Formal tables definition

Vibrations may be decomposed into three orthogonal components Ta (a = x, y, z) in three directions. These displacements have the same symmetry properties as cartesian coordinates. Likewise, any rotation may be decomposed into components Ra. The i.r. spanned by translations and rotations must clearly follow the appropriate symmetry type of the point-group character table. In quantum formalism, a transition will be allowed only if the symmetry product of the initial and final-state wave functions contains the symmetry species of the operator appropriate to the transition process. Definition of the symmetry product will be explained in terms of a simple example. 

In addition to a narrative, it is useful to have an index of the vocabulary. The glossary s purpose is to link the formal terms back to the real world. Rather than a single monolithic table at the end of the document, the definitions could be introduced as needed in the context of the document structure, together with additional explanation.2... 

The dictionary is the interface between the informal and the formal. It has an entry for the name of each named feature together with both its formal and its informal definitions. Whether or not the dictionary is presented as an actual table (either in printed matter or on screen—a good tool should be able to extract it and present it if you so require), the dictionary is the notional holder of all the features defined within the package. 

One way of getting rid of distortions and basis set dependence could be that one switches to the formalism developed by Bader [12] according to which the three-dimensional physical space can be partitioned into domains belonging to individual atoms (called atomic basins). In the definition of bond order and valence indices according to this scheme, the summation over atomic orbitals will be replaced by integration over atomic domains [13]. This topological scheme can be called physical space analysis. Table 22.3 shows some examples of bond order indices obtained with this method. Experience shows that the bond order indices obtained via Hilbert space and physical space analysis are reasonably close, and also that the basis set dependence is not removed by the physical space analysis. 

A short list of invariant lattice complex symbols is reported in the following. For a complete list, for a more systematic description and formal definition, see the International Tables of Crystallography, Hahn (2002). 

What are the empirical inductive laws on which thermodynamics rests For future reference, Table 2.1 lists the six general statements IL-l-IL-6 of observational experience on which the present exposition will be based. Several of these require additional definitions or explanations before they can be properly understood. Each will be introduced explicitly in the text as its definitional basis is properly laid and its logical role in the formal construction of the theory becomes apparent. 

With definitions of work w and heat q established, we proceed to formal statement(s) of the first law of thermodynamics (cf. IL-5 Table 2.1). Although the first law is sometimes stated colloquially as Energy is conserved (or, somewhat more satisfactorily, Energy is conserved if heat is taken into account ), a proper statement requires the introduction of a new quantity, internal energy U, that can be distinguished from energy as used in the mechanical framework ... 

It follows that the functions in the left-hand column of the preceding table are the integrals of the functions in the right-hand column. More formally, they are the indefinite integrals of the function, in contrast to the definite integrals described next. Tables of indefinite integrals in reference books may be consulted for more complex examples, or mathematical software can be used to evaluate them. 

The next two entries to Table 3 are cited for completeness. Nitrogenase is treated in Chapter 7 and CO dehydrogenase in Chapter 9. Nitrogenase contains a very complex iron-sulfur cluster that includes another metal, molybdenum or vanadium. The crystal structure of the Mo variant has been determined. There is a third variant, alternative nitrogenase [92], whose cluster apparently does not contain any heterometal. That cluster would thus be a perfect candidate for our definition of a redox-catalytic iron-sulfur cluster. Unfortunately, this third nitrogenase has thus far been characterized to a much lesser extent than the other two forms. For all nitrogenases holds that the binding of N2 to the cluster has not been established [53] therefore, formally these enzymes have not yet been positively identified as redox iron-sulfur catalysts. 

It is remarkable that the integral equation (3.693) proved to be formally exact [51], so that all theories differ only in their definition of the kernel, given as E(f) or V(.v). Sometimes the kernel is explicitly defined in the original works, but more often the reduction to the integral form and extraction of the kernel is a separate problem solved in Ref. 46. In a few cases this procedure was nontrivial and required rather long and sophisticated calculations that are of no interest except for the final results represented by the Laplace transformed kernels S in Table V. 

The tables are arranged by subject. The five columns in each table give the name of the quantity, the recommended symbol(s), a brief definition, the symbol for the coherent SI unit (without multiple or submultiple prefixes, see p.74), and footnote references. When two or more symbols are recommended, commas are used to separate symbols that are equally acceptable, and symbols of second choice are put in parentheses. A semicolon is used to separate symbols of slightly different quantities. The definitions are given primarily for identification purposes and are not necessarily complete they should be regarded as useful relations rather than formal definitions. For dimensionless quantities a 1 is entered in the SI unit column. Further information is added in footnotes, and in text inserts between the tables, as appropriate. 

OF THE FORMAL ESTIMATE AND THE IN-HOUSE, PRELIMINARY ESTIMATE WAS WITHIN 1% OF THE DEFINITIVE ESTIMATE (See Table M.4.)... 

Biomimicry. While there is no formal definition of biomimicry, for the purpose of this section, biomimicry refers to (1) uncovering the chemical configuration of natural materials and the network of natural processes involved in their production and use, and (2) designing new chemical industry products and processes based on these natural models. Applied, industrial-scale outcomes are likely to be engineered hybrids - a compromise between nature s designs and man-made creations. Ideally, these applications would exploit the environmental and health benefits of the natural chemistry model (Table 7.1). 

The constituents of the compound to be named are divided into formally electropositive and formally electronegative constituents. There must be at least one electropositive and one electronegative constituent. Cations are electropositive and anions electronegative, by definition. Electropositive elements occur later in Table VI than electronegative elements by... 

Developing an FMEA is a DFSS team effort. The numerical ratings for severity, detectability and frequency must be agreed on by the team, so that everyone agrees on the definitions. Otherwise one team member may consider the effect to be mild, another moderate, and yet another severe. As with QFD, the numbers here are semi quantitative, and round table team discussions help to achieve a more accurate picture of the true risks associated with the specific failure modes. If possible, the team should formalize the rating system as much as possible. A sample of such a formalized scale is given in Fig. 14. 

Table III shows that definitions A, i = I-IIl, and Aj,, y = I-IV, have the same commutation properties as definitions A and A, respectively. This arises because the operators A, i = I-III, and j = I-IV, have the same matrix representation as, respectively, A and one particular operator A and, thus, are formally equivalent to the latter (although computationally the procedures are different). These equivalences are most simply established in the basis sets of the dgenvectors used to projluce the A. Table I shows that o( ...

Section II.C demonstrates that only one a definition emerges from any formalism that involves norm-preserving mappings. Many such formalisms are mentioned in Section V, and most of the corresponding a definitions have been suggested in the literature. All these definitions belong to category A of Table I. 

Formally, the key difference in the definition of by-tuple semantics from that of by-table semantics is that a consistent target instance is defined by a mapping sequence that assigns a (possibly different) mapping in m to each source tuple in Ds. (Without losing generality, to compare between such sequences, we assign some order to the tuples in the instance.)... 

The concept of entropy is a difficult one. An understanding of it comes with repeated encounters in different contexts. Although a formal definition will be given later, it is sufficient for the time being to associate it pictorially with the degree of randomness of a material, as exemplified in the processes discussed above. It is possible to assign absolute numerical values to this entropy, 5°, which refer to individual materials in a particular physical state. In Table 4.1 values are given for a few compounds. 

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