Constraint index, definition

The common underlying principle in the approaches for characterizing the solvability of a DAE system is to obtain, either explicitly, or implicitly, a local representation of an equivalent ODE system, for which available results on existence and uniqueness of solutions are applicable. The derivation of the underlying ODE system involves the repeated differentiation of the algebraic constraints of the DAE, and it is this differentiation process that leads to the concept of a DAE index that is widely used in the literature. For the semi-explicit DAE systems (A. 10) that are of interest to us here, the index has the following definition. 

Next, it is necessary to define the concept of orthogonality of sign vectors. Two sign vectors a and b are said to be orthogonal (a T b) if either (1) the supports of a and b have no indices in common, or (2) there is an index i for which a, and bi have the same signs and there is another index j (j i) for which aj and bj have opposite signs. Given these definitions, the thermodynamic constraint may be stated as ... 

The above definition of molecular chaperone is entirely fnnctional and contains no constraints on the mechanisms by which different chaperones may act. The term noncovalent is nsed to exclude those proteins that carry out posttranslational covalent modifications. Protein disulfide isomerise may seem to be an exception, bnt it is both a covalent modification enzyme and a molecular chaperone. It is helpful to think of a molecnlar chaperone as a fnnction rather than as a molecnle. Thns, no reason exists why a chaperone function shonld not be a property of the same molecnle that has other fnnctions. Other examples include peptidyl-prolyl isomerase, which possesses both enzymatic and chaperone activities in different regions of the molecnle, and the alpha-crystallins, which combine two essential fnnctions in the same molecnle in the lens of the eye-contribnting to the transparency and the refractive index reqnired for vision as well... 

CONSTRAINT identifier index domain definition CONSTRAINT identifier index domain definition ENDSECTION ... 

This method consists of characterizing the design features, especially in the safety system architecture, that are likely to pose problems in the operation, notably during the degraded situations in which the plant safety strongly depends on human reliability. The characterization of the intrinsic physical behaviour of the plant processes (safety functions), of the operating constraints of the safety systems and finally of the interrelations between these entities (most complexity theories consider these interrelations to be the main contributors to the complexity of a system), lead to the definition of an operational complexity index and to the identification of the sources of the operational constraints bearing on the operation crews. 

In order to use this transformation for the Hamiltonian as represented by Eq. (6.68), microscopic density terms that are quadratic in nature need to be written in the form given on the left-hand side in Eqs. (6.77) and (6.78). Electrostatic terms in He are already in the appropriate form. It is only the terms in Hw that needs to be rewritten. This can be achieved by rewriting in terms of order parameters and total density. For an n component system, all microscopic densities can be described by n—1 independent order parameters (due to the incompressibility constraint serving as the nth relation among the densities). There are many different ways of defining these order parameters. One convenient definition, which makes mathematics simple, is the deviation of densities of solutes from the solvent density, that is, defining c )j(r) = Qj(r)—Qj(r) forj = 1,2,... (n—1), wherej is the index for different solutes (monomers, counterions, and the salt ions). Using the transformation for each quadratic term in the Hamiltonian (cf. Eq. (6.68)), the partition function becomes... 

Saturation index calculations made as part of a species distribution problem allow an assessment to be made of the effect of organic acids on the likely state of heterogeneous equilibria in an aqueous system (see Drever 1988, for discussion and definitions). By comparing saturation indices for minerals in systematically different waters we can predict the likely behavior of these minerals in the presence of organic acids. The predictions about mineral stability vary with the precise constraints that are placed on the calculations, in particular whether the cations are constrained to be in equilibrium with a mineral phase or set as a total concentration, the temperature, the partial pressure of CO2, and the anionic composition of the water. Conclusions that differ from those presented here may be possible, nevertheless, some consistent trends emerge that are related to observations made in the preceding section about speciation. 

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