Extension principle

The extension principle The extension principle, introduced by Zadeh, is one of the basic ideas related to fuzzy set theory. This principle provide an extension of the mathematical relationship between fuzzy sets and crisp sets. In this paper only the concepts associated with addition, multiplication, subtraction and division will be considered. 

In this paper we presented an approach where fuzzy logic and BBN concepts are combined to estimate human error probability. This combination leads to a fuzzy Bayesian network approach based on the concept o fuzzy number and on extension principles applied to discrete fuzzy probabilities calculation. 

In the context of interest here, namely that of fault tree analysis, Lindley Singpurwalla (1986) present a formal probabilistic (Bayesian) procedure for the use of expert opinions, assuming expert input in the form of mean and standard deviation of lognormally distributed failure rates. In Tanaka et al. (1983), Liang Wang (1991) and Huang et al. (2001), basic event probabilities (chances) are treated as trapezoidal fuzzy numbers and the extension principle is applied to compute the probability (chance) of occurrence of the top event. In order to deal with repeated basic events in fault tree analysis. Soman Misra (1993) provide a simple method for fuzzy fault tree analysis based on the a-cut method, also known as resolution identity. Another approach to fuzzy fault tree analysis based on the treatment of the system state as a fuzzy variable has been proposed by Huang et al. (2004). 

Propagation of the uncertainty in the hybrid approach is based on the combination of the Monte Carlo technique (Kalos Withlock 1986) with the extension principle of fuzzy set theory (Zadeh 1965),... 

Soman, K.P. Misra, K.B. (1993) Fuzzy fault tree analysis using resolution identity and extension principle. International Journal of Fuzzy Mathematics 1 193-212. 

A more general definition supplies the max- composition. An outstanding special case of a composition of fuzzy relations is the so-called extension principle. That is the most practiced way of extending algebraic operations from crisp to fuzzy numbers. ... 

A possibility to apply the mapping / is the application of the extension principle (Zadeh 1975 Yager 1986) ... 

The computation of x d) needs to consider Eq. 4. The extension principle is accompanied by a high numerical effort, which is not feasible for practical applications. Therefore, an approximate numerical algorithm, called a-level optimization, was adopted in Moller and Beer (2004). 

The aim of the a-level optimization is the computation of the membership function of the fuzzy result value f. Therefore, the min-max operator of the extension principle (Eq. 26) is replaced by optimization tasks. The necessary discretization of the fuzzy input quantities into a-level is defined in Eq. 5. The main idea of this approach is that the necessary information is the minimum and maximum value for the response only, which can be reached at each a-level a.k G (0,1]. The membership values of the result are... 

Tintner G (1960) A note on stochastic linear programming. Econometrica 28(2) 490 95 Yager RR (1986) A characterization of the extension principle. Fuzzy Set Syst 18(3) 205-217 Zadeh L (1975) The concept of a linguistic variable and its application to approximate reasoning. Inform Sci 8(3) 199-249... 

Two common ways of defining the extended operation are based on the a-cut representation of fuzzy intervals and on the extension principle of FST (Kaufman and Gupta (1985), Klir and Yuan (1995)). When a-cut representation is employed, arithmetic operations on fuzzy intervals are defined in terms of arithmetic operations on closed intervals. To define the individual... 

CaH803. Fine white needles, m.p. 82°C, b.p. 285°C, strong vanilla odour, characteristic taste. It occurs extensively in nature, and is the odoriferous principle of the vanilla pod it can be obtained from the glucoside coniferin. Vanillin is made commercially from the ligno-sulphonic acid obtained as a by-product in the manufacture of wood pulp. It is one of the most important flavouring and perfuming... 

In this paper, we have exposed a solution to improve the resolution in Low Frequency Ultrasonic Tomography. Since the basic principle of ultrasonic reflection tomography prohibits the inspection of objects with strong contrast and large extension, we turn down the frequency of the transducer, in order to increase the penetration length of the wave and the validity of the method. But this is done at the expense of resolution. 

Equation XVII-127 connects the functions 0(F, T), d(Q,P, T) and f Q) and, in principle, if any two are known or can be assumed, the remaining one can be calculated. As may be imagined, many choices of such pairs of functions have been examined, often designed so that Eq. XVII-127 can be handled analytically alternatively, various schemes of successive approximations may be used. The field has become somewhat of a happy hunting ground for physical chemists and there are numerous reviews of the now-extensive literature (see Refs. 144-147 the last is a personalized account). For this reason only some generic approaches will be discussed here. 

