Indicator function definitions

In view of the term containing activity coefficients, the acidity function depends on the ionic type of the indicator. The definition of H0 is combined with the assumption that the ratio Yb/Ybh+ is constant for all indicators of the same charge type (in the present case the base is electroneutral hence the index 0 in //0). Thus, the acidity function does not depend on each individual indicator but on the series of indicators. 

For simplicity of notation, we will here discuss the circumstance that the distinguished molecule is present at the lowest concentration. The occupants of the inner shell will be of one type only, solvent of type denoted by w w = H20 for example. Then our discussion can be more economical, though the ideas do have broader relevance. For a distinguished molecule of type a, we will encode the definition of the inner shell by an indicator function ba (k) which is one when the /cth solvent molecule occupies the defined inner shell, and zero otherwise. Then the PDT formula can be recast as... 

Hirt and Nichols [12] demonstrated the volume of fluid (VOF) method and started a new trend in multiphase flow simulation. It relies on the definition of an indicator function y. This function allows us to know whether one fluid or another occupies the cell, or a mix of both. In the conventional volume of fluid method [12], the transport equation for an indicator function y, representing the volume fraction of one phase, is solved simultaneously with the continuity and momentum equations as follows ... 

By definition the phase indicator function Xj. cannot be decomposed into a mean component and a fluctuation component, thus the only way this relation can be fulfilled is by defining = 0. 

Fio. 10. A molecule undergoing an intramolecular addition (cyclization) reaction, indicating the definitions of functional groups A, B, and C. 

Function tests of thyroid activity have little or no value in establishing the diagnosis of lymphocytic thyroiditis because of the wide spectrum of thyroid activity that may be seen in this condition. Thyroid function is likely to be normal, low normal, or subnormal, but on occasions increased function may be encountered. The diagnosis of this condition rests on a clinical suspicion of its existence, the presence of microsomal or thyroglobulin antibodies at high dilution with purposefully insensitive techniques (as discussed in Section 4.3). Where indicated, the definitive diagnosis is made by biopsy. 

Although this member function definition resembles an ordinary C procedure, there are some important differences that should be discussed. The definition starts with the notation Complex which is a way of reopening the context of the class declaration. The pair of colons, which is called the scope resolution operator, is placed after the name of a class to indicate a definition of something in the class. After the constructor header is a pair of initialization expressions. The first initialization expression is realpart(r), which states that the class member named realpart should have its initial value taken from the parenthesized expression, r. The second initialization expression sets the imagpart member to zero. The body of the constructor, like the body of all C procedures, is delimited by a pair of curly braces, but in this particular example there is nothing to do in the constructor body because all the work has been done in the initialization section of the constructor. 

In this section we first introduce the definition of the f-insensitive loss function, then show that the same quadratic optimization technique that was used in Section 2.3 for constructing approximations to indicator functions provides an approximation to real-valued functions, involving the linear case and nonlinear case. 

In order to mathematically derive the expression for failure probability, consider the definition of Pf as per Eq. 11. Note that the domain of integration is specifically over the failiue region, defined by G < 0. Consider a mathematical indicator function as follows ... 

For indicator functions defined according to (18), Ha and are simply related hy hs = I-Ha and definition of either Ha or hs would suffice. Later, however, we wUl need regions A and B which are not complementary to each other and therefore we write aU expressions using hA and ho such that they will remain valid later. 

The eode in this example illustrates several language features such as die do. end loop, the if... then. .. end loop and a function definition. The code also illustrate some features of the lexical scope of Lua variables wifli the printO statement on line 18. It is noted that variables x, y and z are first introduced within the body of the tst() function definition. However, y is declared as a local variable and is thus known only within the scope of the code block in which it is defined, which in this case is within lines 6 through 13 of the code. The attempt to print the value of y on line 18 thus results in the printed value of nil indicating an unknown value. This illustrates the fact that in Lua, variables are known globally unless restricted to a code block by the local keyword, i.e. die default is that all variables have global scope. 

For a simplified case, one can obtain the rate of CL emission, =ft GI /e, where /is a function containing correction parameters of the CL detection system and that takes into account the fact that not all photons generated in the material are emitted due to optical absorption and internal reflection losses q is the radiative recombination efficiency (or internal quantum efficiency) /(, is the electron-beam current and is the electronic charge. This equation indicates that the rate of CL emission is proportional to q, and from the definition of the latter we conclude that in the observed CL intensity one cannot distii pish between radiative and nonradiative processes in a quantitative manner. One should also note that q depends on various factors, such as temperature, the presence of defects, and the... 

Instead of the definition in Eq. (7-82), the selectivity is often written as log k,). Another way to consider a selectivity-reactivity relationship is to compare the relative effects of a series of substituents on a pair of reactions. This is what is done when Hammett plots are made for a pair of reactions and their p values are compared. The slope of an LEER is a function of the sensitivity of the process being correlated to structural or solvent changes. Thus, in a family of closely related LFERs, the one with the steepest slope is the most selective, and the one with the smallest slope is the least selective.Moreover, the intercept (or some arbitrarily selected abscissa value, usually log fco for fhe reference substituent) should be a measure of reactivity in each reaction series. Thus, a correlation should exist between the slopes (selectivity) and intercepts (reactivity) of a family of related LFERs. It has been suggested that the slopes and intercepts should be linearly related, but the conditions required for linearity are seldom met, and it is instead common to find only a rough correlation, indicative of normal selectivity-reactivity behavior. The Br nsted slopes, p, for the halogenation of a series of carbonyl compounds catalyzed by carboxylate ions show a smooth but nonlinear correlation with log... 

The antisymmetrization of the trial function has a definite effect on the generalized density matrices P(x x 2.. . x x1x2. . . xv) defined by Eq. II.9 since, except for the first-order matrix, they will now all be antisymmetric in each set of the indices. For p = 2, we have in particular ... 

The state w, f>s is an eigenstate of N with eigenvalue N, and N is called the total population operator. Because the vector , Os is a function of the time, it is necessary to specify the time at which the creation or annihilation operators are applied, and in some discussions it may be advisable to indicate the time explicitly in the symbol for the operator. For our present discussion it will be sufficient to keep this time dependence in mind. In an expression such as Eq. (8-109), all the creation operators are applied at the same time, and since they all commute, this presents no logical problem. The order of the operators in the definition Eq. (8-107) is important however the opposite order produces a different operator ... 

This is an indication of the collective nature of the effect. Although collisions between hard spheres are instantaneous the model itself is not binary. Very careful analysis of the free-path distribution has been undertaken in an excellent old work [74], It showed quite definite although small deviations from Poissonian statistics not only in solids, but also in a liquid hard-sphere system. The mean free-path X is used as a scaling length to make a dimensionless free-path distribution, Xp, as a function of a free-path length r/X. In the zero-density limit this is an ideal exponential function (Ap)o- In a one-dimensional system this is an exact result, i.e., Xp/(Xp)0 = 1 at any density. In two dimensions the dense-fluid scaled free-path distributions agree quite well with each other, but not so well with the zero-density scaled distribution, which is represented by a horizontal line (Fig. 1.21(a)). The maximum deviation is about... 

The size-dependence of the intensity of single shake-up lines is dictated by the squares of the coupling amplitudes between the Ih and 2h-lp manifolds, which by definition (22) scale like bielectron integrals. Upon a development based on Bloch functions ((t>n(k)), a LCAO expansion over atomic primitives (y) and lattice summations over cell indices (p), these, in the limit of a stereoregular polymer chain consisting of a large number (Nq) of cells of length ao, take the form (31) ... 

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