Function explicit

Series Expansion When an explicit function cannot be found, the integration can sometimes be carried out by a series expansion. 

Note that this list of dimensionless parameters is by no means unique. A set of variables in which each variable in the new set is a combination of the abovementioned set is also permissible and is in principle completely equivalent to the original set. In fact, an infinite number of sets of the dimensionless parameters exist, each of which could be justified as the "original" set. The normal approach at this point is to find an explicit functional relation among one set of variables... 

This level of simplicity is not the usual case in the systems that are of interest to chemical engineers. The complexity we will encounter will be much higher and will involve more detailed issues on the right-hand side of the equations we work with. Instead of a constant or some explicit function of time, the function will be an explicit function of one or more key characterizing variables of the system and implicit in time. The reason for this is that of cause. Time in and of itself is never a physical or chemical cause—it is simply the independent variable. When we need to deal with the analysis of more complex systems the mechanism that causes the change we are modeling becomes all important. Therefore we look for descriptions that will be dependent on the mechanism of change. In fact, we can learn about the mechanism of... 

For the PPM, corresponding to the free energy of the CVM is the Path Probability Function (hereafter PPF), P t t -t- At), which is an explicit function of time and is defined as the product of three factors Pj, P2 and P3. Each factor is provided in the following in the logarithmic expression. 

The Laplace transformation is based upon the Laplace integral which transforms a differential equation expressed in terms of time to an equation expressed in terms of a complex variable a + jco. The new equation may be manipulated algebraically to solve for the desired quantity as an explicit function of the complex variable. 

But P(u) is not a function of t if the differential operator P is not an explicit function of t, which we shall assume here. Then... 

One may attempt to approximate to such an experimental situation by considering a subsystem with small dimensions in the direction of the flow, so that a single temperature may be sufficiently precise in describing it. In this model one would have to provide a time-dependent hamiltonian operating in such a way as to feed energy into the system at one boundary and to remove energy from the other boundary. We would therefore be obliged to discuss systems with hamiltonians that are explicitly functions of time, and also located on the boundaries of the macrosystem. 

This equation allows us to calculate the partial derivative (0X/dY)z when it may be difficult or impossible to obtain an explicit relationship X — X(Y, Z), but Z = Z(X, Y) can be written explicitly and its derivatives can be evaluated. We will find this to be useful when we consider equations of state like the van der Waals equation that cannot be solved for V as an explicit function of T and p, and we are interested in (dV/dT)por (dV/dp)T. 

The above procedure may be followed using any other equation of state, provided that y can be expressed as an explicit function of shear stress Rs. 

Both equations 3.176 and 3.177 are easy to use as they give R/pu2 as explicit functions of ReMR and n. Equation 3.175 should be used with caution, particularly if the fluid exhibits any elastic properties, bearing in mind the conclusions of HEYWOOD and CHENG(25). 

A TBma,) explicit functions of the available concentrations of the other nutrients. This approach allows for a pronounced interdependence between the fluxes of the different nutrients but it does not ensure that the Redfield ratios are maintained. In the second approach the contents of the nutrients in the biota reservoir are forced to remain close to the Redfield ratios. This method was used by Mackenzie et al. (1993) in their study of the global cycles of C, N, P, and S and their interactions. They were able to demonstrate how a human perturbation in one of these element cycles could influence the cycles of the other elements. 

Volume changes also can be mechanically determined, as in the combustion cycle of a piston engine. If V=V(i) is an explicit function of time. Equations like (2.32) are then variable-separable and are relatively easy to integrate, either alone or simultaneously with other component balances. Note, however, that reaction rates can become dependent on pressure under extreme conditions. See Problem 5.4. Also, the results will not really apply to car engines since mixing of air and fuel is relatively slow, flame propagation is important, and the spatial distribution of the reaction must be considered. The cylinder head is not perfectly mixed. 

Within a given animal species, in addition, the selection of target organs, at least under certain dosing schedules, may also be an explicit function of molecular structure. 

The choice of the objective function is very important, as it dictates not only the values of the parameters but also their statistical properties. We may encounter two broad estimation cases. Explicit estimation refers to situations where the output vector is expressed as an explicit function of the input vector and the parameters. Implicit estimation refers to algebraic models in which output and input vector are related through an implicit function. 

At this point, we come back to our original problem finding a better way for the determination of the kinetic energy. The very clever idea of Kohn and Sham was to realize that if we are not able to accurately determine the kinetic energy through an explicit functional,... 

Since 7 , is not an experimentally measurable quantity, it is useful to insert the solution for Ts (from the Clausius-Clapeyron equation) and solve for W h as an explicit function of RH0 and RHC. VanCampen et al. showed (using sample algebraic approximations and conversion factors) that substituting for Ts in Eq. (35) gives the useful solution... 

In general the SIMPLEX program adjusts variables subject to supplied constraints to minimize a given object function. The object function must be an explicit function of the variables. In the case where paint cure uniformity was optimized, the oven was divided into three zones and the variables were the three oven... 

Wesolowski and Warshel197 introduced a DFT based approach in which all short-range terms in the effective Hamiltonian (Eq. 4.25) were derived entirely from density functional theory and were involved in the construction of the Fock matrix195 196. In this approach, the H croEnv is expressed using explicit functionals of the electron density ... 

To obtain the explicit functional form of the convenience kinetics rate equation, recall the Bi-Bi random-order mechanism already considered in Section III.C.5 and depicted again below ... 

The constraints on m1 and m4 are explicit. The lower limit of m, however, does not depend on the other flow rate ratios, whereas the upper limit of m4 is an explicit function of the flow rate ratios m2 and m3 and of the feed composition respectively [25]. The constraints on m2 and m3 are implicit (see Eq. 4), but they do not depend on m1 and m4. Therefore, they define a unique region of complete separation in the (m2, m3) plane, which is the triangle-shaped region abw in Fig. 4. The boundaries of this region can be calculated explicitly in terms of the adsorption equilibrium parameters and the feed composition as follows [25] ... 

The concentrations of the differently protonated species, as a function of pH, are calculated with the explicit function we developed in Special Case Explicit Calculation for Polyprotic Acids, (p.64). A data matrix Y is constructed as before. Data eqAH2a. m generates the data, it is called by Main eqAH2a. m. 

As an alternative to this procedure, the data used for the generation of equation 3.13 may be expressed to give Re 0 as an explicit function of (R /pu Re 1, which from... 

Thus, we would know that the volume of an ideal gas is a thermodynamic property, even if we had not been aware previously of an explicit function for... 

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