Main Results of Chapter

If only mass is balanced we can formally admit a change in accumulation of mass in a node as a fictitious stream oriented towards die environment see Fig. 3-1. The conservation of mass is then expressed as Eq.(3.1.6) thus Cm = 0 where m is the (column) vector of mass flowrates and C the reduced incidence matrix of G, thus without row Hq. We assume G connected, hence C is of full row rank (3.1.5) thus N where M denotes generally the number of elements of set M. 

The problems of solvability arise when certain mass flowrates are given a priori see Section 3.2. The graph structure allows one to analyze the problems completely by graph operations. The arc set J is partitioned 

Recall that a connected component can also consist of one (isolated) node, with empty subset of incident arcs. Having merged the nodes of each N,j in G we have the reduced graph G see Fig. 3-7. The K nodes of G correspond uniquely to the K subsets N, of N. By the graph reduction (merging), we have deleted all the arcs j e J°, and some arcs j g J in addition (subset J c J ). The arc set of G , denoted by J, consists of the remaining (not deleted) arcs j e J. Thus J is partitioned 

In the first step, having re-arranged the variables and equations according to the graph decomposition, we have thus rewritten the balance equation Cm = 0 (3.1.6) in equivalent form (3.2.4). The last of the equations (3.2.4) reads 

The structure of the graph G along with the partition (3.6.1) allows one to classify all the variables (mass flowrates) with respect to the solvability see Section 3.3. According to the standard terminology, the variables mj, i J are called measured, the remaining (j J°) unmeasured. This classification will be used henceforth any a priori fixed variable of a model will be called measured , else unmeasured . The classification of the measured variables follows immediately from the partition (3.6.3) if i e J then m is redundant, else (i G J ) nonredundant. The nonredundant variables are also characterized by the property that i g J if and only if arc i closes a circuit in G with certain unmeasured streams (/ g J°). The nonredundant variables are unaffected by the solvability condition. 

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The results of this section have shown that neither the assumption of Michaelis-Menten functional response nor the assumption of only two competitors is essential for the main results of Chapter 1 to hold. In much of what follows we retain the Michaelis-Menten kinetics, since the parameters are readily measurable in the laboratory. 

Similarly, our analysis of the variable-yield model in Chapter 8 is limited to two competing populations because we rely on the techniques of monotone dynamical systems theory. One would expect the main result of Chapter 8 to remain valid regardless of the number of competitors, just as it did for the simpler constant-yield model treated in Chapters 1 and 2. Perhaps the LaSalle corollary of Chapter 2 can be used to carry out such an extension, using arguments similar to those used in [AM] (described in Chapter 2). As noted in [NG], a structured model in which... 

Sastry s approach has some resemblance to the one in [H], but there are a number of new techniques involved in the construction of Dualizing Complexes. In short. Chapter 6 of [H] is localized, generalized, and extended to the context of formal schemes in [LNS] and then, among other things, the main results of Chapter 7 of [H], are extended to this context in [S ]. 

The advantage of Raman spectromicroscopy is that very small specimens can be studied while still allowing the determination of the second and fourth moments of the ODF. However, the expressions for the Raman intensities are more complex since the optical effects induced by the microscope objective have to be considered. Although the corrections may be small, they are not necessarily negligible [59]. This problem was first treated by Turrell [59-61] and later by Sourisseau and coworkers [5]. Turrell has mathematically quantified the depolarization of the incident electric field in the focal plane of the objective and the collection efficiency of the scattered light by high numerical aperture objectives. For brevity, only the main results of the calculations will be presented. Readers interested in more details are referred to book chapters and reviews of Turrell or Sourisseau [5,59,61]. The intensity in Raman spectromicroscopy is given by [59-61]... 

The theory of electron-transfer reactions presented in Chapter 6 was mainly based on classical statistical mechanics. While this treatment is reasonable for the reorganization of the outer sphere, the inner-sphere modes must strictly be treated by quantum mechanics. It is well known from infrared spectroscopy that molecular vibrational modes possess a discrete energy spectrum, and that at room temperature the spacing of these levels is usually larger than the thermal energy kT. Therefore we will reconsider electron-transfer reactions from a quantum-mechanical viewpoint that was first advanced by Levich and Dogonadze [1]. In this course we will rederive several of, the results of Chapter 6, show under which conditions they are valid, and obtain generalizations that account for the quantum nature of the inner-sphere modes. By necessity this chapter contains more mathematics than the others, but the calculations axe not particularly difficult. Readers who are not interested in the mathematical details can turn to the summary presented in Section 6. 

We consider below the possibilities for simplification of overall reaction rate equations and introduce the main result of this chapter — the hypergeometric series for reaction rate. 

To explain how these problems arise, let me first summarize the main argument of chapter III. An action, to be rational, must bc the final result of three optimal decisions. First, it must be the best means of realizing a person s desire, given his beliefs. Next, these beliefs must themselves be optimal, given the evidence available to him. Finally, the person must collect an optimal amount of evidence - neither too much nor too little. That amount depends both on his desires - on the importance he attaches to the decision - and on his beliefs about the costs and benefits of gathering more information. The whole process, then, can be visualized as depicted in Fig. IV. 1. 

