scholarly journals On matrix exponential distributions

2007 ◽  
Vol 39 (01) ◽  
pp. 271-292 ◽  
Author(s):  
Qi-Ming He ◽  
Hanqin Zhang
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Matrix Exponential ◽  
Exponential Distributions ◽  
Hankel Matrices ◽  
Phase Type ◽  
Types Of Information ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Derivatives Of

In this paper we introduce certain Hankel matrices that can be used to study ME (matrix exponential) distributions, in particular to compute their ME orders. The Hankel matrices for a given ME probability distribution can be constructed if one of the following five types of information about the distribution is available: (i) an ME representation, (ii) its moments, (iii) the derivatives of its distribution function, (iv) its Laplace-Stieltjes transform, or (v) its distribution function. Using the Hankel matrices, a necessary and sufficient condition for a probability distribution to be an ME distribution is found and a method of computing the ME order of the ME distribution developed. Implications for the PH (phase-type) order of PH distributions are examined. The relationship between the ME order, the PH order, and a lower bound on the PH order given by Aldous and Shepp (1987) is discussed in numerical examples.

scholarly journals On matrix exponential distributions

2007 ◽  
Vol 39 (1) ◽  
pp. 271-292 ◽  
Author(s):  
Qi-Ming He ◽  
Hanqin Zhang
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Matrix Exponential ◽  
Exponential Distributions ◽  
Hankel Matrices ◽  
Phase Type ◽  
Types Of Information ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Derivatives Of

In this paper we introduce certain Hankel matrices that can be used to study ME (matrix exponential) distributions, in particular to compute their ME orders. The Hankel matrices for a given ME probability distribution can be constructed if one of the following five types of information about the distribution is available: (i) an ME representation, (ii) its moments, (iii) the derivatives of its distribution function, (iv) its Laplace-Stieltjes transform, or (v) its distribution function. Using the Hankel matrices, a necessary and sufficient condition for a probability distribution to be an ME distribution is found and a method of computing the ME order of the ME distribution developed. Implications for the PH (phase-type) order of PH distributions are examined. The relationship between the ME order, the PH order, and a lower bound on the PH order given by Aldous and Shepp (1987) is discussed in numerical examples.

THE INDEPENDENCE OF FUZZY VARIABLES WITH APPLICATIONS TO FUZZY RANDOM OPTIMIZATION

2007 ◽  
Vol 15 (supp02) ◽  
pp. 1-20 ◽  
Author(s):  
YIAN-KUI LIU ◽  
JINWU GAO
Keyword(s):  
Distribution Function ◽  
Sufficient Conditions ◽  
Possibility Distribution ◽  
Fuzzy Variables ◽  
Fundamental Properties ◽  
Random Optimization ◽  
Fuzzy Random Programming ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Fuzzy Random

This paper presents the independence of fuzzy variables as well as its applications in fuzzy random optimization. First, the independence of fuzzy variables is defined based on the concept of marginal possibility distribution function, and a discussion about the relationship between the independent fuzzy variables and the noninteractive (unrelated) fuzzy variables is included. Second, we discuss some properties of the independent fuzzy variables, and establish the necessary and sufficient conditions for the independent fuzzy variables. Third, we propose the independence of fuzzy events, and deal with its fundamental properties. Finally, we apply the properties of the independent fuzzy variables to a class of fuzzy random programming problems to study their convexity.

Concentrated matrix exponential distributions with real eigenvalues

2021 ◽  
pp. 1-17
Author(s):  
András Mészáros ◽  
Miklós Telek
Keyword(s):  
Stochastic Models ◽  
Laplace Transformation ◽  
Matrix Exponential ◽  
Phase Type Distribution ◽  
Exponential Distributions ◽  
Inverse Laplace Transformation ◽  
Complex Eigenvalues ◽  
Phase Type ◽  
The Matrix ◽  
The Laplace Transform

Abstract Concentrated random variables are frequently used in representing deterministic delays in stochastic models. The squared coefficient of variation ( $\mathrm {SCV}$ ) of the most concentrated phase-type distribution of order $N$ is $1/N$ . To further reduce the $\mathrm {SCV}$ , concentrated matrix exponential (CME) distributions with complex eigenvalues were investigated recently. It was obtained that the $\mathrm {SCV}$ of an order $N$ CME distribution can be less than $n^{-2.1}$ for odd $N=2n+1$ orders, and the matrix exponential distribution, which exhibits such a low $\mathrm {SCV}$ has complex eigenvalues. In this paper, we consider CME distributions with real eigenvalues (CME-R). We present efficient numerical methods for identifying a CME-R distribution with smallest SCV for a given order $n$ . Our investigations show that the $\mathrm {SCV}$ of the most concentrated CME-R of order $N=2n+1$ is less than $n^{-1.85}$ . We also discuss how CME-R can be used for numerical inverse Laplace transformation, which is beneficial when the Laplace transform function is impossible to evaluate at complex points.

