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## Über dieses Buch

Uncertainty theory is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms. Uncertainty is any concept that satisfies the axioms of uncertainty theory. Thus uncertainty is neither randomness nor fuzziness. It is also known from some surveys that a lot of phenomena do behave like uncertainty. How do we model uncertainty? How do we use uncertainty theory? In order to answer these questions, this book provides a self-contained, comprehensive and up-to-date presentation of uncertainty theory, including uncertain programming, uncertain risk analysis, uncertain reliability analysis, uncertain process, uncertain calculus, uncertain differential equation, uncertain logic, uncertain entailment, and uncertain inference. Mathematicians, researchers, engineers, designers, and students in the field of mathematics, information science, operations research, system science, industrial engineering, computer science, artificial intelligence, finance, control, and management science will find this work a stimulating and useful reference.

## Inhaltsverzeichnis

### Uncertainty Theory

Abstract
Some information and knowledge are usually represented by human language like “about 100km”, “approximately 39°C”, “roughly 80kg”, “low speed”, “middle age”, and “big size”. How do we understand them? Perhaps some people think that they are subjective probability or they are fuzzy concepts. However, a lot of surveys showed that those imprecise quantities behave neither like randomness nor like fuzziness. This fact provides a motivation to invent another mathematical tool, namely uncertainty theory.
Baoding Liu

### Uncertain Programming

Abstract
Uncertain programming was founded by Liu [122] in 2009 as a type of mathematical programming involving uncertain variables. This chapter provides a general framework of uncertain programming, including expected value model, chance-constrained programming, dependent-chance programming, uncertain dynamic programming and uncertain multilevel programming. Finally, we present some uncertain programming models for project scheduling problem, vehicle routing problem, and machine scheduling problem.
Baoding Liu

### Uncertain Risk Analysis

Abstract
The term risk has been used in different ways in literature. Here the risk is defined as the “accidental loss” plus “uncertain measure of such loss”. Uncertain risk analysis was proposed by Liu [126] in 2010 as a tool to quantify risk via uncertainty theory. One main feature of this topic is to model events that almost never occur. This chapter will introduce a definition of risk index and provide some useful formulas for calculating risk index.
Baoding Liu

### Uncertain Reliability Analysis

Abstract
Uncertain reliability analysis was proposed by Liu [126] in 2010 as a tool to deal with system reliability via uncertainty theory. Note that uncertain reliability analysis and uncertain risk analysis have the same root in mathematics. They are separately treated for application convenience in practice rather than theoretical demand.
This chapter will introduce a definition of reliability index and provide some useful formulas for calculating reliability index.
Baoding Liu

### Uncertain Process

Abstract
An uncertain process is essentially a sequence of uncertain variables indexed by time or space. The study of uncertain process was started by Liu [121] in 2008. This chapter introduces the basic concepts of uncertain process, including renewal process, martingale, Markov process and stationary process.
Baoding Liu

### Uncertain Calculus

Abstract
Uncertain calculus, invented by Liu [123] in 2009, is a branch of mathematics that deals with differentiation and integration of function of uncertain processes. This chapter will introduce canonical process, uncertain integral, chain rule, and integration by parts.
Baoding Liu

### Uncertain Differential Equation

Abstract
Uncertain differential equation, proposed by Liu [121] in 2008, is a type of differential equation driven by canonical process. Uncertain differential equation was then introduced into finance by Liu [123] in 2009. After that, an existence and uniqueness theorem of solution of uncertain differential equation was proved by Chen and Liu [17], and a stability theorem was showed by Chen [20].
This chapter will discuss the existence, uniqueness and stability of solutions of uncertain differential equations. This chapter will also provide a 99- method to solve uncertain differential equations numerically. Finally, some applications of uncertain differential equation in finance are documented.
Baoding Liu

### Uncertain Logic

Abstract
Uncertain logic is a generalization of mathematical logic for dealing with uncertain knowledge via uncertainty theory. The first model is uncertain propositional logic designed by Li and Liu [96] in which the truth value of an uncertain proposition is defined as the uncertain measure that the proposition is true. An important contribution is the truth value theorem by Chen and Ralescu [18] that provides a numerical method for calculating the truth value of uncertain formulas. The second model is uncertain predicate logic proposed by Zhang and Peng [227] in which an uncertain predicate proposition is defined as a sequence of uncertain propositions indexed by one or more parameters.
One advantage of uncertain logic is the well consistency with classical logic. For example, uncertain logic obeys the law of truth conservation and is consistent with the law of excluded middle and the law of contradiction. This chapter will introduce uncertain propositional logic and uncertain predicate logic.
Baoding Liu

### Uncertain Entailment

Abstract
Uncertain entailment, developed by Liu [124] in 2009, is a methodology for calculating the truth value of an uncertain formula via the maximum uncertainty principle when the truth values of other uncertain formulas are given. In order to solve this problem, this chapter will introduce an entailment model. As applications of uncertain entailment, this chapter will also discuss modus ponens, modus tollens, and hypothetical syllogism.
Baoding Liu

### Uncertain Set Theory

Abstract
Uncertain set theory was proposed by Liu [125] in 2010 as a generalization of uncertainty theory to the domain of uncertain sets. This chapter will introduce the concepts of uncertain set, membership degree, membership function, uncertainty distribution, independence, operational law, expected value, critical values, Hausdorff distance, and conditional uncertain set.
Baoding Liu

### Uncertain Inference

Abstract
Uncertain inference was proposed by Liu [125] in 2010 as a process of deriving consequences fromuncertain knowledge or evidence via the tool of conditional uncertain set. Gao, Gao and Ralescu [42] extended the inference rule to the one with multiple antecedents and with multiple if-then rules.
This chapter will introduce an inference rule, and apply the tool to uncertain system and inference control. The technique of uncertain inference controller is also illustrated by an inverted pendulum system.
Baoding Liu

### Backmatter

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