As on previous occasions, the reader is reminded that no very extensive coverage of the literature is possible in a textbook such as this one and that the emphasis is primarily on principles and their illustration. Several monographs are available for more detailed information (see General References). Useful reviews are on future directions and anunonia synthesis [2], surface analysis [3], surface mechanisms [4], dynamics of surface reactions [5], single-crystal versus actual catalysts [6], oscillatory kinetics [7], fractals [8], surface electrochemistry [9], particle size effects [10], and supported metals [11, 12]. 

Secondary ion mass spectrometry (SIMS) is by far the most sensitive surface teclmique, but also the most difficult one to quantify. SIMS is very popular in materials research for making concentration depth profiles and chemical maps of the surface. For a more extensive treatment of SIMS the reader is referred to [3] and [14. 15 and 16]. The principle of SIMS is conceptually simple When a surface is exposed to a beam of ions... 

In recent years, these methods have been greatly expanded and have reached a degree of reliability where they now offer some of the most accurate tools for studying excited and ionized states. In particular, the use of time-dependent variational principles have allowed the much more rigorous development of equations for energy differences and nonlinear response properties [81]. In addition, the extension of the EOM theory to include coupled-cluster reference fiuictioiis [ ] now allows one to compute excitation and ionization energies using some of the most accurate ab initio tools. 

The program system COBRA [118, 119] can be regarded as a rule- and data-based approach, but also applies the principles of fragment-based (or template-based) methods extensively (for a detailed description sec Chapter 11, Sections 7.1 and 7.2 in the Handbook). COBRA uses a library of predefined, optimized 3D molecular fragments which have been derived from crystal structures and foi ce-field calculations. Each fi agment contains some additional information on... 

Though illustrated here by the Scott and Dullien flux relations, this is an example of a general principle which is often overlooked namely, an isobaric set of flux relations cannot, in general, be used to represent diffusion in the presence of chemical reactions. The reason for this is the existence of a relation between the species fluxes in isobaric systems (the Graham relation in the case of a binary mixture, or its extension (6.2) for multicomponent mixtures) which is inconsistent with the demands of stoichiometry. If the fluxes are to meet the constraints of stoichiometry, the pressure gradient must be left free to adjust itself accordingly. We shall return to this point in more detail in Chapter 11. 

A Iraditional or one-dimensional integral corresponds to the area under the curve between Ihc imposed limit, as illustrated in Figure 1.11. Multiple integrals are simply extensions of llu vc ideas to more dimensions. We shall illustrate the principles using a frmction of two vai ialiles,/(r. yj. The double integral... 

This chapter is divided into two parts The first and major portion is devoted to carbohydrate structure You will see how the principles of stereochemistry and confer matronal analysis combine to aid our understanding of this complex subject The remain der of the chapter describes chemical reactions of carbohydrates Most of these reactions are simply extensions of what you have already learned concerning alcohols aldehydes ketones and acetals... 

At first glance, the contents of Chap. 9 read like a catchall for unrelated topics. In it we examine the intrinsic viscosity of polymer solutions, the diffusion coefficient, the sedimentation coefficient, sedimentation equilibrium, and gel permeation chromatography. While all of these techniques can be related in one way or another to the molecular weight of the polymer, the more fundamental unifying principle which connects these topics is their common dependence on the spatial extension of the molecules. The radius of gyration is the parameter of interest in this context, and the intrinsic viscosity in particular can be interpreted to give a value for this important quantity. The experimental techniques discussed in Chap. 9 have been used extensively in the study of biopolymers. 

The mote general food processing appHcations requite data on dielectric and thermal properties (139). Considerable effort has been expended by food companies in the design of food for the microwave oven. These principles have been reviewed (140). The microwave oven at 2450 MH2, used for reheating, cooking, and thawing foods, may also be used for drying (qv), eg, flowers or food materials (141). Commercial microwave ovens ate used extensively in restaurants and fast-food estabUshments. 

Continuous paper machines have undergone extensive mechanical developments since the 1950s, although the principles employed have changed Utde. Cylinder machines stUl are operated and involve multiples of five to seven cylinders they are used to produce heavy multi-ply boards. Fourdriniers are common in the industry and can be used to produce virtually any grade of paper or paperboard. They vary from 1 to 10 meters in width and, including the press and dryer sections, may be more than 200 m long. 

---