Non-uniqueness

E. V. Albano. Irreversible phase transitions into non-unique absorbing states in a multicomponent reaction system. Physica A 274 426-434, 1995. 

Due is a certain non-uniqueness in the symmetry assignments in D2h symmetry, some authors, e.g. in ref [9], instead designate the compressed ground state as B2g and the elongated one as ig. In the present paper, we follow the convention set by the Gaussian program... 

Since the limit j/i —> 0 and t/2 —> 0 is not unique, we can either choose a procedure to make it unique, e.g. fix that there is always a border of an interval at z = 0, or - what is more realistic - we accept the non-uniqueness and hence an incomplete information and average over the indefinite phase in some consistent way. Leaving the phase unspecified we get... 

Unlike prisms, in this class of bodies uniqueness requires knowledge of the density. This theorem was proved by P. Novikov. The simplest example of starshaped bodies is a spherical mass. Of course, prisms are also star-shaped bodies but due to their special form, that causes field singularities at corners, the inverse problem is unique even without knowledge of the density. It is obvious that these two classes of bodies include a wide range of density distributions besides it is very possible that there are other classes of bodies for which the solution of the inverse problem is unique. It seems that this information is already sufficient to think that non-uniqueness is not obvious but rather a paradox. 

For higher-order reactions, the fluid-element concentrations no longer obey (1.9). Additional terms must be added to (1.9) in order to account for micromixing (i.e., local fluid-element interactions due to molecular diffusion). For the poorly micromixed PFR and the poorly micromixed CSTR, extensions of (1.9) can be employed with (1.14) to predict the outlet concentrations in the framework of RTD theory. For non-ideal reactors, extensions of RTD theory to model micromixing have been proposed in the CRE literature. (We will review some of these micromixing models below.) However, due to the non-uniqueness between a fluid element s concentrations and its age, micromixing models based on RTD theory are generally ad hoc and difficult to validate experimentally. 

In the mixtures with initial H20/C0 ratios greater than 4, unique critical points were found for all e covering the range from zero to 100% conversion of CO. In this range F(e) varies from - to +< , hence must cross zero at some e. There was only one such zero crossing, however, indicating that non-uniqueness, if it occurs, will arise only because of the branching of F(e). 

It is obvious from these experiments that the absorption spectrum of the Martian red surface can be simulated reasonably well by a non-unique variety of Fe rich phases or their mixtures as can the weak magnetism, so that a positive identification will probably only be possible, following further in situ analyses and/or sample return and analysis in the lab.Two Mars Exploration Rovers (MERs) are due to arrive at Mars in 2004 and will attempt to analyze rocks and soils on the surface using several small spectrometers, including PanCAM (an extended visible region spectrometer), MiniTES (a thermal emission spectrometer), APXS (alpha proton X-ray spectrometer measuring the major elements), Mossbauer (run at current local temperature), as well as a 5-level magnet array similar to that on-board the Pathfinder Lander. 

Walter E, Piet-Lahannier H, Happel J. 1986. Estimation of non-uniquely identifiable parameters via exhaustive modeling and membership set theory. Math Computers Simulation 28 479 90. 

According to Justi and Gilbert (40), a model (i) is a non-unique partial representation of an object, an event, a process or an idea (ii) can be changed ... 

If we assume that the phases present are formed at non-unique conditions of pressure and temperature, i.e., slight variations of one or the other will not change the kind or number of phases present, the degrees of freedom for the intensive physical variables (N) is 2 (F 2). As each component becomes completely mobile, it is withdrawn from C and becomes C. If we insist upon non-unique conditions for the intensive variables,... 

It is evident then that the number of phases present can be reduced by restricting the intensive variables to non-unique or inter-dependent conditions and thus a general case" is valid. As the number of chemical components which are perfectly mobile (intensive variables) increases, the number of phases will be decreased. Thus a rock banded in mono-mineralic zones, such as is commonly found in hydrothermal deposits, indicates a system with few inert components. The zonation most commonly represents gradients of the extensive variables temperature and chemical activity of ionic species in aqueous solution. However, it is not always easy to determine which variables are active in a given sequence of mineral bands related to an alteration source. 

M. Shapiro If the resonances overlap to such an extent that we can no longer break the frequency resolved spectrum to a sum of Fano lines in a unique way, then my analysis would not be unique. However this is an extreme situation and even in this case one can try to fit the spectrum (admittedly in a non-unique way) to a sum of Fano lines (or complex energy poles). 

As x grows, the number of extremal polycycles in the non-unique case grows. For example, for x = 13,14, and 15, the polycycles C + ), C + Ez, and Ci + E3 are also extremal. It would be interesting to extend the notion of densities to infinite (r, )-polycycles, but it is sometimes impossible to define densities for infinite settings (see [FeKuKu98, FeKu93] for a possible methodology in the hyperbolic case). 

FIGURE 6 Demonstration of the relation between the function and the shape of the phase plane, (a) Phase plane contours of the function d> = (I - 0 - ft) = A. (b) Phase portrait for the special case of negligible desorption and identical absorption rates (< i = a2 = A -y, = y2 = 0). (c) Shape of the non-unique steady-state reaction rates for variations in reactant partial pressures for the case of negligible desorption. 

Non-uniqueness of the Pseudopotential.—The Phillips-Kleinman pseudopotential contains operators 2( v- k) k)<.p which shifts the eigenvalues of the core func-... 

The non-uniqueness problem is more apparent in the case where the pseudopotential is to be defined by localizing the non-local Hartree-Fock (HF) potential due to... 

However, it is known that the open-shell hamiltonian itself is non-unique.34 In particular this means that the eigenvalues are dependent on the particular form of the hamiltonian for which the orbitals are canonical. Any attempt to use the extended form of the Phillips-Kleinman projection operator, which for open-shell cases can be written... 

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