Kolmogorov Probability Theory

2017 ◽  
pp. 1-54
Author(s):  
Renáta Bartková ◽  
Beloslav Riečan ◽  
Anna Tirpáková
Keyword(s):  
Probability Theory ◽  
20Th Century ◽  
Set Theory ◽  
Measure Theory ◽  
Basic Ideas

Kolmogorov probability theory based on set theory belongs to the most important results of mathematics of the 20th century. Naturally, its main advantage is the possibility to use results of the modern measure theory. However, this fact sometimes does not allow larger considerations. In this chapter we want to show this paradox can be eliminated. Of course, we present only some basic ideas. Understanding them enables one to study further results and applications.

Fundamental Concepts

2019 ◽  
pp. 27-77
Keyword(s):  
Decision Making ◽  
Probability Theory ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Probability Distributions ◽  
Possibility Theory ◽  
Multi Objective ◽  
Basic Ideas ◽  
Fuzzy Stochastic Programming

In this chapter, the authors discuss some basic concepts of probability theory and possibility theory that are useful when reading the subsequent chapters of this book. The multi-objective fuzzy stochastic programming models developed in this book are based on the concepts of advanced topics in fuzzy set theory and fuzzy random variables (FRVs). Therefore, for better understanding of these advanced areas, the authors at first presented some basic ideas of probability theory and probability density functions of different continuous probability distributions. Afterwards, the necessity of the introduction of the concept of fuzzy set theory, some important terms related to fuzzy set theory are discussed. Different defuzzification methodologies of fuzzy numbers (FNs) that are useful in solving the mathematical models in imprecisely defined decision-making environments are explored. The concept of using FRVs in decision-making contexts is defined. Finally, the development of different forms of fuzzy goal programming (FGP) techniques for solving multi-objective decision-making (MODM) problems is underlined.

Probability Theory on Fuzzy Sets

2017 ◽  
pp. 55-83
Author(s):  
Renáta Bartková ◽  
Beloslav Riečan ◽  
Anna Tirpáková
Keyword(s):  
Probability Theory ◽  
Fuzzy Sets ◽  
20Th Century ◽  
Set Theory ◽  
Significant Role ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Multivalued Logic ◽  
Probability Spaces

Similarly as the Kolmogorov probability theory in the first half of the 20th century, the Zadeh fuzzy set theory played a significant role in the second half of the 20th century. In this chapter we present probability theory on intuitionisic fuzzy sets as well as probability spaces on multivalued logic.

Arguments from Providence and from Miracles

2018 ◽  
Author(s):  
Timothy McGrew
Keyword(s):  
Probability Theory ◽  
20Th Century ◽  
Probabilistic Analysis ◽  
Interdisciplinary Collaboration ◽  
Formal Analysis ◽  
Empirical Science ◽  
The Subject

The mid-20th century consensus regarding Hume’s critique of reported miracles has broken down dramatically in recent years thanks to the application of probabilistic analysis to the issue and the rediscovery of its history. Progress from this point forward is likely to be made along one or more of three fronts. There is wide room for interdisciplinary collaboration, work that will bring together scholars with expertise in religion, psychology, philosophy, and empirical science. There is a great deal of work still to be done in formal analysis, making use of the tools of modern probability theory to model questions about testimony and inference. And the recovery and study of earlier works on the subject—works that should never have been forgotten—can significantly enrich our understanding of the underlying issues.

scholarly journals Probability Axioms and Set Theory Paradoxes

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 179
Author(s):  
Ari Herman ◽  
John Caughman
Keyword(s):  
Probability Theory ◽  
Set Theory ◽  
Common Sense ◽  
Weak Form

In this paper, we show that Zermelo–Fraenkel set theory with Choice (ZFC) conflicts with basic intuitions about randomness. Our background assumptions are the Zermelo–Fraenekel axioms without Choice (ZF) together with a fragment of Kolmogorov’s probability theory. Using these minimal assumptions, we prove that a weak form of Choice contradicts two common sense assumptions about probability—both based on simple notions of symmetry and independence.

