Mathematical description

Note The direction of the polarization vector P may be different from those of E1 and 2. The components xijk are generally complex and the phase of the polarization differs from that of the driving fields. 

Taking into account that the field amplitudes Ei, E2 are vectors and that the second-order susceptibility is a tensor of rank 3 with components Xijk depend- 

The components P, (i = x, y, z) of the induced polarization are determined by the polarization characteristics of the incident wave (i.e., which of the components Ex, Ey, Pz are nonzero), and by the components of the susceptibility tensor, which in turn depend on the symmetries of the nonlinear medium. 

Let us first discuss the linear part of (6.6), which can be written as 

One can always choose a coordinate system (, r, g) in which the tensor becomes diagonal (principal axis transformation). If we align the crystal in such a way that the (, t], -)-axes coincide with the (x, y, z)-axes, (6.7a) simplifies in the principal axes system to  

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Identification of the material properties as an estimation of transfer function (TF) for the black box model. In this case the problem of identification is solving according to the results of the input (IN) and output (OUT) actions. There is a transfer of notion of mathematical description of TF on characterization of the material. This logical substitution gives us an opportunity to formalize testing procedure and describe the material as a set of formulae, which can be used for quantitative and qualitative characterization of the materials. 

At first we tried to explain the phenomenon on the base of the existence of the difference between the saturated vapor pressures above two menisci in dead-end capillary [12]. It results in the evaporation of a liquid from the meniscus of smaller curvature ( classical capillary imbibition) and the condensation of its vapor upon the meniscus of larger curvature originally existed due to capillary condensation. We worked out the mathematical description of both gas-vapor diffusion and evaporation-condensation processes in cone s channel. Solving the system of differential equations for evaporation-condensation processes, we ve derived the formula for the dependence of top s (or inner) liquid column growth on time. But the calculated curves for the kinetics of inner column s length are 1-2 orders of magnitude smaller than the experimental ones [12]. 

When a system is not in equilibrium, the mathematical description of fluctuations about some time-dependent ensemble average can become much more complicated than in the equilibrium case. However, starting with the pioneering work of Einstein on Brownian motion in 1905, considerable progress has been made in understanding time-dependent fluctuation phenomena in fluids. Modem treatments of this topic may be found in the texts by Keizer [21] and by van Kampen [22]. Nevertheless, the non-equilibrium theory is not yet at the same level of rigour or development as the equilibrium theory. Here we will discuss the theory of Brownian motion since it illustrates a number of important issues that appear in more general theories. 

The mathematical description of the echo intensity as a fiinction of T2 and for a repeated spin-echo measurement has been calculated on the basis that the signal before one measurement cycle is exactly that at the end of the previous cycle. Under steady state conditions of repeated cycles, this must therefore equal the signal at the end of the measurement cycle itself For a spin-echo pulse sequence such as that depicted in Figure B 1.14.1 the echo magnetization is given by [17]... 

The term theoretical chemistry may be defined as the mathematical description of chemistry. The term computational chemistry is generally used when a mathematical method is sufficiently well developed that it can be automated for implementation on a computer. Note that the words exact and perfect do not appear in these definitions. Very few aspects of chemistry can be computed exactly, but almost every aspect of chemistry has been described in a qualitative or approximately quantitative computational scheme. The biggest mistake a computational chemist can make is to assume that any computed number is exact. However, just as not all spectra are perfectly resolved, often a qualitative or approximate computation can give useful insight into chemistry if the researcher understands what it does and does not predict. 

One of the most commonly used constructs is a model. A model is a simple way of describing and predicting scientific results, which is known to be an incorrect or incomplete description. Models might be simple mathematical descriptions or completely nonmathematical. Models are very useful because they allow us to predict and understand phenomena without the work of performing the complex mathematical manipulations dictated by a rigorous theory. Experienced researchers continue to use models that were taught to them in high school and freshmen chemistry courses. However, they also realize that there will always be exceptions to the rules of these models. 

Approximations are another construct that is often encountered in chemistry. Even though a theory may give a rigorous mathematical description of chemical phenomena, the mathematical difficulties might be so great that it is... 

Quantum mechanics gives a mathematical description of the behavior of electrons that has never been found to be wrong. However, the quantum mechanical equations have never been solved exactly for any chemical system other than the hydrogen atom. Thus, the entire held of computational chemistry is built around approximate solutions. Some of these solutions are very crude and others are expected to be more accurate than any experiment that has yet been conducted. There are several implications of this situation. First, computational chemists require a knowledge of each approximation being used and how accurate the results are expected to be. Second, obtaining very accurate results requires extremely powerful computers. Third, if the equations can be solved analytically, much of the work now done on supercomputers could be performed faster and more accurately on a PC. 

Electrostatics is the study of interactions between charged objects. Electrostatics alone will not described molecular systems, but it is very important to the understanding of interactions of electrons, which is described by a wave function or electron density. The central pillar of electrostatics is Coulombs law, which is the mathematical description of how like charges repel and unlike charges attract. The Coulombs law equations for energy and the force of interaction between two particles with charges q and q2 at a distance rn are... 

Thermodynamics is one of the most well-developed mathematical descriptions of chemistry. It is the held of thermodynamics that dehnes many of the concepts of energy, free energy and entropy. This is covered in physical chemistry text books. 

