Extended formulation, definition

However, serious difficulties appeared later when efforts were made to attack more general problems not necessarily of the nearly-linear character. In terms of the van der Pol equation this occurs when the parameter is not small. Here the progress was far more difficult and the results less definite moreover there appeared two distinct theories, one of which was formulated by physicists along the lines of the theory of shocks in mechanics, and the other which was analytical and involved the use of the asymptotic expansions (Part IV of this chapter). The latter, however, turned out to be too complicated for practical purposes, and has not been extended sufficiently to be of general usefulness. 

Almost all problems that require knowledge of free energies are naturally formulated or can be framed in terms of (1.15) or (1.16). Systems 0 and 1 may differ in several ways. For example, they may be characterized by different values of a macroscopic parameter, such as the temperature. Alternatively, they may be defined by two different Hamiltonians, 3%o and 3%, as is the case in studies of free energy changes upon point mutation of one or several amino acids in a protein. Finally, the definitions of 0 and 1 can be naturally extended to describe two different, well-defined macroscopic states of the same system. Then, Q0 is defined as ... 

While the Dove formulation has undergone relatively minor changes since its launch in the U.S. in 1955, the marketing concept has seen major changes. Today s Dove is marketed on a platform of extreme mildness to skin as evidenced by its neutral pH, and has been extended into different skin cleansing product forms (e.g. Dove Body Wash) as well as other personal care products (e g. Dove Shampoo, Dove Deodorant) [3], However, the original problem definition had a very different focus. 

In the next section we shall recall the definitions of the chemical concepts relevant to this paper in the framework of DFT. In Section 3 we briefly review Strutinsky s averaging procedure and its formulation in the extended Kohn-Sham (EKS) scheme. The following section is devoted to the presentation and discussion of our results for the residual, shell-structure part of the ionization potential, electron affinity, electronegativity, and chemical hardness for the series of atoms from B to Ca. The last section will present some conclusions. 

Bioavailability studies quantify rate and extent of absorption. They compare the efficiency of the disposition of several drug formulations, e.g. immediate-release vs. extended-release or capsule vs. tablet or tablet A vs. tablet B etc., or they compare the disposition of different routes of administration, e.g. oral vs. subcutaneous or oral vs. intravenous. According to the definition, a comparison to the intravenous bolus injection yields the absolute bioavailability. 

In some points of the previous analysis we have used formulations based on the Hartree-Fock (HF) expression of the quantum problem, mainly for simplicity of exposition. As a matter of fact, there are no formal reasons to limit continuum solvent approach to the HF level. Actually, PCM solvation procedures have been extended to MCSCF (Aguilar et al., 1993b), Cl (Persico and Tomasi, 1984), MBPT (Olivares del Valle et al., 1991, 1993), CASSCF, MR-SDCI (Aguilar et al., 1993b), DFT (Fortunelli and Tomasi, 1994) levels of the quantum description. The other continuum solvation methods have, at least in principle, the same flexibility in the definition of the quantum theory level to be used in computations. 

Another issue is reproducibility. The formulation may work perfectly in an in vitro test system, but the dosage form requires aerosolization, and lung deposition is a function of the characteristics of the aerosol (dose, mass concentration, droplet/particle size, etc.) and the nature of the inspiratory maneuver, a factor that the patient has control over. These factors can influence performance to a far greater extent than can be built into a particle, and thus the term controlled does not seem a defensible objective for pulmonary delivery. The vagaries of the deposition profile and of the amount that will deposit also imply that sustaining a certain drug concentration is a difficult proposition, but the loosest definition extended release, seems an acceptable goal within the boundaries set by the clearance mechanisms. 

To describe polydispersed multiphase systems the Boltzmann equation can be extended by including the dependency of the internal property coordinates such as the particle size and shape in the definition of the distribution function. In this way a statistical balance formulation can be obtained by means of a distribution function on the form p, r,v, c,t)d dr dv dc, defined as the probable number of particles with internal properties in the range about with a velocity range in property space dv about v, located in the spatial range dr about the position r, with a velocity range dc about c, at time t. The particular Boltzman type of equation is given by ... 

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