Extended general, definition

Any text on acids and bases would not be deemed complete if mention were not made of the extended definition of acids and bases that is embodied in the Lowry-Bronsted theory. The theory basically proposed a more general definition of acids and bases to overpower the limitations of the theory arising from the Arrhenius concept. 

Syntactic foams are obtained by mixing hollow micro- or macro-balloons, generally made of glass but sometimes of polymer, directly with a liquid resin, generally a thermoset. Some extend this definition to all composites with a foamed core. 

In this account we extend the definition to include the concept of systematic, synthetic proliferation within an abiotic molecular structure starting from an origin core initiator and continuing through a succession of well-defined, stoichiometric compounds and structures. In general, this synthetic strategy involves the following criteria ... 

There is a more general definition of reversibility which extends nicely to higher-order systems. Consider any mapping 7 (x) of the phase space to itself that satisfies In other words, if the map-... 

A more general definition of acid-base equilibrium in molecular solvents was proposed independently by Brpnsted and Lowry, who extended the term base . According to them, the acid is a donor of protons, and they defined a base only as an acceptor of H+. According to the Brpnsted-Lowry definition, the dissociation of an acid results in the formation of a proton and a conjugate base ... 

The actual discussion has the aim to adopt this previous spirit, but obviously choosing a much more modest point of view, attached to Quantum Similarity Measures (QSM). This work is focused to explore the various possible extensions for the study of DF, the auxiliary building block elements of QSM [16-38]. In order to fulfil such a purpose, this study will start analysing a sound formal basis as a first step to understand the role of momentum operators in computational Quantum Chemistry. From this introductory position, it will be finally obtained a general pattern enveloping the whole area of DF study, beginning at the basic aspects and ending over the final applications of extended DF definitions. 

Risks The general definition of risk is the danger of injury, harm, or loss. With respect to soil and groundwater remediation, this definition can be extended to the danger of failure or the likelihood of an unwanted situation occurring. The probability that one or more of these calamities will actually happen is the risk factor. 

Eqs. (4.183) and (4.184) are, therefore, not based on the electroneutrality equation (4.181) because they also include weak acids and bases which are not protolyzed in solution at the given pH (that means in non-ionic form), but contribute to acid or base titration. The first attempt to determine airborne acidity was done by careful titration inserting microliter quantities of a NaOH solution (Brosset 1976) using Gran s titration method (Gran 1950). The reference points, however, are not objective criteria. Zobrist (1987) extended this definition to the general equations ... 

With the general definition in Equation 12.14, it becomes possible to extend the use of the electrochemical potential in kinetics, which is not possible with the classical definition limited to simple coupling situations in fact. This point will be detailed in case study J2. 

The quantity 0 may be called the phase defined on C, and its value is only determined to an integer multiple of T, It would be very inconvenient, however, if the definition of phase were restricted to C. This is because arbitrarily small perturbations could generally kick the state point out of C, so that without definition of 0 outside C we could no longer say anything about the phase of perturbed oscillations. It is therefore desirable to extend the definition of 0 so that we could say, e.g., how the phase, or its rate of change, is influenced by the perturbation. Since the perturbation is assumed to be weak, 0 need only be defined in the vicinity of C. 

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. ... 

Any Pi and the corresponding qi are said to be conjugate variables. By extending this definition to include any pair of variables which satisfy equations of the form 2-39 the concept becomes more general. For example, if a system of F degrees of freedom is described by the coordinates qi Qf and the time with t being formally treated on the same basis as the g s, the analysis shows that for systems in which... 

However, serious difficulties appeared later when efforts were made to attack more general problems not necessarily of the nearly-linear character. In terms of the van der Pol equation this occurs when the parameter is not small. Here the progress was far more difficult and the results less definite moreover there appeared two distinct theories, one of which was formulated by physicists along the lines of the theory of shocks in mechanics, and the other which was analytical and involved the use of the asymptotic expansions (Part IV of this chapter). The latter, however, turned out to be too complicated for practical purposes, and has not been extended sufficiently to be of general usefulness. 

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