Maximal linked systems on families of measurable rectangles

2021 ◽  
pp. 77-104
Author(s):  
Aleksandr G. Chentsov
Keyword(s):  
Probability Theory ◽  
Measure Theory ◽  
Cartesian Product ◽  
Wide Sense ◽  
The Family ◽  
Principal Property ◽  
Measurable Structure ◽  
Stone Type ◽  
By Products ◽  
Algebra Of Sets

Linked and maximal linked systems (MLS) on π -systems of measurable (in the wide sense) rectangles are considered (π-system is a family of sets closed with respect to finite intersections). Structures in the form of measurable rectangles are used in measure theory and probability theory and usually lead to semi-algebra of subsets of cartesian product. In the present article, sets-factors are supposed to be equipped with π-systems with “zero” and “unit”. This, in particular, can correspond to a standard measurable structure in the form of semialgebra, algebra, or σ-algebra of sets. In the general case, the family of measurable rectangles itself forms a π -system of set-product (the measurability is identified with belonging to a π - system) which allows to consider MLS on a given π -system (of measurable rectangles). The following principal property is established: for all considered variants of π -system of measurable rectangles, MLS on a product are exhausted by products of MLS on sets-factors. In addition, in the case of infinity product, along with traditional, the “box” variant allowing a natural analogy with the base of box topology is considered. For the case of product of two widely understood measurable spaces, one homeomorphism property concerning equipments by the Stone type topologies is established.

Study on Fuzzy Reliability of Shaft Fixing Grinding Wheel in Internal Grinding

2006 ◽  
Vol 304-305 ◽  
pp. 218-221
Author(s):  
Chun Xiang Ma ◽  
Z.H. Qu ◽  
Q.J. Wu ◽  
Zhen Ruan ◽  
Li Ming Xu
Keyword(s):  
Probability Theory ◽  
Fatigue Strength ◽  
Set Theory ◽  
Fuzzy Set Theory ◽  
Grinding Wheel ◽  
Reliability Model ◽  
Fuzzy Probability ◽  
Fuzzy Reliability ◽  
Internal Grinding ◽  
Fuzzy Probability Theory

The fuzzy allowable intervals of the deflection, the vibration frequency and the fatigue strength of the shaft fixing grinding wheel are proposed in present paper based fuzzy set theory. According to fuzzy probability theory, the fuzzy reliability model of the rigidity, the fuzzy reliability model of the vibration and the fuzzy reliability model of the fatigue for the shaft fixing grinding wheel in internal grinding are given. The method is put into real example.

Ecopedagogy and Education

2021 ◽  
Author(s):  
Ruyu Hung
Keyword(s):  
Human Nature ◽  
20Th Century ◽  
Environmental Issues ◽  
Ecological Approach ◽  
Late 20Th Century ◽  
The World ◽  
Fundamental Understanding ◽  
Basic Ideas ◽  
Ecological Worldview

The neologism ecopedagogy was coined in the late 20th century to represent the joining of ecology and pedagogy. However, ecopedagogy is not an education about ecology but an education through ecology, meaning that it is an education based on an ecological worldview. A worldview is the fundamental understanding of life and the world. Ecological worldview means the ecological approach to the understanding of life and the world. The basic ideas of the ecological worldview come from the science of ecology, of which there are two interpretations: ecology of stability and ecology of instability. Both provide a general, shared outline of the world and how it works but each offers distinctive values of philosophy, ethics, culture, and society with regard to the ecosystem. Ecopedagogy, which encompasses both ecological worldview and education, develops into two broad movements: philosophical ecopedagogy and critical ecopedagogy. For the former, referred to as ecosophy, focuses on the metaphysical investigation of the human-nature relationship and related issues in education. For the latter, ecojustice, the mission is to critique the injustice and oppression involved in environmental issues and to construct a utopian society of planetary civilization.

The Wiener-Hopf Technique in Applied Probability

1975 ◽  
Vol 12 (S1) ◽  
pp. 145-156 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Probability Theory ◽  
Applied Probability ◽  
Short Introduction ◽  
Basic Ideas

In this study some of the basic ideas needed for the application of the Wiener-Hopf Technique in solving problems occurring in applied probability theory are discussed; the paper aims to give a short introduction. The method is illustrated by applying it to two problems; one, although basic in probability theory, is rather simple to handle by this method. The second is much more intricate, but shows clearly the power of the method.

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