Moment functions on hypergroup joins

2019 ◽  
Vol 10 (3) ◽  
pp. 215-220
Author(s):  
Kedumetse Vati ◽  
László Székelyhidi
Keyword(s):  
Probability Theory ◽  
Functional Equations ◽  
Spectral Synthesis ◽  
Natural Generalization ◽  
Moment Function ◽  
Function Classes ◽  
Moment Functions ◽  
Basic Blocks ◽  
Generalized Moment

Abstract Moment functions play a basic role in probability theory. A natural generalization can be defined on hypergroups which leads to the concept of generalized moment function sequences. In a former paper we studied some function classes on hypergroup joins which play a basic role in spectral synthesis. Moment functions are also important basic blocks of spectral synthesis. All these functions can be characterized by well-known functional equations. In this paper we describe generalized moment function sequences on hypergroup joins.

Article

1999 ◽  
Vol 77 (11) ◽  
pp. 1775-1781 ◽  
Author(s):  
Christopher D Daub ◽  
Bryan R Henry ◽  
Martin L Sage ◽  
Henrik G Kjaergaard
Keyword(s):  
Dipole Moment ◽  
Ab Initio Calculations ◽  
Ab Initio ◽  
Electron Correlation ◽  
Moment Function ◽  
Polynomial Method ◽  
Small Contribution ◽  
Least Squares Regression ◽  
Vibrational Overtone ◽  
Moment Functions

Two studies of aspects of modelling dipole moment functions of XH bonds in small molecules for use in calculating overtone intensities have been undertaken. The first study deals with the fitting of ab initio calculations of the dipole moment at discrete points to a functional form. The two methods that are compared are the use of least-squares regression and the use of interpolating polynomials. The interpolating polynomial method is deemed superior due to its greater efficiency in terms of the number of points necessary to obtain reasonable results. The second study attempts to explain the indifference of calculated overtone intensities to the inclusion of electron correlation in the ab initio calculation of the dipole moment function. It is found that in most cases the influence of electron correlation can be modelled as a function with a very small matrix element, which results in a very small contribution to the overtone intensity.Key words: dipole moment function, vibrational overtone intensities, electron correlation, ab initio calculations.

scholarly journals Moment functions and exponential monomials on commutative hypergroups

2021 ◽  
Author(s):  
Żywilla Fechner ◽  
Eszter Gselmann ◽  
László Székelyhidi
Keyword(s):  
Linear Combination ◽  
Linear Subspace ◽  
One Dimensional ◽  
Moment Functions ◽  
Commutative Hypergroup ◽  
Sine Functions ◽  
Generalized Moment

AbstractThe purpose of this paper is to prove that if on a commutative hypergroup an exponential monomial has the property that the linear subspace of all sine functions in its variety is one dimensional, then this exponential monomial is a linear combination of generalized moment functions.

scholarly journals The moments of the distribution for normal samples of measures of departure from normality

1930 ◽  
Vol 130 (812) ◽  
pp. 16-28 ◽  
Keyword(s):  
Mean Value ◽  
Direct Application ◽  
Moment Function ◽  
Symmetrical Distribution ◽  
Moment Functions ◽  
The Mean ◽  
Departure From Normality ◽  
Simple Combination

If x 1 ... x n are the values of a variate observed in a sample of n , from any population, we may evaluate a series of statistics ( K ) such that the mean value of k p will be the p th cumulative moment function of the sampled population; the first three of these are defined by the equations; k 1 = 1/ n S ( x ), k 2 = 1/ n -1 S ( x - k 1 ) 2 , k 3 = n /( n -1) ( n -2) S ( x - k 1 ) 3 ; then it has been shown (fisher, 1929) that the cumulative moment functions of the simultaneous distribution, in samples, of k 1 , k 2 , k 3 ,..., may be obtained by the direct application of a very simple combination procedure. The simplest measure of departure from normality will the be γ = k 3 k 2 -3/2 , a quantity which is evidently independent of the units of measurements, and in samples from a symmetrical distribution will have a distribution symmetrical about the value zero. In testing the evidence provided by a sample, of departure from normality, the distribution of this quantity in normal samples is required.

scholarly journals Moment Functions on Groups

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Żywilla Fechner ◽  
Eszter Gselmann ◽  
László Székelyhidi
Keyword(s):  
Commutative Group ◽  
Bell Polynomials ◽  
Exponential Polynomials ◽  
Bell Polynomial ◽  
Rank One ◽  
Moment Functions ◽  
Binomial Type ◽  
Higher Rank ◽  
Generalized Moment

AbstractIn this paper generalized moment functions are considered. They are closely related to the well-known functions of binomial type which have been investigated on various abstract structures. The main purpose of this work is to prove characterization theorems for generalized moment functions on commutative groups. At the beginning a multivariate characterization of moment functions defined on a commutative group is given. Next the notion of generalized moment functions of higher rank is introduced and some basic properties on groups are listed. The characterization of exponential polynomials by means of complete (exponential) Bell polynomials is given. The main result is the description of generalized moment functions of higher rank defined on a commutative group as the product of an exponential and composition of multivariate Bell polynomial and an additive function. Furthermore, corollaries for generalized moment function of rank one are also stated. At the end of the paper some possible directions of further research are discussed.

Łukasiewicz Logic and the Divisible Extension of Probability Theory

2021 ◽  
Vol 78 (1) ◽  
pp. 119-128
Author(s):  
Roman Frič
Keyword(s):  
Probability Theory ◽  
Fuzzy Sets ◽  
Random Variables ◽  
Natural Generalization ◽  
Boolean Logic ◽  
Added Value ◽  
Probability Measures ◽  
Łukasiewicz Logic ◽  
Lukasiewicz Logic ◽  
Random Events

Abstract We show that measurable fuzzy sets carrying the multivalued Łukasiewicz logic lead to a natural generalization of the classical Kolmogorovian probability theory. The transition from Boolean logic to Łukasiewicz logic has a categorical background and the resulting divisible probability theory possesses both fuzzy and quantum qualities. Observables of the divisible probability theory play an analogous role as classical random variables: to convey stochastic information from one system to another one. Observables preserving the Łukasiewicz logic are called conservative and characterize the “classical core” of divisible probability theory. They send crisp random events to crisp random events and Dirac probability measures to Dirac probability measures. The nonconservative observables send some crisp random events to genuine fuzzy events and some Dirac probability measures to nondegenerated probability measures. They constitute the added value of transition from classical to divisible probability theory.

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