Reduction principle

Markovitz,H. The reduction principle in linear viscoleasticity. J. Phys. Chem. 69,671 (1965). 

Fluorinated -alkenes and -cycloalkenes have a special relationship with their hydrocarbon analogues, usually exhibiting a chemistry that is complementary. For example, the fluorinated systems are frequently susceptible to nucleophilic attack, in some cases dramatically so, and therefore reactions of nucleophiles with fluorinated alkenes often reveal unique new chemistry. This chapter covers electrochemical reduction, principles governing orientation and reactivity of fluorinated alkenes towards nucleophiles, fluoride ion as a nucleophile and the mirror-image relationship of this chemistry with that of proton-induced reactions, reactions with nitrogen-, oxygen-, carbon- centred nucleophiles etc., and, finally, chemistry of some oligomers of fluorinated -alkenes and -cycloalkenes. 

Fishburn (1971) has suggested strengthening the Pareto principle to require that removing a Pareto dominated alternative from Y does not change the social choice set. He calls this condition the reduction principle, RP. 

While silver is a good model cation for fundamental studies leading to better understanding of the phenomena affecting combined oxidation-reduction photocatalytic processes, the use of silver is not recommended for full exploitation of the oxidation-reduction principle. Silver can also be very toxic and difficult to remove. 

It should be noted that the electrokinetic-chemical reduction principles are the same as electrokinetic-PRB using iron filing as reactive media, but one has to wait for the contaminated water to pass through the PRB for the remediation to occur. 

Rupture of a tensile test piece may be regarded as catastrophic tearing at the tip of a chance flaw. The success of the WLF reduction principle for fracture energy, G, in tearing thus implies that it will also hold for tensile rupture properties. Indeed, a/, and may be calculated from the appropriate value of G at each rate and temperature, using relations analogous to Eqs. (10.6) and (10.7). The rate of extension at the crack tip will, however, be much greater than the rate of extension of the whole test piece, and this discrepancy in rates must be taken into account (Bueche and Halpin, 1964). 

Requirements 1 to 7 can be met by the application of either qualitative or quantitative hazard and risk analysis techniques as per part 5 of the Standard. The example to date has applied a quantitative analysis employing a number of techniques. In terms of qualitative assessment, an "unlikely failure but with "catastrophic outcome represents an extreme risk necessitating "necessary risk reduction and application of ALARP (as low as reasonably practicable), good practice and continuous risk reduction principles... 

This completes the study of the basic principles of hazards and risk reduction principles. The key points are sununarized here ... 

The essential map carries most of the information on the global saddle-node bifurcations. As already mentioned, its degree m defines the topological type of W. If m = 1, then is smooth if, and only if, /(< ) does not have critical points (see (12.2.10) and (12.2.13)). Below (Theorem 12.4), we give a precise formulation to the following reduction principle ... 

The above reduction principle was applied explicitly in [151] for the case m = 1. An earlier study in [97] was essentially based on the same idea. 

Theorem 12.4. (Reduction principle) The Poincare map T — TqoT is written as... 

Note the difference between the transition to chaos under the big lobe condition and without it in the second case the intervals Ai of chaotic dynamics may, in principle, interchange with the intervals where the system has only finitely many saddle and stable periodic orbits [151]. According to the reduction principle (Theorem 12.4), this occurs if, within some interval of u, the essential map... 

As we mentioned, the importance of this result is that the flight time from any cross-section x = constant to any other cross-section of this form is independent on the initial point on this cross-section. Thus, when the system is brought to the form (12.5.5), Lemma 12.2 is valid which, in turn, implies the reduction principle of Theorem 12.4. 

In this section, we will discuss some algorithms for constructing normal forms. Due to the reduction principle, it is sufficient to construct the normal forms for the system on the center manifold only. Therefore, in order to consider bifurcations of an equilibrium state with a single zero characteristic root, we need a one-dimensional normal form. If it has a pair of zero characteristic exponents, one should examine the corresponding family of two-dimensional normal forms, and so on. 

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