Two Important Theorems

The AR is composed entirely of mixing surfaces and manifolds ofPFR trajectories 

The boundary of the AR exhibits a simple structure, irrespective of the kinetics used. As a demonstration, refer back to Chapter 5 and count the number of occurrences where the optimal reactor structure terminated with a PFR. (You will find that all examples considered in Chapter 5 resulted in a final optimal reactor structure terminating with a PFR.) Theorem 1 helps prove that this behavior is not a coincidence, for the final approach to all optimal reactor structures on the boundary of the AR terminates with a PFR. 

From Section 6.3.2 it is understood that extreme points on the AR boundary result from either feed points or reaction surfaces (specifically protrusions). Using theorem 1, this statement may be refined to say that exposed points from reaction originate specifically from PFR trajectories only. This result is summarized compactly by the following theorem, which is adapted from Feinberg and Hildebrandt (1997)  

Theorem 6.1 (exposed points on the AR boundary are either PFR trajectories or feed points) Suppose that we have a specified feed set F in R and a convex set of achievable points given by C, also contained in R . The rate function r(C) associated with this region is assumed to be continuously differentiable and also defined on R . Furthermore, the set of concentrations in C is assumed to comply with the complement principle. If it is found that all rate vectors on the boundary ofC do not point outward, then any protrusion in C that is separate (disjoint) from F is the union of PFR trajectory segments. The solution trajectories then satisfy the PFR equation dC/dr =r(C). 

Due to theorem 1, the final approach to any optimal reactor network on the AR boundary, where reaction and mixing are employed, must involve a PFR. 

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It is easy to show that every polynomial f x) (not divisible by x) over a finite field J g is a factor of 1—cc , for some power n . The order (sometimes also called the period or exponent) of f x), denoted by ord(/), is the least such n . If f x) = p x) is an irreducible polynomial (other than x) with d[f] = n then ord(/) must divide pTi i There are two important theorems concerning the orders of prime factors and products of relatively prime polynomials over Fg ... 

We shall now prove two important theorems concerning the behavior of characters. 

Equn. (3.9) is a special case of the representation theorem or closure theorem, which is one of two important theorems that are used frequently in formal quantum mechanics. 

Two theorems of Gibbs and of Konovalow —Under what circumstances shall we observe such a state of indifferent equilibrium Two important theorems, discovered by J. Willard Gibbs, found anew by D. Konovalow, give us this information. Here are these two theorems ... 

Application of the first theorem to mixtures of volatile liquids.—Let us consider in detail the consequences of these two important theorems beginning with the first. 

The above equations enable us to derive two important theorems first enunciated by Gibbs, and later rediscovered by Konovalow and Duhem. 

The integration domain and the weight function n( ) uniquely define the family of polynomials PaiO)- A polynomial is defined as monic when its leading coefficient (i.e. kap) is equal to unity. Below two important theorems (without proof) are reported. 

There are two important theorems which determine whether a spectrum is likely to be seen for various numbers of electrons in an ion. 

For the future discussion of the liquid crystal structure we need two important theorems. The first of them, the theorem of convolution is formulated as follows a Fourier transform of convolution of two functions/i(x) and/2(x) is a product of their Fourier transforms Fi q)-F2 q) -... 

Two important theorems in vector analysis involving inverse spatial operators are included in Graph 5.9. The first is the Stokes theorem ... 

Section 3.3 continues with formal aspects of Hartree-Fock theory. We derive and discuss two important theorems associated with the Hartree-Fock equations Koopmans theorem and Brillouin s theorem. The first... 

We consider in this section the variation principle in molecular electronic-structure theory. Having established the particular relationship between the Schrddinger equation and the variational condition that constitutes the variation principle, we proceed to examine the variation method as a computational tool in quantum chemistry, paying special attention to the application of the variation method to linearly expanded wave functions. Next, we examine two important theorems of quantum chemistry - the Hellmann-Feynman theorem and the molecular virial theorem - both of which are closely associated with the variational condition for exact and approximate wave functions. We conclude this section by presenting a mathematical device for recasting any electronic energy function in a variational form so as to benefit to the greatest extent possible from the simplifications associated with the fulfilment of the variational condition. 

