanics ◽  
Particle Distribution ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Distribution Functions ◽  
Relativistic Dynamics ◽  
Direct Computation ◽  
Lagrangian Equations ◽  
The One

Relativistic statistical mechanics should be manifestly Lorentz covariant. In the absence of a Hamiltonian formalism in relativistic dynamics, a different approach which is based on the (Lagrangian) equations of motion is presented. Without any Liouville equation, this approach provides the direct computation of all the reduced n-particle distribution functions. The trajectories in the fully interacting system and ensemble averages are defined with respect to the parameters that fix the trajectories in the interaction-free limit. Irreversibility may emerge from microscopic dynamics due to the choice as to which part of the particles’ history — past or future — contributes to the interaction. Irreversibility is explicitly demonstrated in the evolution of the one-particle distribution function.

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 Some subordination involving polynomials induced by lower triangular matrices

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Saiful R. Mondal ◽  
Kottakkaran Sooppy Nisar ◽  
Thabet Abdeljawad
Keyword(s):  
Unit Disk ◽  
Triangular Matrices ◽  
Cesàro Mean ◽  
Cesaro Mean ◽  
Lower Triangular ◽  
Lower Triangular Matrices

Abstract The article considers several polynomials induced by admissible lower triangular matrices and studies their subordination properties. The concept generalizes the notion of stable functions in the unit disk. Several illustrative examples, including those related to the Cesàro mean, are discussed, and connections are made with earlier works.

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