The Formal Definition

Changing to the generalized state variable Z, we recall from Chapter 2 (equation 

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Orthogonal transformations preserve the lengths of vectors. If the same orthogonal transformation is applied to two vectors, the angle between them is preserved as well. Because of these restrictions, we can think of orthogonal transfomiations as rotations in a plane (although the formal definition is a little more complicated). 

Comparing this expression with (3.81), one obtains the formal definition of the quasiclassical tunneling matrix element... 

Because proteins are not composed of identical repeating units, but of different amino acids, they do not fall within the formal definition of polymers given at the start of this chapter. They are nevertheless macromolecular and techniques developed for the study of tme polymers have been applied to them with success. However, for the most part they are outside the scope of this book and accordingly will receive very little attention in the chapters that follow. 

The formal definition of a partition function implies that it is a summation over an infinitely high number of terms. Explain why the partition function... 

The hazardous waste listings, the hazardous waste characteristics, and the mixture and derived-from rules are all essential parts of the definition of hazardous waste, but these key elements are all described in different sections of the RCRA regulations. Only one regulatory section unites all four elements to establish the formal definition of hazardous waste. This section is entitled Definition of Hazardous Waste, which states that all solid wastes exhibiting one of the four hazardous characteristics are hazardous wastes. This section also states that all solid wastes listed on one of the four hazardous waste lists are hazardous wastes. Finally, this section explains in detail the mixture and derived-from rules and the regulatory exemptions from these rules. Thus, although the section is entitled Definition of Hazardous Waste, it serves primarily as a guide to the mixture and derived-from rules. 

The rectangular pulse can be generated by subtracting a step function with dead time T from a step function. We can derive the Laplace transform using the formal definition... 

On the negative real axis (-180°), find the "distance" of GCGP from (-1,0). This is the gain margin, GM. The formal definition is... 

Find the frequency where the magnitude Gc(jto)Gp(jco) is 1. This particular frequency is the phase crossover frequency, to = tocp. We then find the angle between GcGp and -180°. This is the phase margin, PM. The formal definition is... 

In this abstraction each edge corresponds to a pipeline network element and each vertex corresponds to a junction connecting two or more elements. It is often convenient to refer to the formal definition of a graph G as the sets... 

Now we must explain the semantics of the logical connectives. Informally, they have meanings you are probably acquainted with. Connective " a " is "AND", connective " v " is "OR", " 1 " is "NOT", " = " is "IMPLIES", while " = " is "IF AND ONLY IF". Occasionally we use the English words rather than the formal symbols as a matter of convenience. We now give the formal definitions of these concepts. In this definition, recall that we allow members of the domain of I to play the same role as constants or free (but not bound) variable in wffs. 

The formal definition of the electronic chemical hardness is that it is the derivative of the electronic chemical potential (i.e., the internal energy) with respect to the number of valence electrons (Atkins, 1991). The electronic chemical potential itself is the change in total energy of a molecule with a change of the number of valence electrons. Since the elastic moduli depend on valence electron densities, it might be expected that they would also depend on chemical hardness densities (energy/volume). This is indeed the case. 

The formal definition of this quality factor, Q, is the amount of power stored in the resonator divided by the amount of power dissipated per cycle (at 9.5 GHz a cycle time is l/(9.5 x 109) 100 picoseconds). The dissipation of power is through the resonator walls as heat, in the sample as heat, and as radiation reflected out of the resonator towards the detector. The cycle time is used in the definition because the unit time of one second would be far too long for practical purposes within one second after the microwave source has been shut off, all stored power has long been dissipated away completely. 

Students often get the terms, heat, specific heat and temperature confused. Here are the formal definitions ... 

The effective molarity (EM) is formally the concentration of the catalytic group (RCOO- in [5]) required to make the intermolecular reaction go at the observed rate of the intramolecular process. In practice many measured EM s represent physically unattainable concentrations, and the formal definition is probably relevant only in reactions (which will generally involve very large cyclic transition states) where the formation of the ring or cyclic transition state per se is enthalpically neutral, or in diffusion-controlled processes. For the formation of small and medium-sized rings and cyclic transition states the EM as defined above contains, and may indeed be dominated by, the enthalpy of formation of the cyclic form. This topic has been discussed briefly by Illuminati et al. (1977) and will be treated at greater length in a future volume in this series. 

