Equivalence Relationships

We can start with the other Gibbs equations to obtain equivalent relationships involving H, A, and G. 

The kernel of 5) is a linear subspace of B,), which we use to define an equivalence relationship on... 

To make the above discussion more concrete, consider the example branching structure of Fig. 3. In this structure, we have identified an x and y, which have the same objective function values, and their ancestors, (x, y) which are characterized by the same set of symbols, (i.e., batches). Furthermore, we can see that the children of x,y do indeed satisfy the requirements of Condition-a and Condition-b and hence, (x, y) would be considered as candidates to develop a new equivalence relationship. If we examine the partial schedules (x, y) as depicted in Fig. 6, our knowledge... 

In these cases there is no well defined notion of a looser constraint, the choice is then either to force those variables to be equal in x and y, or to find some path from their value to a constraint on another inter- or intrasituational variable and thus be able to show that their values in jc, y should obey some ordering based on these other constraints. This topic is the subject of current research, but is not limiting in the flowshop example, since no such constraints exist. Lastly, it is not enough to assert conditions on the state variables in x and y, since we have made no reference to the discrete space of alternatives that the two solutions admit. Our definition of equivalence and dominance constrains us to have the same set of possible completions. For equivalence relationships the previous statement requires that the partial solutions, x and y, contain the same set of alphabet symbols, and for dominance relations the symbols of JC have to be equal to, or a subset of those of y. Thus our sufficient theory can be informally stated as follows ... 

The tern "weak equivalence" is somewhat unfortunate since weak equivalence is not in fact an equivalence relationship in the usual meaning of the tern. It is obviously symmetric but it is not transitive, since every scheme is weakly equivalent to any scheme that never halts. For example, consider P-, and below ... 

Now weak equivalence, finite equivalence, recursive equivalence and strong equivalence are all reasonable equivalences although the first is not a true equivalence relationship and the last three are. Reasonable equivalences stand between weak and strong - they must hold between schemes that behave identically for all interpretations and must fail between schemes which can be demonstrated to produce different results on the same interpretation. Total equivalence is not a reasonable equivalence because it does not hold for schemes which can diverge, and... 

Equivalent relationships are obtained by differentiation of the entropic form of the fundamental relation, formulated as... 

Figure 5 shows the equivalent relationship for a monostatic geometry. The resolutions are inversely proportional to the sampled extents Au and Av in Fourier space, thus... 

This is a linear relationship between the free energy of activation for homologous series of reaction. An equivalent relationship... 

The product of pressure p and pipe diameter d for a particular gas at a certain temperature can serve as a characterizing quantity for the various types of flow. Using the numerical values provided in Table III, Chapter 9, the following equivalent relationships exist for air at 20 °C ... 

Subtracting equation 7.180 from equation 7.179 gives the equivalent relationship in terms of deviation variables (Section 7.S.2), viz. ... 

Using the equivalency relationships in Table 3.1, propose an equivalent electrical circuit diagram for an enzyme thermistor operated (a) in direct detection mode and (b) in the feedback, push-pull mode. 

The current densities (i = I/A) obtained for disc and microspheres of the same radius for reversible electrode processes at any value of the applied potential follow the equivalence relationship given by [70, 71] ... 

In view of the expressions of the stationary current-potential responses of microspherical and microdisc electrodes (Eqs. (3.74) and (3.95), respectively), it is clear that an equivalence relationship between disc and hemispherical microelectrodes, like that shown for fast charge transfer processes (see Eq. (2.170) of Sect. 2.7), cannot be established in this case. 

Equation (6.42) clearly shows that the CV stationary responses of disc and spherical electrodes hold the same equivalence relationship as that observed for a simple charge transfer process ... 

To keep this proposition in focus, equivalence relationships, in the sense of symmetries in the physical world, may be defined in terms of a metric for the state space of a system. The metric [14] is a real non-negative function d(, ) with the following properties for all states u, v, w ... 

The tendency to establish equivalency relationships between functional groups, previously treated as unrelated, is one of the major motivations in the... 

We suggest the use of the symbol o to designate the equivalency relationships between reagent and synthon. In mathematical logic, this symbol means the equivalency of assertions. 

Substituting this relationship into equations (2.86) to (2.89) yields the equivalent relationships relating surface average sizes with the surface geometric mean diameter. 

We note that T(x2) is proportional to Ip(Sr), therefore, Ipfs ) is directly obtained by measuring with a slit collimated system. It should be realized that an equivalent relationship does not exist if the intensity is measured along the cylinder-axis with the incident beam being infinitely perpendicular to this axis. 

Topological information indices are graph invariants, based on information theory and calculated as - information content of specified equivalence relationships on the molecular graph. 

The information content is zero, i.e. no equivalence relationships are known if all the elements are different from each other, i.e. there are G = n different equivalence classes. On the contrary, the information content is maximal if all the elements of the set are recognized as belonging to the same class (G = 1). This quantity is called maximal information content and represents the information content needed to characterize all of the n alternatives, i.e. the elements of the considered set ... 

You make unit conversions everyday when you determine how many quarters are needed to make a dollar or how many feet are in a yard. One unit that is often used in calculations in chemistry is the mole. Chapter 11 shows you equivalent relationships among mole, grams, and the number of representative particles (atoms, molecules, formula units, or ions). For example, one mole of a substance contains 6.02 X 10 representative particles. Try the next example to see how this information can be used in a conversion factor to determine the number of atoms in a sample of manganese. 

Considering an orthorhombic crystal system, the equivalency relationships established above (section 2.10.3) also satisfy Eq. 2.1J4 i.e. 

Calculate the weight of sample that will react with 1 mL of the titrant. This figure is called the equivalent relationship or sometimes the equivalent and is the most important part of the calculation. 

The equation in bold type is the equivalent relationship and tells us that for every 1 mL of titrant added, we can expect to react slightly more than 64 mg of citric acid. Note also that the equivalent is derived for a precisely 1 m solution, i.e. f = 1.000. 

The concentration of metal ions in electrolyte preparations is often stated in millimoles per litre or sometimes millimoles per mL, where a millimole is simply one thousandth of a mole. This means that the method of deriving the equivalent relationship needs to be altered slightly from that previously stated. Using calcium ions as an example ... 

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