Infrared spectra and substituent effects in 2-(3-, and 4-substituted phenylmethylene)-1,3-cycloheptanediones

1983 ◽  
Vol 48 (2) ◽  
pp. 586-595 ◽  
Author(s):  
Alexander Perjéssy ◽  
Pavol Hrnčiar ◽  
Ján Šraga
Keyword(s):  
Substituent Effects ◽  
Phenyl Group ◽  
Wave Number ◽  
Vibrational Coupling ◽  
Wave Numbers ◽  
Stretching Vibration ◽  
Stretching Vibrations ◽  
The Relationship ◽  
Derivatives Of ◽  
Effect Of Structure

The wave numbers of the fundamental C=O and C=C stretching vibrations, as well as that of the first overtone of C=O stretching vibration of 2-(3-, and 4-substituted phenylmethylene)-1,3-cycloheptanediones and 1,3-cycloheptanedione were measured in tetrachloromethane and chloroform. The spectral data were correlated with σ+ constants of substituents attached to phenyl group and with wave number shifts of the C=O stretching vibration of substituted acetophenones. The slope of the linear dependence ν vs ν+ of the C=C stretching vibration of the ethylenic group was found to be more than two times higher than that of the analogous correlation of the C=O stretching vibration. Positive values of anharmonicity for asymmetric C=O stretching vibration can be considered as an evidence of the vibrational coupling in a cyclic 1,3-dicarbonyl system similarly, as with derivatives of 1,3-indanedione. The relationship between the wave numbers of the symmetric and asymmetric C=O stretching vibrations indicates that the effect of structure upon both vibrations is symmetric. The vibrational coupling in 1,3-cycloheptanediones and the application of Seth-Paul-Van-Duyse equation is discussed in relation to analogous results obtained for other cyclic 1,3-dicarbonyl compounds.

Private and Public Law

2020 ◽  
pp. 574-592
Author(s):  
Thomas W. Merrill
Keyword(s):  
Law And Economics ◽  
Public Law ◽  
Private Law ◽  
General Criterion ◽  
The Public ◽  
Private Ordering ◽  
Public Rights ◽  
Private And Public ◽  
The Relationship ◽  
Derivatives Of

This chapter explores the relationship between private and public law. In civil law countries, the public-private distinction serves as an organizing principle of the entire legal system. In common law jurisdictions, the distinction is at best an implicit design principle and is used primarily as an informal device for categorizing different fields of law. Even if not explicitly recognized as an organizing principle, however, it is plausible that private and public law perform distinct functions. Private law supplies the tools that make private ordering possible—the discretionary decisions that individuals make in structuring their lives. Public law is concerned with providing public goods—broadly defined—that cannot be adequately supplied by private ordering. In the twentieth and twenty-first centuries, various schools of thought derived from utilitarianism have assimilated both private and public rights to the same general criterion of aggregate welfare analysis. This has left judges with no clear conception of the distinction between private and public law. Another problematic feature of modern legal thought is a curious inversion in which scholars who focus on fields of private law have turned increasingly to law and economics, one of the derivatives of utilitarianism, whereas scholars who concern themselves with public law are increasingly drawn to new versions of natural rights thinking, in the form of universal human rights.

On derivatives of fuzzy multi-dimensional mappings and applications under generalized differentiability

2021 ◽  
pp. 1-19
Author(s):  
M. Miri Karbasaki ◽  
M. R. Balooch Shahriari ◽  
O. Sedaghatfar
Keyword(s):  
Linear Function ◽  
Level Set ◽  
Fuzzy Numbers ◽  
Sufficient Condition ◽  
Continuous Linear ◽  
Generalized Differentiability ◽  
Relationship Of ◽  
The Relationship ◽  
Derivatives Of ◽  
Dimensional Mapping

This article identifies and presents the generalized difference (g-difference) of fuzzy numbers, Fréchet and Gâteaux generalized differentiability (g-differentiability) for fuzzy multi-dimensional mapping which consists of a new concept, fuzzy g-(continuous linear) function; Moreover, the relationship between Fréchet and Gâteaux g-differentiability is studied and shown. The concepts of directional and partial g-differentiability are further framed and the relationship of which will the aforementioned concepts are also explored. Furthermore, characterization is pointed out for Fréchet and Gâteaux g-differentiability; based on level-set and through differentiability of endpoints real-valued functions a characterization is also offered and explored for directional and partial g-differentiability. The sufficient condition for Fréchet and Gâteaux g-differentiability, directional and partial g-differentiability based on level-set and through employing level-wise gH-differentiability (LgH-differentiability) is expressed. Finally, to illustrate the ability and reliability of the aforementioned concepts we have solved some application examples.

scholarly journals Permutation Matrices, Their Discrete Derivatives and Extremal Properties

2020 ◽  
Vol 48 (4) ◽  
pp. 719-740
Author(s):  
Richard A. Brualdi ◽  
Geir Dahl
Keyword(s):  
Permutation Matrix ◽  
Permutation Matrices ◽  
Discrete Derivative ◽  
The Relationship ◽  
Derivatives Of ◽  
Extremal Properties

AbstractFor a permutation π, and the corresponding permutation matrix, we introduce the notion of discrete derivative, obtained by taking differences of successive entries in π. We characterize the possible derivatives of permutations, and consider questions for permutations with certain properties satisfied by the derivative. For instance, we consider permutations with distinct derivatives, and the relationship to so-called Costas arrays.

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