Given the definition of the geometry of the transition states in TST as the highest energy point in the minimum energy pathway from reactants to products, the formal definition of MEP is as follows. The MEP is, in one direction, the path of steepest descents from the transition state to reactants while, in the other direction, it is the path of steepest descents from transition state to products. For reasons which will not be discussed here, the formal definition of MEP includes the statement that the pathway is expressed in mass scaled Cartesian coordinates of the position of the atoms (introduced in Chapter 3, e.g. x is replaced by x = ). This simplifies... 

We surreptitiously introduced reciprocal space in Figure 4.1, in an attempt to show its usefulness before giving the formal definitions. It is so very helpful in the interpretation of many diffraction experiments that we need to understand it more fully. We shall use it extensively in the discnssion of triple-axis experiments, in which reciprocal space mapping is an essential technique. 

These operators satisfy the formal definition for spin-adapted operators in Eqs. (79) and (78). Inserting these four operators into Eq. (8), we can generate four... 

Geminal functional theory is a very promising research area. The different varieties of antisymmetrized products are very flexible and inherently handle difficult problems, like multideterminantal molecules. The computational effort is low compared to the quality of the solutions. The perturbation theoretical approach to SSG should essentially be possible for AGP and UAGP as well. The formal definition of GFT is a flexible framework that opens up many new opportunities for exploring the nature of solutions to the Schrodinger equation. 

Actually, by starting with Equations (C.8) and (C.9) as the definitions, all the properties of the spherieal modified Bessel functions can be obtained, without tracing back to the formal definition. Equations (C.6) and (C.7). 

The quantities G, //, and S are called extensive thermodynamic functions because the magnitude of the quantity in each case depends on the amount of substance in the system. The change in Gibbs free energy under addition of unit concentration of component / at constant concentrations of the other components is called the partial Gibbs free energy of the /-component, i.e., the chemical potential of the /-component in the system. The chemical potential is an intensive thermodynamic quantity, like temperature and concentrations. The formal definition is... 

Each of the groups we introduce in this text is a Lie group. We give the formal definition in terms of manifolds however, readers unfamiliar with differential geometry may think of a manifold as analogous to a nicely parametrized surface embedded in R. More to the point for our purposes, a manifold is a set on which differentiability is well defined. Since all the mamfolds we will consider are nicely parameterized, we can define differentiability in terms of the parameters. 

Substituting the formal definition of E — Eo from Eq. 8.76 into Eq. 8.122 and taking the derivative with respect to temperature, after some algebra we obtain... 

POISE (P). A unit of dynamic viscosity. The unit is expressed in dyne second per square centimeter The centipoise (cP) is more commonly used The formal definition of viscosity arises from the concept put forward by Newton that under conditions of parallel flow, the shearing stress is proportional to the velocity giadieut. If lire force acting on each of two planes of aiea A parallel to each oilier, moving parallel lo each other with a relative velocity V, and separated by a perpendicular distance X, be denoted by F. the shearing stress is F/A and the velocity gradient, which will be linear for a true liquid, is V/X. Thus, Ft A = q V/X, where the constant if is the viscosity coefficient or dynamic viscosity of the liquid. The poise is the CGS unit of dynamic viscosity. 

Following Atkins [68], the propagation of particles follows a path dictated by Newton s laws, equivalent to Hamilton s principle, that particles select paths between two points such that the action associated with the path is a minimum. Therefore, Fermat s principle for light propagation is Hamilton s principle for particles. The formal definition of action is an integral identical in structure with the phase length in physical optics. Therefore, particles are associated with wave motion, the wave-particle dualism. Hamilton s principle of least... 

An important aspect of this definition is that the studies are part of the formal, definitive, storage program. In contrast, ICH, in Annex 1, Glossary and Information of the revised stability guideline (6) defined stress testing (drug substance) as ... 

First, we provide the formal definitions of classical absorption and photodissociation cross sections and subsequently we describe a practical way in order to calculate them. 

The copper half-reaction is an oxidation reaction, while the sulfur reaction is a reduction reaction. The formal definitions are (/ ) oxidation is the result of a loss of electrons and (ii) reduction is the result of a gain of electrons. A trick for keeping the terminology straight is to consider that a reduction reaction includes the drop in charge (reduction in charge, S° S2-). Since the half-reactions must be paired, the other reaction, the oxidation... 

Equation 11.2-25 is the formal definition of half-life, the time the second-order reaction takes to reduce the initial concentration to half the initial cq. Following the methodology of... 

Fractionation factors for hydrogen-containing solutes other than the hydrogen ion are represented by the symbol , the formal definition being... 

The formal definitions of the nonlocal operators x ar d e can be expressed in the form of their application to a generic F(r) function ... 

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