Subscripts ambiguity

For simplicity of writing we will omit the subscript h if this does not cause an ambiguity. The equation we must solve is of the form... 

Our primary interest is in a chemical system rather than its surroundings. Eor convenience, the subscript denoting the system usually is omitted in thermodynamic equations. Thus, an energy term without a subscript always refers to a system. To avoid ambiguity, we never omit the subscript indicating the surroundings. 

It may be shown that when the polymer concentration is large, the perturbation tends to be less. In particular, in a bulk polymer containing no diluent a = l for the molecules of the polymer. Thus the distortion of the molecular configuration by intramolecular interactions is a problem which is of concern primarily in dilute solutions. In the treatment of rubber elasticity—predominantly a bulk polymer problem—given in the following chapter, therefore, the subscripts may be omitted without ambiguity. 

The definition of P(n m) is identical with that of (n m) except that condition (b) is to be omitted. (In writing these equations, and in many of those that follow, we can omit the subscript 2 since no ambiguity will arise in practice.) Examples of products occurring in ( (n m) and P(n m) are shown in Fig. 5. 

A serious ambiguity arises for compounds such as the active tartaric acids. If the amino-acid convention is used, (+)-tartaric acid falls in the d series by the sugar convention, it has the l configuration. One way out of this dilemma is to use the subscripts, v and g to denote the amino-acid or carbohydrate conventions, respectively. Then the absolute configuration of (+)-tartaric acid can be designated as either Ds-(+)-tartaric acid of lb-(+)-tartaric acid. 

The subscript to q is omitted when no ambiguity can arise. The quantity e2Qq/h is the nuclear quadrupole coupling constant, usually quoted in MHz. 

When there is no ambiguity the slash is omitted in the subscripts, as in ALm, MmL, or M f (between substrate R and medium m through intervening layers). For a finite number of layers, refer to material layers A , A2,..., Ay of thickness a , a2,..., ay on half-space L Bi, B2,..., Bj of thickness bi, b2,..., by on half-space R. Indices j or ] count away from the central medium m. For repeating layers and multilayers, a single layer of material B, thickness b on half-space R, is successively followed by N pairs of material B, thickness b, and B, thickness b. ... 

The Elementary Partial Derivatives.—We can set up a number of familiar partial derivatives and thermodynamic formulas, from the information which we already have. We have five variables, of which any two are independent, the rest dependent. We can then set up the partial derivative of any dependent variable with respect to any independent variable, keeping the other independent variable constant. A notation is necessary showing in each case what are the two independent variables. This is a need not ordinarily appreciated in mathematical treatments of partial differentiation, for there the independent variables are usually determined in advance and described in words, so that there is no ambiguity about them. Thus, a notation, peculiar to thermodynamics, has been adopted. In any partial derivative, it is obvious that the quantity being differentiated is one of the dependent variables, and the quantity with respect to which it is differentiated is one of the independent variables. It is only necessary to specify the other independent variable, the one which is held constant in the differentiation, and the convention is to indicate this by a subscript. Thus (dS/dT)P, which is ordinarily read as the partial of S with respect to T at constant P, is the derivative of S in which pressure and temperature are independent variables. This derivative would mean an entirely different thing from the derivative of S with respect to T at constant V, for instance. 

To avoid ambiguity or poor appearance, add an apostrophe and a lowercase s to form the plurals of lowercase abbreviations, single-capital-letter abbreviations, abbreviations ending in a subscript or superscript, and abbreviations ending in an italic letter. 

It is sometimes convenient to divide all extensive quantities by amount of substance, so that all quantities become intensive the subscript m may then be omitted if this convention is stated and there is no risk of ambiguity. (See also the symbols recommended for partial molar quantities in section 2.11, p.49, and Examples of the use of these symbols , p.51.)... 

When there is no risk of ambiguity the subscript 0 denoting vacuum is often omitted. 

In the above examples, we started from a precisely known expression such as y = Oq + a,x, added Gaussian noise, and then extracted from the data the estimates Oo.caic and i,Caic- This allowed us to judge how closely we can reconstruct the true values of a, and av In practice, however, the experimenter has no such luxury, since the true parameter values are generally not known, so that we will only have parameter estimates. In practice, then, there is little need to distinguish between the true parameters and their estimates, so that from now on the subscripts calc will be deleted whenever that can be done without introducing ambiguity. 

Some of the molecular frame R-axis quantum numbers (Sr,Ir,sr and Jr) correspond to quantum numbers C = Ir and S = Sr that have been in widespread use in the literature. There is, however, an ambiguity concerning onto which molecule frame rotation axis (J+,N+,or R), the projection quantum number is defined. The subscript- notation eliminates this ambiguity and we therefore recommend its use. Also, the lower-case forms of Ir and sr conform to the convention that single-electron quantum numbers appear as lower-case letters. 

Subscripts will be used for components whenever any ambiguity might arise. In Eq. (50) v is the center of mass velocity and in Eq. (51) X is a diffusion constant and a chemical potential jXi and are the chemical potentials and mass, per molecule, of component i. Except for the generahzation to the critical region, the presentation of the equations of motion follows that given by Landau and Lifshitz. Accompanying the flux i is an entropy production s given by... 

For systems as potentially complex as multiple emulsions, it is very important that a clear and consistent system of nomenclature be employed. For a w/o/w system, for example, in which the final continuous phase is aqueous, the primary emulsion will be a w/o emulsion, which is then emulsified into the final aqueous phase. The surfactant or emulsifier system used to prepare the primary emulsion is denoted as the primary surfactant. To avoid further ambiguity as to components or their locations in the system, subscripts may be used. For example, in a w/o/w system the aqueous phase of the primary emulsion would be denoted as wi and the primary emulsion as Wi/o. After the primary emulsion is further dispersed in the second aqueous phase W2, the complete system may be denoted W1/0/W2. In the case of an o/w/o multiple emulsion in which the oil phases are different, the notation becomes Oi/w/02. Additional refinements to fit even more complex systems, including the order of multiple emulsions, have been suggested. 

Regarding the use of upper and lower case subscripts it must be borne in mind that in some instances it is not without ambiguity, as in the case of displacements. Strictly speaking, the distinction is mandatory only for the coordinates themselves in combination with the comma notation for derivatives. In all other contexts upper case indices serve as a useful reminder that a material frame representation is used. This reminder, of course, is superfluous if only Unear effects are considered. 

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