Nonuniqueness

While this is disappointing, the nonuniqueness theorem also shows that if some empirical potential is able to predict correct protein folds then many other empirical potentials will do so, too. Thus, the construction of empirical potentials for fold prediction is much less constrained than one might think initially, and one is justified in using additional qualitative theoretical assumptions in the derivation of an appropriate empirical potential function. 

A. Neumaier, A nonuniqueness theorem for empirical protein potentials, in preparation. 

Much of what we currently understand about the micromechanics of shock-induced plastic flow comes from macroscale measurement of wave profiles (sometimes) combined with pre- and post-shock microscopic investigation. This combination obviously results in nonuniqueness of interpretation. By this we mean that more than one micromechanical model can be consistent with all observations. In spite of these shortcomings, wave profile measurements can tell us much about the underlying micromechanics, and we describe here the relationship between the mesoscale and macroscale. 

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

The antisymmetric part of nonlinear transport matrix is not uniquely defined (due to the nonuniqueness of fh). However, the most likely terminal position is given uniquely by any that satisfies Eq. (122). 

Assuming that the pj (t) and Qj (t) can be interpreted as a TS trajectory, which is discussed later, we can conclude as before that ci = ci = 0 if the exponential instability of the reactive mode is to be suppressed. Coordinate and momentum of the TS trajectory in the reactive mode, if they exist, are therefore unique. For the bath modes, however, difficulties arise. The exponentials in Eq. (35b) remain bounded for all times, so that their coefficients q and q cannot be determined from the condition that we impose on the TS trajectory. Consequently, the TS trajectory cannot be unique. The physical cause of the nonuniqueness is the presence of undamped oscillations, which cannot be avoided in a Hamiltonian setting. In a dissipative system, by contrast, all oscillations are typically damped, and the TS trajectory will be unique. 

It fix) and g(x) are nonconvex, additional difficulties can occur. In this case, nonunique, local solutions can be obtained at intermediate nodes, and consequently lower bounding properties would be lost. In addition, the nonconvexity in g(x) can lead to locally infeasible problems at intermediate nodes, even if feasible solutions can be found in the corresponding leaf node. To overcome problems with nonconvexities, global solutions to relaxed NLPs can be solved at the intermediate nodes. This preserves the lower bounding information and allows nonlinear branch and bound to inherit the convergence properties from the linear case. However, as noted above, this leads to much more expensive solution strategies. 

To demonstrate nonuniqueness, we pose here three problems in geochemical modeling that each have two physically realistic solutions. In the first example, based on data from an aluminum solubility experiment, we assume equilibrium with an alumina mineral to fix the pH of a fluid of otherwise known composition. Setting pH by mineral equilibrium is a widespread practice in modeling the chemistry of... 

Repeating the calculations using minerals such as kaolinite [Al2Si205(0H)4] or muscovite [KAl3Si30io(OH)2] to fix pH also produces nonunique results. 

Analogous examples of nonuniqueness can be constructed using any mineral or gas of intermediate oxidation state. Buffering the fugacity of N2(g) or S02(g), for example, would be a poor choice for constraining oxidation state, since the gases can either oxidize to NO3 and SO4 , respectively, or reduce to NH4 and H2S(aq) species. 

The examples in the previous section demonstrate that nonunique solutions to the equilibrium problem can occur when the modeler constrains the calculation by assuming equilibrium between the fluid and a mineral or gas phase. In each example, the nonuniqueness arises from the nature of the multicomponent equilibrium problem and the variety of species distributions that can exist in an aqueous fluid. When more than one root exists, the iteration method and its starting point control which root the software locates. 

In each of the cases, the dual roots differ from each other in terms of pH, sulfide content, or ionic strength, so that in a modeling study the correct root could readily be selected. The danger of nonuniqueness is that a modeler, having reached an inappropriate root, might not realize that a separate, more meaningful root to the problem exists. 

This set is not unique and does not necessarily imply anything about the way in which reaction occurs. Thus, from a stoichiometric point of view, (A), (B), and (C) are properly called equations and not reactions. The nonuniqueness is illustrated by the fact that any... 

Figure 4.12 corresponds to objective functions in well-posed optimization problems. In Table 4.2, cases 1 and 2 correspond to contours of /(x) that are concentric circles, but such functions rarely occur in practice. Elliptical contours such as correspond to cases 3 and 4 are most likely for well-behaved functions. Cases 5 to 10 correspond to degenerate problems, those in which no finite maximum or minimum or perhaps nonunique optima appear. 

The anomalous properties of water remain an important subject of inquiry (Errington and Debenedetti, 2001 Mishima and Stanley, 1998). Chaplin (2004) gives a comprehensive overview of 40 anomalous properties of water and suggested explanations. Chaplin (2004) aptly pointed out that whether the properties of water are viewed as anomalous depends on what materials water is compared to and the interpretation of the term anomalous. For example, Angell (2001) included a section on the nonuniqueness of water, stating that. . water is not unique, as is often supposed, but rather water is an intermediate member of a series of substances that form tetrahedral networks of different degrees of flexibility, and that, accordingly, show systematic differences of behavior. Additional references that discuss the properties of water as nonanomalous are Franks (2000), Kivelson and Tarjus (2001), and Netz et al. (2002). 

Nontoxic chlorofluorocarbons, 24 188 Nontronite (iron smectite), 6 664, 696 structure and composition, 6 669 Nonuniqueness, 24 446 Nonvessel operating common carriers (NVOCC), 25 328 Nonvolatile compounds, as taste substances, 11 566 Nonvolatile food components,... 

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