Numerical aspects

As we have seen in this chapter steroids have a number of functions in human physiology Cholesterol is a component part of cell mem branes and is found in large amounts in the brain Derivatives of cholic acid assist the digestion of fats in the small intestine Cortisone and its derivatives are involved in maintaining the electrolyte balance in body fluids The sex hormones responsible for mascu line and feminine characteristics as well as numerous aspects of pregnancy from conception to birth are steroids... 

The numerical aspect, and the lack of explicit velocities, in the Verlet algorithm can be remedied by the leap-frog algorithm. Performing expansions analogous to eqs. (16.28) and (16.29) with half a time step followed by subtraction gives... 

Current estimates suggest that there are around 30,000 alcohol-related deaths a year in the UK. The NHS (National Health Service) spends over 164m a year treating alcohol-related conditions, and one in four male hospital beds is occupied by someone with an alcohol-related illness. Alcohol adversely aflfects numerous aspects of health, even in those who are only moderate drinkers. However, the effects of high volumes over long periods are certainly the most serious and life-threatening. Alcohol passes through the stomach and small intestine and is then absorbed into the blood stream from where it is metabolised by the liver (the first-pass effect Chapters 3 and 9). 

Much of Chapter 20 was devoted to the description of mechanisms of reactions of coordination compounds with emphasis on substitution reactions. Because there are numerous aspects of oxad reactions that are different from those involving substitution, we will address some of the mechanistic aspects of oxad reactions briefly in this section. 

It is clear that pulse sequences may not only be designed by analytical means, they may also be designed numerically (see, e.g., reviews on numerical aspects of solid-state NMR in [54, 65, 66]) using standard nonlinear optimization to well-defined analytical expressions [67, 68], by optimizing pulse sequences directly on the spectrometer [69], or by optimal control procedures [70-72] to name but a few of the possibilities. We will in this review restrict ourselves to optimal control design procedures that recently in analytical and numerical form have formed a new basis for efficient NMR experiment design. 

DPMs offer a viable tool to study the macroscopic behavior of assemblies of particles and originate from MD methods. Initiated in the 1950s by Alder and Wainwright (1957), MD is by now a well-developed method with thousands of papers published in the open literature on just the technical and numerical aspects. A thorough discussion of MD techniques can be found in the book by Allen and Tildesley (1990), where the details of both numerical algorithms and computational tricks are presented. Also, Frenkel and Smit (1996) provide a comprehensive introduction to the recipes of classical MD with emphasis on the physics underlying these methods. Nearly all techniques developed for MD can be directly applied to discrete particles models, except the formulation of particle-particle interactions. Based on the mechanism of particle-particle interaction, a granular system may be modeled either as hard-spheres or as soft-spheres. ... 

Virtual screening applications based on superposition or docking usually contain difficult-to-solve optimization problems with a mixed combinatorial and numerical flavor. The combinatorial aspect results from discrete models of conformational flexibility and molecular interactions. The numerical aspect results from describing the relative orientation of two objects, either two superimposed molecules or a ligand with respect to a protein in docking calculations. Problems of this kind are in most cases hard to solve optimally with reasonable compute resources. Sometimes, the combinatorial and the numerical part of such a problem can be separated and independently solved. For example, several virtual screening tools enumerate the conformational space of a molecule in order to address a major combinatorial part of the problem independently (see for example [199]). Alternatively, heuristic search techniques are used to tackle the problem as a whole. Some of them will be covered in this section. 

Duosol Process. The Duosol process developed by the Max B. Miller Co. (28) is an outstanding example of commercial adoption of a double solvent extraction process. Patents (27) for this process date from May 1933 and cover numerous aspects of the problem including a variety of paraffinic solvents (ethane, propane, butane, petroleum ether) and naphthenic solvents (wood tar acids, cresols, creosote, and phenol). Present commercial application utilizes propane and Selecto (a mixture of phenol and cresylic acid, normally ranging in composition from 20 to 80% phenol). 

It is our objective in this chapter to outline the basic concepts that are behind sedimentation and diffusion. As we see in this chapter, gravitational and centrifugal sedimentation are frequently used for particle-size analysis as well as for obtaining measures of solvation and shapes of particles. Diffusion plays a much more prevalent role in numerous aspects of colloid science and is also used in particle-size analysis, as we see in Chapter 5 when we discuss dynamic light scattering. The equilibrium between centrifugation and diffusion is particularly important in analytical and preparative ultracentrifuges. 

When given the active substance characteristics determined during development acceptance criteria for the validation studies can be established. These criteria will demonstrate the consistency of the dried material processed within a proven acceptable range in the development phase and adequacy of the scale-up to manufacturing. To be comprehensive in this presentation, numerous aspects, although not necessarily applicable to all products, are presented as illustrations in the following sections. 

The aim of this chapter is to assist the reader to generate an intuitive understanding of the mechanism of asymmetric phase-transfer catalysis, together with a practical guide for the design of such processes. More detailed studies related to the physical and numerical aspects of phase-transfer catalysis may be consulted elsewhere [3]. 

---