Fuzzy principles

This book presents systematic and comprehensive modeling of uncertainty, vagueness, or imprecision through fuzzy principles and procedures for problem solving in engineering. There are several chapters for introduction to fuzzy logic... 

The principle of applying fuzzy logic to matching of spectra is that, given a sample spectrum and a collection of reference spectra, in a first step the reference spectra are unified and fuzzed, i.e., around each characteristic line at a certain wavenumber k, a certain fuzzy interval [/ o - Ak, + Afe] is laid. The resulting fuzzy set is then intersected with the crisp sample spectrum. A membership function analogous to the one in Figure 9-25 is applied. If a line of the sample spec-... 

All these methods and the methods of the preceding section have one characteristic in common an object may be part of only one cluster. Fuzzy clustering applies other principles. It permits objects to be part of more than one cluster. This leads to results such as those illustrated by Fig. 30.16. Each object i is given a value... 

Bayesian probability theory and methods that are based on fuzzy-set theory. The principles of both theories are explained in Chapter 16 and Chapter 19, respectively. Both approaches have advantages and disadvantages for the use in expert systems and it must be emphasized that none of the methods, developed up to now are satisfactory [7,11]. 

Fuzzy logic is often presented as an extension in books that cover expert systems. Few texts exist in which the applications of fuzzy logic to scientific problems are described, but several texts include more general discussions of the principles and practical implementation of this method. Among the best is Negnevitsky s text on intelligent systems.9... 

They demonstrate a certain fuzziness, which leaves scope for flexible concretion (subtypes of guiding principles) thus also favouring the integration of practical and specific content. Guiding principles themselves in most cases do not contain any operative goals, objectives and indicators. They can thus also not... 

There is probably no other concept that contributed to the development of chemistry so remarkably as the ill-defined, qualitative concept of similarity. Not despite but rather because of a certain fuzziness, the applicability of this concept is extremely broad and touches practically all areas of chemistry [1, 2]. An example would be the Mendeleev periodic law, the disclosure of which was closely connected with the effort to systematise the similarities in the properties of elements. From the intuitively interpreted meaning of similarity arises also one of the most powerful chemical principles - the principle of analogy, on the basis of which a wealth of fundamental chemical notions were introduced. 

The generally accepted notion of wave-particle duality, which predates Heisenberg and Bohr, could be reconciled with the probability interpretation, but the fuzziness associated with waves remained unexplained in the orthodox tradition. The proclamation of the quantum-mechanical uncertainty principle was intended to take care of the oversight. A more serious indictment of the orthodox tradition is hard to imagine, short of the blunt statement by Nobel physicist, Murray Gell-Mann [28] ... 

Density functional theory provides the means for defining fragment energies, based on the following, simple principle the fuzzy electron density fragment of a functional group, together with the associated set of nuclei, is treated as a complete... 

The first model, describing the isolated region approach, can be derived easily from the AFDF principle. Molecular regions are described by fuzzy electron densities analogous to densities of complete molecules, and the local shape analysis of regions follows the same principles as the shape analysis of... 

An unambiguous algorithm. The intent of this principle is to ensure the transparency of the modeling algorithm. Sometimes, it is a difficult task to satisfy this principle, particularly when complex methods like neural networks or fuzzy logic techniques are used for modeling. 

The partition of the total system into a spin part and a lattice part is, in principle, fuzzy. Other nuclear or electronic spins can either be included in the spin space or into the lattice. In a perturbation treatment, it depends on the interaction strength and on the time-scales of different processes. If nuclear spins are interacting over long periods (e.g. within a small molecule), the different spins cannot be considered separately. However, electron spins can in many cases be put in the lattice space since the difference in time scales for the electron spin and nuclear spin dynamics is prohibitive an effective coupling. 

The term fuzzy logic is often interpreted in two ways. In a broad interpretation, fuzzy logic is viewed as a system of concepts, principles, and methods for dealing with modes of reasoning that are approximate rather than exact. In a narrow interpretation, it is viewed as a generalization of the various many-valued logics. This narrow interpretation of fuzzy logic is not of interest in this chapter. 

Reasoning based on fuzzy propositions of the four types, possibly quantified by various fuzzy quantifiers, is usually referred to as approximate reasoning. Although approximate reasoning is currently a subject of intensive research, its basic principles are already well established. For example, the most common inference rules of classical logic, such as modus ponens,... 

Chirality, an important shape property of molecules, can be regarded as the lack of certain symmetry elements. Chirality measures are in fact measures of symmetry deficiency. These principles, originally used for crisp sets, also apply for fuzzy sets. Considering the case of three-dimensional chirality, the lacking point symmetry elements are reflection planes a and rotation-reflections 82 of even indices. Whereas the lacking symmetry elements can be of different nature in different dimensions, nevertheless, all the concepts, definitions, and procedures discussed in this section have straightforward generalizations for any finite dimension n. 

Following the principles of the ZPA approach, these symmetry deficiency measures are generalizations of the folding-unfolding approach, equally applicable to crisp continuum sets and fuzzy sets, for example, to entire electron density distributions of molecules and various molecular fragments representing fuzzy functional groups. 

In this work a fuzzy matching procedure is suggested which takes the foregoing uncertainties into account at least in principle. The approach is based on a soft definition of a surface that is defined in terms of membership functions (see Fig. 8) These functions are and, .(r) and they measure to what extent a given space point belongs to the surface and the bulk of a molecule, respectively. The matching of two molecules A and B can then be calculated in many different ways. In a first attempt we used the intersection of two fuzzy sets... 

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