A definition of subjective possibility

Based on the setting of exchangeable bets, this paper proposes a subjectivist view of numerical possibility theory. It relies on the assumption that when an agent constructs a probability measure by assigning prices to lotteries, this probability measure is actually induced by a belief function representing the agent’s actual state of knowledge. We also assume that the probability measure proposed by the agent in the course of the elicitation procedure is constructed via the so-called pignistic transformation (mathematically equivalent to the Shapley value in game theory). We pose and solve the problem of finding the least informative belief function having a given pignistic probability. We prove that it is unique and consonant, thus induced by a possibility distribution. This result exploits a simple informational ordering, in agreement with partial orderings between belief functions, comparing their information content. The obtained possibility distribution is subjective in the same sense as in the subjectivist school in probability theory. However, we claim that it is the least biased representation of the agent’s state of knowledge compatible with the observed betting behaviour.     

On the use of zadeh's probabilistic definition for testing statistical hypotheses from fuzzy information

A statistical hypothesis is an assertion about the distribution of an experiment. We consider the study of the problem of testing a statistical hypothesis (that is, the problem of concluding whether or not the hypothesis is correct) on the basis of data from the experiment, when its outcomes do not provide exact but rather fuzzy information.For establishing optimality criteria of testing we will use the definition of probability of a fuzzy event, given by Zadeh, in order to extend both Neyman-Pearson and Bayes theories, to the fuzzy framework.Then, we will analyze several properties for the new criteria. Particularly, the goodness of optimal procedures in both the fuzzy and the nonfuzzy situation, will be compared for each criterion.Finally, we will apply the extended criteria for testing simple hypotheses. This application leads us to prefer Bayesian procedures to Neyman-Pearson procedures in the fuzzy context.     

The use of fuzzy variables in pert

In this paper we present a method—called Fpert—for estimating a project completion time in the situation when activity duration times in the project network model are given in the form of fuzzy variables—fuzzy sets on time space. Theoretical foundations of the method as well as results of calculations derived from a simple example are included.     

The possibility of reconciling quantum mechanics with classical probability theory

We describe a scheme for constructing quantum mechanics in which a quantum system is considered as a collection of open classical subsystems. This allows using the formal classical logic and classical probability theory in quantum mechanics. Our approach nevertheless allows completely reproducing the standard mathematical formalism of quantum mechanics and identifying its applicability limits. We especially attend to the quantum state reduction problem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 457–472, December, 2006.     

Semideviations of reduced fuzzy variables: a possibility approach

This paper proposes new methods to reduce the uncertain information embedded in the secondary possibility distribution of a type-2 fuzzy variable. Based on possibility measure, we define the lower value-at-risk (VaR) and upper VaR of a regular fuzzy variable, and develop the VaR-based reduction methods for type-2 fuzzy variables. The proposed VaR-based reduction methods generalize some existing reduction methods by introducing possibility level parameter in distribution functions. For VaR reduced fuzzy variables, we employ Lebesgue–Stieltjes (L–S) integral to define three $n$ th semideviations to gauge the risk resulted from asymmetric fuzzy uncertainty. Furthermore, we compute the mean values and semideviations of the VaR reduced fuzzy variables, and derive some useful analytical expressions. The theoretical results obtained in this paper have potential applications in practical risk management and engineering optimization problems.     

Application of possibility theory to investment decisions

Carlson and Fuller (2001, Fuzzy Sets and Systems, 122, 315–326) introduced the concept of possibilistic mean, variance and covariance of fuzzy numbers. In this paper, we extend some of these results to a nonlinear type of fuzzy numbers called adaptive fuzzy numbers (see Bodjanova (2005, Information Science, 172, 73–89) for detail). We then discuss the application of these results to decision making problems in which the parameters may involve uncertainty and vagueness. As an application, we develop expression for fuzzy net present value (FNPV) of future cash flows involving adaptive fuzzy numbers by using their possibilistic moments. An illustrative numerical example is given to illustrate the results.     

On the possibility of gravitational collapse in the relativistic theory of gravity

In the example of dust matter, it is shown that a gravitational attractive force does not lead to the formation of “black holes” in the relativistic theory of gravity. It is proved that in the absence of matter, the gravitational field is also absent. Therefore, a vacuum is not a source of a gravitational field. The mechanism of energy production in the process of the accumulation of matter into massive objects is discussed.     

Applications of the theory of fuzzy sets

This survey summarizes the current state of the theory of fuzzy sets. The use of the theory in applications is illustrated with examples of control and modelling problems, as well as an application to the design of expert systems. Translated from Itogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 29, pp. 83–151, 1990.     

General negations in fuzzy set theory

All possible negation operations which are optimal in some precise sense are fully described. They turn out to be Lowen's fuzzy complements, Yager's intuitionistic negation, and a dual to the latter.     

The use of vector-valued martingales in risk theory

A completely dependent risk process with perturbation and phase-type distributed claim sizes is analyzed. Claim arrivals are modeled by a Markovian arrival process. Using a vector-valued martingale, the Laplace transform of the time to ruin is derived algorithmically. The conditional memoryless property of the phase-type distribution yields the distribution of the deficit at ruin as a corollary.     

The use of jacobi's lemma in unimodularity theory

The paper applies Jacobi's fundamental result on minors of the adjoint matrix to obtain properties on determinants of unimodular matrices.     

The use of integral operators in number theory

It is known that all of the non-real zeros of the Riemann zeta-function lie in the critical strip $\text{L}$={s=σ+it: O<σ<1} and important arithmetic consequences would follow from the assertion that all such zeros, in fact, lie on the line $\text{σ =}\text{1}\text{2}$. This assertion, known as the Riemann Hypothesis, is not yet established. In the present article, it is shown that the knowledge of the spectrum of any member of a certain class of self-adjoint, integral operators yields an explicit region of $\text{l}$ devoid of zeros of the Riemann zeta-function. Moreover, if a certain non-linear functional, Φ, is unbounded, then the Riemann hypothesis would follow.     

Associative monotonic operations in fuzzy set theory

The properties of binary operations in a real interval are considered and used in the discussion of generalized operations on fuzzy sets, on fuzzy numbers and on fuzzy probabilistic sets.     

On the use of divergence distance in fuzzy clustering

Clustering algorithms divide up a dataset into a set of classes/clusters, where similar data objects are assigned to the same cluster. When the boundary between clusters is ill defined, which yields situations where the same data object belongs to more than one class, the notion of fuzzy clustering becomes relevant. In this course, each datum belongs to a given class with some membership grade, between 0 and 1. The most prominent fuzzy clustering algorithm is the fuzzy c-means introduced by Bezdek (Pattern recognition with fuzzy objective function algorithms, 1981), a fuzzification of the k-means or ISODATA algorithm. On the other hand, several research issues have been raised regarding both the objective function to be minimized and the optimization constraints, which help to identify proper cluster shape (Jain et al., ACM Computing Survey 31(3):264–323, 1999). This paper addresses the issue of clustering by evaluating the distance of fuzzy sets in a feature space. Especially, the fuzzy clustering optimization problem is reformulated when the distance is rather given in terms of divergence distance, which builds a bridge to the notion of probabilistic distance. This leads to a modified fuzzy clustering, which implicitly involves the variance–covariance of input terms. The solution of the underlying optimization problem in terms of optimal solution is determined while the existence and uniqueness of the solution are demonstrated. The performances of the algorithm are assessed through two numerical applications. The former involves clustering of Gaussian membership functions and the latter tackles the well-known Iris dataset. Comparisons with standard fuzzy c-means (FCM) are evaluated and discussed.