We have found three distinet irredueible representations for the C3V symmetry group two different one-dimensional and one two dimensional representations. Are there any more An important theorem of group theory shows that the number of irredueible representations of a group is equal to the number of elasses. Sinee there are three elasses of operation, we have found all the irredueible representations of the C3V point group. There are no more. 

It is usefid to know the sensitivity and specificity of a test. Once a researcher decides to use a certain test, two important questions require answers If the test results are positive, what is the probability that the researcher has the condition of interest If the test results are negative, what is the probability that the patient does not have the disease Bayes theorem provides a way to answer these questions. 

Here an explanation is provided on the structure of the near-field solution with the help of some fundamental theorems. These theorems provide the basis for interpreting both the near- and far-field solutions. These are due to Abel and Tauber and their utility was highlighted by Van der Pol Bremmer (1959) in connection with the properties of bilateral Laplace transform. In exploring relationships between the original in the physical plane and the image or transform in the spectral plane these two important... 

The proof of this important theorem which establishes the necessary conditions for holonomicity is provided in two Appendices, given in Sections 1.11 and 1.12. Readers not interested in the mathematical niceties of this proof may assume its correctness, and hence, that holonomicity implies inaccessibility and vice versa. The stage is then set for the... 

The Hohenberg-Kohn theorem was of great importance for the development of density-functional theory, but a practical implementation of DF theory was first presented by Kohn and Sham (1965). This paper contains two important advances of the theory (1) the exact conversion of the many-electron problem to an effective one-electron problem, and (2) the local-density approximation. 

A major difference between competitive and cooperative systems is that cycles may occur as attractors in competitive systems. However, three-dimensional systems behave like two-dimensional general autonomous equations in that the possible omega limit sets are similarly restricted. Two important results are given next. These allow the Poincare-Bendix-son conclusions to be used in determining asymptotic behavior of three-dimensional competitive systems in the same manner used previously for two-dimensional autonomous systems. The following theorem of Hirsch is our Theorem C.7 (see Appendix C, where it is stated for cooperative systems). 

The proof of this important theorem is provided in the next two sections. 

If two operators commute, there is no restriction on the accuracy of their simultaneous measurement. For example, the x- and y-coordinates of a particle can be known at the same time. An important theorem states that two commuting observables can have simultaneous eigenfunctions. To prove this, write the eigenvalue equation for an operator A ... 

This important theorem refers to two- and three-dimensional structures. It expresses the fact that tiling of the Euclidean plane by regular polygons can be achieved only with the triangle, the square and the hexagon. A four-dimensional periodic structure can allow other symmetry operations. 

It appears possible to make the following two important generalizations concerning the relative rates of mass transfer to the catalyst pellet and diffusion into the pellet (a) Mass transfer to the external catalyst surface is always faster than diffusion into the internal catalyst surface. This is because turbulence in the fluid stream enhances the effective diffusion coeflacient in the flowing fluid to much larger values than those possible inside a catalyst pellet. Even in the absence of turbulence, the presence of small pores in catalysts depresses the diffusion coefficient to (Knudsen) values lower than the bulk values in the flowing stream, (b) Hence, whenever mass transfer to the external catalyst surface is influencing reaction rate, then the internal surface area can be only partly available to the reaction. We thus get the elementary theorem that whenever a catalyst is sufficiently active so that the reaction rate is influenced by mass transfer (diffusion) to the catalyst pellet, then the internal surface area of that catalyst can be only partially available to the reaction. 

Recall the two important equations before we start to discuss the conservation laws. The first one is the Gauss-Ostrogradsky theorem . It relates the surface and volume integrals of the type... 

The complement principle serves an important role in the development of two fundamental theorems in AR theory, which are described in Chapter 6. We briefly describe the principle here, which is an adaptation from Feinberg and Hildebrandt (1997). 

Electrostatics. Two important laws of electrostatics are directly deduced from this scheme of spatial reduction the Poisson equation and the Gauss theorem (case studies B4 and B5 in this chapter). The mutual influence between two of these poles is modeled within the framework of an electrostatic dipole using Coulomb s law. 

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