The question of uniqueness

Bethke, C. M., 1992, The question of uniqueness in geochemical modeling. Geo-chimica et Cosmochimica Acta 56,4315 4320. 

Many questions remain open. Key among these are the questions of uniqueness of the interior steady state in the case of Michaelis-Menten dynamics and of sufficient conditions for uniqueness in the general case. The model does not contain any transport terms, although their inclusion would be important to model a moving stream. Periodic coefficients, as discussed in Chapter 7, are certainly relevant to this model and have not been considered. In addition, the case of non-equal diffusion is not considered at all by these methods. Hence further modeling and mathematics are still needed. 

In this case, it is impossible to distinguish these two models from the given data. That is why the question of uniqueness is so important in inversion. 

Now that it is clear that identiliability is concerned with the question of uniqueness of solutions for the basic parameters from the observation function of a given experiment, we have to introduce the various types of identihability that have been defined in the literature. 

The question of uniqueness in the inversion process has been answered (Hildebrand 1965) and is given expression in the understated Lerch s theorem if one fimction fit) corresponding to the known transform F(s) can be found, it is the correct one. Not all functions of are transforms, since continuity and other considerations must be taken into account. But, if Fis) 0 as s 00 and sF(s) is bounded as f - 00, then F(s) is the transform of some function fit), which is at least piecewise continuous in some interval 0 t t and such function is of exponential order. When the initial value of fit) is desired and Fis) is known, the following limit is useful... 

Among the ordinary numbers, only 0 has no inverse. Many matriees have no inverse. The question of whether a matr ix A has or does not have a defined inverse is elosely related to the question of whether a set of simultaneous equations has or does not have a unique set of solutions. We shall eonsider this question more fully later, but for now reeall that if one equation in a pair of simultaneous equations is a multiple of the other. 

Just as artificial life seeks to aii.swer the question of whether life is really an emergent property of the organization of matter and is not just some unique embodiment of its substance, so too it might be said that the goal of finite physics is to see whether physics - what we call reality - is fundamentally a property of the organization of information rather than a unique embodiment of the interaction... 

Thus the current operator indeed transforms like a vector. This must be the case in order that the equation Qdu(x) = ju(x) transform properly, assuming the transformation property (11-267) for Au(x). We now inquire briefly into tike question of the uniqueness of the U(ia) operator, in particular into the question of the phase associated with the fermion field operator. Note that the phase of the photon field operator is uniquely determined (Eq. (11-267)) by the fact that An is a hermitian field which commutes with the total charge operator Q. The negaton-positon field operator on the other hand does not commute with the total charge operator, in fact... 

The connection that has been shown in Section VIII to exist between burn-out in a rod bundle and in an annulus leads to the question of whether or not a link may also exist between, for example, a round tube and an annulus. Now, a round tube has its cross section defined uniquely by one dimension—its diameter therefore if a link exists between a round tube and an annulus section, it must be by way of some suitably defined equivalent diameter. Two possibilities that immediately appear are the hydraulic diameter, dw = d0 — dt, and the heated equivalent diameter, dh = (da2 — rf,2)/ however, there are other possible definitions. To resolve the issue, Barnett (B4) devised a simple test, which is illustrated by Figs. 38 and 39. These show a plot of reliable burn-out data for annulus test sections using water at 1000 psia. Superimposed are the corresponding burn-out lines for round tubes of different diameters based on the correlation given in Section VIII. It is clearly evident that the hydraulic and the heated equivalent diameters are unsuitable, as the discrepancies are far larger than can be explained by any inaccuracies in the data or in the correlation used. 

But if the property is unlike any the world has yet seen, then it is not clear how such property should be regarded, let alone protected. In other words, the question of just what sort of property software is has not been satisfactorily answered, which contributes to the debate on the uniqueness question. Nonetheless, devices such as copyrights, patents, encryption, trade secrets, and oaths of confidentiality and standard virtues like trustworthiness and loyalty have been tried to protect ownership and the right to property [17, 23]. 

The fast reactions of ions between aqueous and mineral phases have been studied extensively in a variety of fields including colloidal chemistry, geochemistry, environmental engineering, soil science, and catalysis (1-6). Various experimental approaches and techniques have been utilized to address the questions of interest in any given field as this volume exemplifies. Recently, chemical relaxation techniques have been applied to study the kinetics of interaction of ions with minerals in aqueous suspension (2). These methods allow mechanistic information to be obtained for elementary processes which occur rapidly, e.g., for processes which occur within seconds to as fast as nanoseconds (j0. Many important phenomena can be studied including adsorption/desorption reactions of ions at electri fied interfaces and intercalation/deintercalation of ions with minerals having unique interlayer structure. 

Having established the effect of substitution on the rates of both monomer isomerization and polymerization, we addressed the question of polymer structure. Specifically, are norbornenyl imide units incorporated into the fully cured polymer with their norbornyl rings intact If so, does the polymer also reflect the equilibrium ratio of exo and endo ring fused monomers For our parent monomers, PN and PX, this question has been unanswerable. We have not found any direct probe that allows an unambiguous assessment of specific substructures within the cured polymer. We do, however, have some evidence bearing on this question for the phenyl substituted monomer. This evidence is attributable in part to our discovery of an unexpected side-reaction in the cure of the phenyl substituted monomer, and in part to the presence of a unique NMR diagnostic for phenyl substituted, endo norbornyl N-phenyl imides. Both of these results are detailed below. 

The second question of unique importance concerns the investigator s selection of a rotor for the Lab Plan. Whereas, in the Optimal Plan, SpinPro selects the rotor in the Lab Plan, the investigator selects the rotor. The investigator, however, is not required to... 

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