Mathematical Basis

The integrated intensity I of reflection hkl for phase a in a multi-phase mixture measured on a flat-plate sample of infinite thickness can be calculated from  

For Bragg-Brentano geometry, the path lengths of the incident and diffracted beams are equal for all values of 26 and, consequently, the effect of increased sample absorption is a decrease in the overall intensity of the pattern. 

However, for some instrument geometries, there is an angular dependence on observed intensity due to sample absorption, which must be allowed for in the calculation of I. This is the ease where a flat plate sample is examined using a fixed angle of incidence or where a capillary sample is used. 

A fixed angle of incidenee arises when a flat plate sample is examined in an instrument such as the Inel powder diffractometer incorporating the CPS120 position sensitive deteetor (detector produced by Inel, Z.A.-C.D. 405, 45410 Artenay, France. http //www.inel.fr/en/accueil/). In this ease, the absorption term above takes the form  

Where a is the angle between the incident beam and the sample surfaee and jS is the angle between the diffracted beam and the sample surfaee. In this geometry, a is set to a fixed value and jS varies with diffraetion angle aeeording to p = 26-a.. In Bragg-Brentano geometry a = p = 6 and Equation (3) reduces to the expression in Equation (2). 

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In this chapter, recent advances in the theory of conical intersections for molecules with an odd number of electrons are reviewed. Section II presents the mathematical basis for these developments, which exploits a degenerate perturbation theory previously used to describe conical intersections in nonrelativistic systems [11,12] and Mead s analysis of the noncrossing rule in molecules with an odd number of electrons [2], Section III presents numerical illustrations of the ideas developed in Section n. Section IV summarizes and discusses directions for future work. 

MATHEMATICAL BASIS AND SOFTWARE FOR SINGLE GRAIN MILLIPROBE XRF TRACE ELEMENT ANALYSIS (XRF-MP/SG) OF ACCESSORY MINERALS... 

For erosive wear. Rockwell or Brinell hardness is likely to show an inverse relation with carbon and low alloy steels. If they contain over about 0.55 percent carbon, they can be hardened to a high level. However, at the same or even at lower hardness, certain martensitic cast irons (HC 250 and Ni-Hard) can out perform carbon and low alloy steel considerably. For simplification, each of these alloys can be considered a mixture of hard carbide and hardened steel. The usual hardness tests tend to reflect chiefly the steel portion, indicating perhaps from 500 to 650 BHN. Even the Rockwell diamond cone indenter is too large to measure the hardness of the carbides a sharp diamond point with a light load must be used. The Vickers diamond pyramid indenter provides this, giving values around 1,100 for the iron carbide in Ni-Hard and 1,700 for the chromium carbide in HC 250. (These numbers have the same mathematical basis as the more common Brinell hardness numbers.) The microscopically revealed differences in carbide hardness accounts for the superior erosion resistance of these cast irons versus the hardened steels. 

To discover and analyze the mathematical basis for the generation of complexity, one must identify simple mathematical systems that capture the essence of the process. Cellular automata are a candidate class of such systems. Cellular automata promise to provide mathematical models for a wide variety of complex phenomena, from turbulence in fluids to patterns in biological growth. 

This book has been written in an attempt to provide students with the mathematical basis of chemistry and physics. Many of the subjects chosen are those that I wish that I had known when I was a student It was just at that time that the no-mans-land between these two domains - chemistry and physics - was established by the Harvard School , certainly attributable to E. Bright Wilson, Jr., J. H. van Vleck and the others of that epoch. I was most honored to have been a product, at least indirectly, of that group as a graduate student of J. C. Decius. Later, in my post-doc years. I profited from the Harvard-MIT seminars. During this experience I listened to, and tried to understand, the presentations by those most prestigious persons, who played a very important role in my development in chemistry and physics. The essential books at that time were most certainly the many publications by John C. Slater and the Bible on mathematical methods, by Margeneau and Murphy. They were my inspirations. 

Some of the mathematical basis for the equations is due to the geneticist Moto Kimura, who is considered a resolute proponent of the neutral theory of evolution , which states that statistical, chance fluctuations are more important in the formation of new species than is Darwinian natural selection. Evolution via such chance fluctuations is referred to as genetic drift . Dyson considers that both forms of evolution are important (Dyson, 1999). 

Molecular mechanics is a useful and reliable computational method for structure, energy, and other molecular properties. The mathematical basis for molecular models in MM3 has been described, along with the limitations of the method. One of the major difficulties associated with molecular mechanics, in general, and MM3 in particular is the lack of accurately parameterized diverse functional groups. This lack of diverse functional groups has severely limited the use of MM3 in pharmaceutical applications. 

Let us begin by describing the logic and technique behind the most prominent taxometric analytic procedure, MAXCOV-HITMAX (Meehl Yonce, 1996), to which we refer as MAXCOV. MAXCOV is by far the oldest taxometric procedure. For example, the mathematical basis for MAXCOV was established in 1965 (Meehl), and the original version of the technique was described in 1973 (Meehl). Learning the principles employed by this classic method can facilitate understanding of the newer procedures, which are described later in the chapter. 

