Uniqueness of the solution

There are cases for which more than one solution is found, and it is possible that both may possess physical reality under certain conditions [12] (this will arise again in chapter 11). Furthermore, the Hartree-Fock method can be made multiconfigurational, i.e. several configurations can be mixed or superposed. An electron is then shared between different states, which goes beyond the independent particle approximation. The self-consistent method allows the mixing coefficients to be determined, but the configurations to be included must be specified at the outset, and there is no simple prescription as to which ones should be chosen or left out. 

Another important question, which we outlined above, is the uniqueness of the solution. Again, it seems that mother nature dislikes simple solutions as a rule, the solutions of geophysical inverse problems are not unique  

In analysing the non-uniqueness problem, we should distinguish betw een the two classes of inverse problems we introduced above the inverse model (or inverse scattering) problem and the inverse source problem. The advantage of the inverse 

The inverse source problem is more ambiguous because there usually exists a source distribution that generates a zero external field. This type of source is called a nonradiating source. The detailed analysis of the analytical properties of the nonradiating source was given in a classical paper by Bleistein and Cohen (1976). Following this paper, we can easily demonstrate this fact as applied to inverse source problems for an acoustic field. 

Let us assume that we are seeking a nonradiating source for an acoustic wave equation  

Let W v,t) be a function which is zero outside some finite domain Dq  

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The algebraic solution is the classical fitting technique, as exemplified by the linear regression (Chapter 2). The advantage lies in the clear formulation of the numerical algorithm to be used and in the uniqueness of the solution. If one is free to choose the calibration concentrations and the number of... 

Fitting luminescence decay data to sums of exponentials, even with rather good statistics, can present very serious problems in data fitting and in uniqueness of the solutions. This difficulty can severely cloud interpretation of data from... 

There are a number of unsatisfactory features about this procedure which it is important to examine. The first is the uniqueness of the solution. From a fundamental viewpoint, we may believe that the Uniqueness Theorem in electromagnetism suggests that there is indeed only one possible perfect match between experiment and simulation. However, even if this is the case, we can never have sufficiently perfect data for this stringent condition to be valid. All data are intrinsically statistically noisy, have a non-zero background and a finite range of wavevector covered. In practice, there can be no traly unique solution and this immediately leads to the second problem, that of local minima. 

The uniqueness of the solution will be assured if the Jacobian, J = dxrld r , does not vanish and this will be shown in the following theorem. 

Proof of the Uniqueness of the Solution of the Equations of the Law of Mass Action... 

With this we also establish the uniqueness of the solution of the LMA equations for an ideal system at constant pressure. 

It can be seen that the solution of the problem of the energy-optimal guiding of the system from a chaotic attractor to another coexisting attractor requires the solution of the boundary-value problem (33)-(34) for the Hamiltonian dynamics. The difficulty in solving these problems stems from the complexity of the system dynamics near a CA and is related, in particular, to the delicate problems of the uniqueness of the solution, its behaviour near a CA, and the boundary conditions at a CA. 

Note also that for biochemical systems, uniqueness of the solutions prevails over the whole range of systems with Michaelis-Menten kinetics. 

This method is absolutely analogous to one-site CPA on replacing the scalar quantities, such the self-energy, with nxn matrices. Therefore, one may show158 that the MCPA is also analytic and satisfies the necessary elementary conditions positivity of the spectrum, uniqueness of the solution, sum rules, etc. The self-consistency relation is obtained in a manner analogous to that of the one-site CPA (Section IV.A.4) it will be detailed in the calculation of the n x n matrix elements. 

As for the non-uniqueness of the solution, there is no method that can bypass this inherent problem. In inverse problems, one of the common practices to overcome the stability and non-uniqueness criteria is to make assumptions about the nature of the unknown function so that the finite amount of data in observations is sufficient to determine that function. This can be achieved by converting the ill-posed problem to a properly posed one by stabilization or regularization methods. In the case of groundwater pollution source identification, most of the time we have additional information such as potential release sites and chemical fingerprints of the plume that can help us in the task at hand. 

To analyze a physical problem analytically, we must obtain the governing equations that model the phenomenon adequately. Additionally, if the auxiliary equations pertaining to initial and boundary conditions are prescribed those are also well-posed, then conceptually getting the solution of the problem is straightforward. Mathematicians are justifiably always concerned with the existence and uniqueness of the solution. Yet not every solution of the equation of motion, even if it is exact, is observable in nature. This is at the core of many physical phenomena where ohservahility of solution is of fundamental importance. If the solutions are not observable, then the corresponding basic flow is not stable. Here, the implication of stability is in the context of the solution with respect to infinitesimally small perturbations. 

Alternatives Challenge the uniqueness of the solution. Is this the only way to provide the needed feature/functionality, or are there alternatives ... 

The solution of a problem in linear viscoelasticity requires the determination of the stress, strain, and displacement histories as a function of the space coordinates. The uniqueness of the solution was proved originally by Volterra (11). The analysis carried out in this chapter refers exclusively to isotropic materials under isothermal conditions. As a rule, it is not possible to give a closed solution to a viscoelastic problem without previous knowledge of the material functions. The experimental determination of such functions and the relationships among them have been studied in a specific way in separate chapters, and therefore the reader s knowledge of them is assumed. At the same time, the methods of analysis carried out in this chapter and in Chapter 17 will allow us to optimize the calculation of the material functions. 

The question of the uniqueness of the solution can be illustrated by the following formulae. Assume that we have two different models, mj and m2, and two different sources, Si and S2, which generate the same data do ... 

We will now prove the uniqueness of the solution of the three initial-value problems formulated at the beginning of this section. Let us assume that any of these problems admits two solutions. Then we can obtain the difference between those solutions U (r,t). 

This completes the proof of the uniqueness of the solution of three principal initial-value problems for a bounded domain. 

Uniqueness of the solution of the wave propagation problem based on radiation... 

In the case of the unbounded domain coinciding with the entire space, the radiation conditions ensure the uniqueness of the solution of the wave propagation problem. In order to prove this statement, we will write the internal integral in the Kirchhoff formula (13.112) in the frequency domain, using the convolution theorem, as follows ... 

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