Quantum mechanics (QM) is the correct mathematical description of the behavior of electrons and thus of chemistry. In theory, QM can predict any property of an individual atom or molecule exactly. In practice, the QM equations have only been solved exactly for one electron systems. A myriad collection of methods has been developed for approximating the solution for multiple electron systems. These approximations can be very useful, but this requires an amount of sophistication on the part of the researcher to know when each approximation is valid and how accurate the results are likely to be. A significant portion of this book addresses these questions. 

Until now we have restricted ourselves to consideration of simple tensile deformation of the elastomer sample. This deformation is easy to visualize and leads to a manageable mathematical description. This is by no means the only deformation of interest, however. We shall consider only one additional mode of deformation, namely, shear deformation. Figure 3.6 represents an elastomer sample subject to shearing forces. Deformation in the shear mode is the basis... 

The crystals are assumed to be circular disks. This geometry is consistent with previous thermodynamic derivations. It has the advantage of easy mathematical description. 

Remember that the hump which causes the instability with respect to phase separation arises from an unfavorable AH considerations of configurational entropy alone favor mixing. Since AS is multiplied by T in the evaluation of AGj, we anticipate that as the temperature increases, curves like that shown in Fig. 8.2b will gradually smooth out and eventually pass over to the form shown in Fig. 8.2a. The temperature at which the wiggles in the curve finally vanish will be a critical temperature for this particular phase separation. We shall presently turn to the Flory-Huggins theory for some mathematical descriptions of this critical point. The following example reminds us of a similar problem encountered elsewhere in physical chemistry. 

The model describing interaction between two bodies, one of which is a deformed solid and the other is a rigid one, we call a contact problem. After the deformation, the rigid body (called also punch or obstacle) remains invariable, and the solid must not penetrate into the punch. Meanwhile, it is assumed that the contact area (i.e. the set where the boundary of the deformed solid coincides with the obstacle surface) is unknown a priori. This condition is physically acceptable and is called a nonpenetration condition. We intend to give a mathematical description of nonpenetration conditions to diversified models of solids for contact and crack problems. Indeed, as one will see, the nonpenetration of crack surfaces is similar to contact problems. In this subsection, the contact problems for two-dimensional problems characterizing constraints imposed inside a domain are considered. 

M. Williams, "Measurement and Mathematical Description of Separation Characteristics of Ha2ardous Organic Compounds with Reverse Osmosis Membranes," dissertation. University of Kentucky, Lexiagton, Ky., 1993. 

Electrokinetics. The first mathematical description of electrophoresis balanced the electrical body force on the charge in the diffuse layer with the viscous forces in the diffuse layer that work against motion (6). Using this force balance, an equation for the velocity, U, of a particle in an electric field... 

Capello and Bielsld, Kinetic Systems Mathematical Description of Kinetics in Solution, Wiley, 1972. 

Many models have been proposed for adsorption and ion exchange equilibria. The most important factor in selecting a model from an engineering standpoint is to have an accurate mathematical description over the entire range of process conditions. It is usually fairly easy to obtain correcl capacities at selected points, but isotherm shape over the entire range is often a critical concern for a regenerable process. 

An important question is whether one can rigorously express such an average without referring explicitly to the solvent degrees of freedom. In other words. Is it possible to avoid explicit reference to the solvent in the mathematical description of the molecular system and still obtain rigorously correct properties The answer to this question is yes. A reduced probability distribution P(X) that depends only on the solute configuration can be defined as... 

Tubular reactors have empty spaces only between the catalyst particles. This eliminates one big disadvantage of CSTRs. On the other hand, the mathematical description and analysis of the data become more complicated. For chemical reaction studies it is still useful to detect major changes or differences in reaction mechanism. 

Remarks The aim here was not the description of the mechanism of the real methanol synthesis, where CO2 may have a significant role. Here we created the simplest mechanistic scheme requiring only that it should represent the known laws of thermodynamics, kinetics in general, and mathematics in exact form without approximations. This was done for the purpose of testing our own skills in kinetic modeling and reactor design on an exact mathematical description of a reaction rate that does not even invoke the rate-limiting step assumption. 

O. S. Heavens. Optical Properties of Thin Solid Films. Buttcrworths, 1955. Chapter 4 presents a detailed mathematical description of the Fresnel fringing phenomenon for the transmission of light through thin films. 

A. V. Sokolov. Optical Properties ofMetaL. Elsevier, New York, 1967, Chapters 10 and 11. A very detailed, mathematical description of solutions to the wave equations, with a nice historical perspective. 

Linear elastic fracture mechanics (LEFM) is based on a mathematical description of the near crack tip stress field developed by Irwin [23]. Consider a crack in an infinite plate with crack length 2a and a remotely applied tensile stress acting perpendicular to the crack plane (mode I). Irwin expressed the near crack tip stress field as a series solution ... 

Modeling the pore size in terms of a probability distribution function enables a mathematical description of the pore characteristics. The narrower the pore size distribution, the more likely the absoluteness of retention. The particle-size distribution represented by the rectangular block is the more securely retained, by sieve capture, the narrower the pore-size distribution. 

Due to the combining effects of hydrodynamic and physicochemical factors, the study of cake structure and resistance is extremely complex, and any mathematical description based on theoretical considerations is at best only descriptive. 

A combination of dimensional similitude and the mathematical modeling technique can be useful when the reactor system and the processes make the mathematical description of the system impossible. This combined method enables some of the critical parameters for scale-up to be specified, and it may be possible to characterize the underlying rate of processes quantitatively. 

EquatitHis (8.4)-(8.8) represent a complete mathematical description of the chemical equilibrium between a rich phase and the y th MSA. The simultaneous solution... 

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