Photometric measurements provide the basis for the majority of quantitative methods in biochemistry and are related to the amount of radiation absorbed rather than the nature of such radiation. This relationship is expressed in two experimental laws, which provide the mathematical basis for such quantitative methods. 

In the previous paragraphs a brief account has been given of the fundamental aspects of the crystallographic description of the structures and structure types of solid phases. A number of symbols and names have been defined and their application to intermetallic compounds exemplified. It must, however, be underlined that both for historical reasons and for the need to improve classification and interpretation of the structural characteristics of intermetallic phases, other symbols and nomenclature criteria have been invented. Some of them have a mathematical basis, others are more colloquial. A selection of these criteria will be given in the following. 

The mathematical basis of the distributional approach can be understood by reference to the equations used for multiexponential decay. Integrals replace finite sums of terms, while the discrete parameters inside the sums are replaced by continuous functions of these parameters. The equation... 

The mathematical basis for the exponential series method is Eq. (5.3), the use of which has recently been criticized by Phillips and Lyke.(19) Based on their analysis of the one-sided Laplace transform of model excited-state distribution functions, it is concluded that a small, finite series of decay constants cannot be used to represent a continuous distribution. Livesey and Brouchon(20) described a method of analysis using pulse fluorometry which determines a distribution using a maximum entropy method. Similarly to Phillips and Lyke, they viewed the determination of the distribution function as a problem related to the inversion of the Laplace transform of the distribution function convoluted with the excitation pulse. Since Laplace transform inversion is very sensitive to errors in experimental data,(21) physically and nonphysically realistic distributions can result from the same data. The latter technique provides for the exclusion of nonrealistic trial solutions and the determination of a physically realistic solution. These authors noted that this technique should be easily extendable to data from phase-modulation fluorometry. 

The mathematical basis of classic thermodynamics was developed by J. Willard Gibbs in his essay [1], On the Equilibrium of Heterogeneous Substances, which builds on the earlier work of Kelvin, Clausius, and Helmholtz, among others. In particular, he derived the phase mle, which describes the conditions of equilibrium for multiphase, multicomponent systems, which are so important to the geologist and to the materials scientist. In this chapter, we will present a derivation of the phase rule and apply the result to several examples. 

Studies of dmg absorption, distribution and elimination comprise what is referred to as pharmacokinetics. By contrast, the concentration of a pharmaceutical compound at the site(s) of action in relation to the magnitude of its effect(s) is referred to as pharmacodynamics. Both pharmacokinetics and pharmacodynamics have their roots in physiology, chemical kinetics, biochemistry, and pharmacology. They seek to provide a mathematical basis of the absorption, distribution, metabolisms, and... 

The mathematical basis of the Mie theory is the subject of this chapter. Expressions for absorption and scattering cross sections and angle-dependent scattering functions are derived reference is then made to the computer program in Appendix A, which provides for numerical calculations of these quantities. This is the point of departure for a host of applications in several fields of applied science, which are covered in more detail in Part 3. The mathematics, divorced from physical phenomena, can be somewhat boring. For this reason, a few illustrative examples are sprinkled throughout the chapter. These are just appetizers to help maintain the reader s interest a fuller meal will be served in Part 3. 

Another kind of data analysis, which has much broader application than analysis of variance is called regression. This method has the same mathematical basis as analysis of variance, but in most cases the calculations become very long and tedious. Without computers, regression methods would be very little used. Since the computers... 

D. P. Vaughan and M. Dennis, Mathematical basis for the point-are deconvolution method for determining in vivo input functions. 

For more detail and the mathematical basis and treatment of the relationship between the receptor-ligand interaction and dose-response relationship, the reader is recommended to consult one of the texts indicated at the end of this chapter (5-7). 

The assignment of molecules to the appropriate point groups can be done on a purely formal, mathematical basis. Alternatively, most chemists quickly learn to classify molecules into the common point groups by inspection. The following approach is a combination of the two. 

J.W. Cahn and W.C. Carter. Crystal shapes and phase equilibria A common mathematical basis. Metall. Trans., 27A(6) 1431-1440, 1996. 

This section explores the mathematical basis for the statistical treatment of experimental data. Most measurements required for the completion of the experiments can be made in duplicate, triplicate, or even quadruplicate, but it would be impractical and probably a waste of time and materials to make numerous determinations of the same measurement. Rather, when you perform an experimental measurement in the laboratory, you will collect a small sample of data from the population of infinite values for that measurement. To illustrate, imagine that an infinite number of experimental measurements of the pH of a buffer solution are made, and the results are written on slips of paper and placed in a container. It is not feasible to... 

The simple concepts we have just discussed can be given a sound mathematical basis which is useful in the design of new experiments and instrumentation systems. The practice of interfacial electrochemistry involves analysis of the response of a phase boundary to various stimuli. In this book we examine the most frequently chosen perturbations and their experimental implementations. Some understanding of the properties of the phase boundary between an electrode and a solution can help tie together the seemingly endless variety of electrochemical techniques. 

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