Extended equation

Shannon s method, expands a Boolean function of n variables in minterms consisting of all combinations of occurrences and non-occurrences of the events of interest. Consider a function of n Boolean variables XJ which may be expanded about X, as shown in Equation 2.2-3 where f(l, Xj,..., XJ where 1 replaces X,. This says that a function of Boolean variables equals the function with a variable set to I plus the product of NOT the variable limes the function with the variable set to 0. By extending Equation 2.2-3, a Boolean function may be expanded about all of its... 

Equation (6.60) extends equation (6.59) to calculate [Pg.267]

Eqn. 2.66 is also of interest for the calculation of activity coefficients in other solution media, where as a consequence of considerably different densities (g0) and dielectric constants (a) another value for A is calculated, and also the extended equation log f = - A z+z P/l + I has been applied23. 

Now the alert reader will object that this is circular. For we must define the semantics of such extended equations. This can be done directly but it is probably easiest to do by saying that, for example, an equation FCu, ..., ) =... 

How are we going to perform that calculation Well to start with, we need something to base it on, and the consensus is that the calculation will be based on the errors, since in truth, equation 69-3 is not exactly correct because C will in general NOT be exactly equal bA. Therefore we extend equation 69-3 ... 

Siveral approaches are available in the case of mixed electrolyte solutions. The Guntelberg equation can be used at very high dilutions to avoid the ambiguity in the meaning of aD, the distance of closest approach, when several electrolytes are present. This equation is empirical and has fewer terms than the Debye-Huckel extended equation. I found it to yield poor agreement with experimental results even at m = 0.01 for NaCl at 25°C (y+ caic = 0.8985 and y+ exp = 0.9024). For the Davies equation for m = 0.20 one obtains y+ calc = 0.752 andy+exo = 0.735 also for NaCl at 25°C. 

A second assessment of the enthalpy of formation of 16 is made by extending equation 13 and its reaction enthalpy to include the /V-methylated anilines 16 and 15. 

If there is a nonlinear dependence of kapp on [Lq] then the reaction cannot be a simple bimolecular process. However, the extended equation described inO Section 2.6.1 will give rise to a hyperbolic curve as shown in Figure 10-5b, where Ki is given by k i/ki, and kapp is described by ... 

Hazen and Finger (1979) extended equation 1.110 to mean polyhedral compressibility (mean compressibility of a given coordination polyhedron within a crystal structure), suggesting that it is related to the charge of ions in the polyhedron through an ionicity factor, analogous to what we have already seen for thermal expansion—i.e.. 

In order to extend equation (1) to temperatures other than the critical, the temperature dependence of the constants a, b, c, d and e must be established. For this Joffe assumed, provisionally, that b is independent of the temperatures other than the critical and wrote the equation (1) in the following form ... 

As is illustrated by Fig. 5.3 b, with this extended equation, the fit of the experimental data can be improved significantly. The best fit equation obtained from the experimental data set is ... 

Declare (5.1) valid for all n from — oo to + oo but only consider solutions of this extended equation whose initial values p (0) are zero for n < 0. It is then clear that the p (t) for n< 0 remain zero at all r>0, because the condition r(0) = 0 guarantees that no transitions occur from the state 0 into the state —1. Consequently the general equation (5.1) reduces to (5.2) at n — 0. Thus the boundary merely restricts the possible choice of initial condition, but no longer occurs in the equation itself - which greatly facilitates its solution. 

Extend Equations 4.1 through 4.2 with the term for a chemical reaction, Equation 4.6. Is the additional term consistent with the general structure of Equations 4.1 through 4.2 Where is the A for this term (Hint See the definition for A in Equation 4.4 and for the stoichiometric coefficient y in the nearby text.)... 

Many attempts have been made to extend Equation 3 to higher concentrations. Of these, we have chosen the equation of Guth,... 

In the limit of dipole scattering, the zero-angle scattering Jones matrix for a system of anisotropic particles is simply found by extending equation (4.66) to the case were the polarizability is a tensor. In this case,... 

These fractions obey the set of extended equations (3.559) substituted for Eq. (3.290) ... 

As cells start to adhere to the surface, the uniformity in its characteristics is disturbed. The altered spots on the surface, which consist of adhering cells, present new characteristics concerning the surface potential and Hamaker s constant, and consequently, the rate of cellular deposition is not uniform over the surface. The present model is based on the simplified assumption that the overall rate of deposition is the arithmetic sum of two contributions deposition on the bare surface and deposition on the altered surface. Each of them has its own time constant and depends also on the fraction of the area which is already covered, X. Therefore, by extending equation (19), the overall rate of deposition is given by ... 

The admittance is the inverse or reciprocal of the -+ impedance (Z). The admittance and impedance are related by the extended equation ... 

The extended Van Arkel equation contains the same volume factor as that of Hildebrand-Scatchard. Van Arkel s extended equation is then as follows ... 

Figure 1. Extrapolation of emf (V) in dioxane-water mixtures computed by the Debye-Huckel extended equation. The percentages are those of weight percent of dioxane (5).

For the electrolytes of symmetrical valence type, the extended equation for In y is expressed as... 

Thermodynamic methods were developed based on the extended equation by Ugelstad et al. [169], and have been further dealt with by various researchers [170-181]. Maxwell et al. [170] worked on the partitioning of two monomers between the polymer particle, monomer droplet and aqueous phases in an... 

A graphical solution for all maxima and minima has been determined by Pierce and Maron [93] who, together with Elder [94,95] extended equations (10.21) and (10.22) beyond the Rayleigh-Gans region, (ffj - 1 0) to 1.00 < /M < 1.55, deriving the following formulae ... 

We now define the form of the projection matrix P. Extending equation (4.73) to include perturbations arising from physical processes we get... 

If the aquatic system contains weak bases (or acids) other than CO2 and water, we can extend equation 1 to include these other proton acceptors. For example, in the presence of boric acid-borate we can extend our reference le el to H3BO3 (in addition to H2CO3 and H2O) and write... 

The additivity and separability postulates have two interesting corollaries. The additivity postulate permits us to extend equation 1 to cases where many substituents exist on R. As an example, for the substituents X and Y,... 

A more general relationship for analyzing solvent effects for thermodynamic quantities was given by Koppel and Palm [51]. They extended equation (4.9.9) to include non-specific solvent effects related to solvent permittivity and polarizability. Their equation can be written as... 

In order to connect the above expressions to Onsager s theory, it is necessary to extend equations such as (6.3.1) to three dimensions. This equation shows that a force in one direction leads to velocity changes in the other two spatial directions. A general treatment of fluid flow requires that one identify the components of the pressure tensor associated with the force which leads to fluid flow. The force vector F has, in general, a component in each of the three Cartesian directions. The component in the x-direction, F, gives rise to three pressure components, one in the same direction, P x, and two shear components, Pxy and Pxz- Six more pressure components are obtained from the force components in the y- and z-directions, Fy and F. As a result, there is a second-rank pressure tensor Py with nine components. Analysis on the basis of classical dynamics leads to the conclusion that the off-diagonal elements, Py and Py,-, are equal. As a result, six distinct elements of this tensor must be determined to define it. The second important step is to write an equation of continuity in terms of momentum. Defining the momentum density If as... 

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