To treat the stochastic Lotka and Lotka-Volterra models, we have now to extend the formalism presented in Section 2.2.2, where collective variables-numbers of particles iVA and Vg were used to describe reactions. The point is that this approach neglects local density fluctuations in small element volumes. To incorporate both these fluctuations and their correlations due to diffusive conjunction, we are in position now to reformulate these models in terms of the diffusion-controlled processes - in contrast to the rather primitive birth-death formalism used in Section 2.2.2. It permits also to demonstrate in the non-trivial way a role of diffusion in the autowave processes. The main results of this Chapter are published in [21, 25]. 

This chapter presents an account of the main results of studies of kinetics of some industrial heterogeneous catalytic reactions. The studies have been carried out by the author with his co-workers at the Karpov Institute of Physical Chemistry (Moscow, USSR). The presentation is not chronological the reactions are arranged based on the character of interpretation of their kinetics. 

The relationships describing the tensorial properties of wave functions, second-quantization operators and matrix elements in the space of total angular momentum J can readily be obtained by the use of the results of Chapters 14 and 15 with the more or less trivial replacement of the ranks of the tensors l and s by j and the corresponding replacement of various factors and 3nj-coefficients. Therefore, we shall only give a sketch of the uses of the quasispin method for jj coupling, following mainly the works [30, 167, 168]. For a subshell of equivalent electrons, the creation and annihilation operators a and a(jf are the components of the same tensor of rank q = 1/2... 

The following theorem is the main result of this chapter. (Recall that (L) is... 

In the tenth chapter, we study schemes generated by a set of two involutions. The main result of this chapter is a characterization of schemes which have finite valency and are generated by a Coxeter set of cardinality 2. Referring to results obtained in the previous chapter we obtain, as a consequence, a representation theoretic characterization of such schemes. 

In the course of community development, the biomass and the production of autotrophic and heterotrophic organisms decrease [7]. Meanwhile, the correlation between the biodiversity and productivity of the communities is not yet clear and is still being discussed. The objective of this chapter is to consider the main results of the studies on species diversity in the Black Sea from the point of view of their interrelation to the production processes and to the ecological and physiological characteristics of the phytoplankton community. 

Does the order of retention of the main components of the essential oils correlate with the structural differences in the main components For instance, anethole (main component in anise oil) is retained longer than is eugenol (main component in clove oil) explain this in terms of reverse-phase chromatography. If you cannot explain why the retention is as it occurs, refer to Chapter 4 and 5 or to the results of Chapter 12 (Experiment 5 Reverse-Phase Chromatography) for suggestions. 

This chapter is devoted to describe the impact of metallic nanosphere to the multi-photon excitation fluorescence of Tryptophan, and little further consideration to multi-photon absorption process will be given, as the reader can find several studies in [11-14]. In section II, the nonlinear light-matter interaction in composite materials is discussed through the mechanism of nonlinear susceptibilities. In section III, experimental results of fluorescence induced by multi-photon absorption in Tryptophan are reported and analyzed. Section IV described the main results of this chapter, which is the effect of metallic nanoparticles on the fluorescent emission of the Tryptophan excited by a multi-photon process. Influence of nanoparticle concentration on the Tryptophan-silver colloids is observed and discussed based coi a nonlinear generalization of the Maxwell Garnett model, introduced in section II. The main conclusion of the chapter is given in secticHi IV. 

In the previous chapter the gradostat was introduced as a model of competition along a nutrient gradient. The case of two competitors and two vessels with Michaelis-Menten uptake functions was explored in considerable detail. In this chapter the restriction to two vessels and to Michaelis-Menten uptake will be removed, and a much more general version of the gradostat will be introduced. The results in the previous chapter were obtained by a mixture of dynamical systems techniques and specific computations that established the uniqueness and stability of the coexistence rest point. When the number of vessels is increased and the restriction to Michaelis-Menten uptake functions is relaxed, these computations are inconclusive. It turns out that unstable positive rest points are possible and that non-uniqueness of the coexistence rest point cannot be excluded. The main result of this chapter is that coexistence of two microbial populations in a gradostat is possible in the sense that the concentration of each population in each vessel approaches a positive equilibrium value. The main difference with the previous chapter is that we cannot exclude the possibility of more than one coexistence rest point. 

We can now state the main result of this chapter. It provides sufficient conditions for the coexistence of the two populations in the gradostat and ensures that 2 contains a positive rest point . In fact, it guarantees that the two populations are uniformly persistent in the sense of Appendix D. Briefly, Theorem 4.4 states that coexistence holds if each population can successfully invade its competitor s rest point. 

Obviously, the cases where A12 > 0 and — Ei, or where A21 > 0 and =E, correspond to competitive exclusion and therefore are of less interest. On the other hand, if both A and A21 are positive then Theorem 5.1(iii) implies that both E and are positive. In fact, in this case every solution starting with positive initial values converges to a positive periodic solution by the following result, which is the main result of this chapter. Its proof is given following the proof of Theorem 5.1. 

In this chapter, we have shown why the recent transition path theory (TPT) offers the correct probabilistic framework to understand the mechanism by which rare events occur by analyzing the statistical properties of the reactive trajectories involved in these events. The main results of TPT are the probability density of reactive trajectories and the probability current (and associated streamlines) of reactive trajectories, which also allows one to compute the probability flux of these trajectories and the rate of the reaction. It was also shown that TPT is a constructive theory under the assumption that the reaction channels are local, TPT naturally leads to algorithms that allow to identify these channels in practice and compute the various quantities that TPT offers. 

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