Uniqueness theorems

Ciarlet P.G., Sanchez-Palencia F. (1996) An existence and uniqueness theorem for the two-dimensional linear membrane shell equations. J. Math. Pures Appl. 75 (1), 51-67. 

Once the initial state x(f = 0) of the system is specified, future states, x(t), are uniquely defined for all times t . Moreover, the uniqueness theorem of the solutions of ordinary differential equations guarantees that trajectories originating from different initial points never intersect. 

The second axiom, which is reminiscent of Mach s principle, also contains the seeds of Leibniz s Monads [reschQl]. All is process. That is to say, there is no thing in the universe. Things, objects, entities, are abstractions of what is relatively constant from a process of movement and transformation. They are like the shapes that children like to see in the clouds. The Einstein-Podolsky-Rosen correlations (see section 12.7.1) remind us that what we empirically accept as fundamental particles - electrons, atoms, molecules, etc. - actually never exist in total isolation. Moreover, recalling von Neumann s uniqueness theorem for canonical commutation relations (which asserts that for locally compact phase spaces all Hilbert-space representations of the canonical commutation relations are physically equivalent), we note that for systems with non-locally-compact phase spaces, the uniqueness theorem fails, and therefore there must be infinitely many physically inequivalent and... 

In the treatment of steady-state pipeline network problems so far we have tacitly assumed that there is a unique solution for each problem. For certain types of networks the existence of a unique solution can indeed be rigorously established. The existence and uniqueness theorems for formulation C were proved by Duffin (DIO) and later extended by Warga (Wl). In Warga s derivation the governing relation for each network element assumes the form,... 

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. 

Given the importance of the work of Misra, Prigogine, and co-workers, I am providing in the Appendix a detailed, but simplified, analysis. The main result of these works can be concisely formulated as a unique theorem ... 

As we mentioned before, when a biparticle quantum system AB is in a pure state, there is essentially a unique measure of the entanglement between the subsystems A and B given by the von Neumann entropy S = —Tr[p log2 PaI- This approach gives exactly the same formula as the one given in Eq. (26). This is not surprising since all entanglement measures should coincide on pure bipartite states and be equal to the von Neumann entropy of the reduced density matrix (uniqueness theorem). 

If a solution is found for the initial and boundary conditions, there is a uniqueness theorem that justifies the assumption. Whether a solution can be found using separation of variables depends on whether the boundary conditions follow the symmetry of the separation variables. 

More recently, Boley (B9) shows that the method can be extended to find minimum and maximum bounds for ablation rates. The proof is based on a uniqueness theorem deriving from a theorem due to Picone (P3) and leads to the physically obvious result that the higher the heat input rate, the higher the melting rate. The procedure consists of forming a lower bound for the known arbitrary heat input function in terms of a sequence of constant heat flux periods, for which, as noted above, the solution can be written in terms of integrals of the error function. Upper bounds are constructed in a similar manner. 

A more complete model of equilibrium has been studied by J. H. Hancock T. S. Motzkin [25], who prove some valuable existence and uniqueness theorems. 

Blanchard and Briining [26] bring the history of the calculus of variations into the twentieth century, as the source of contemporary developments in pure mathematics. A search for existence and uniqueness theorems for variational problems engendered deep studies of the continuity and compactness of mathematical entities... 

ORDINARY DIFFERENTIAL EQUATIONS, I.G. Petrovski. Covers basic concepts, some differential equations and such aspects of the general theory as Euler lines, Ariel s theorem, Beano s existence theorem, Osgood s uniqueness theorem, more. 45 figures. Problems. Bibliography. Index, xi + 232pp. 5X 8H. 

When we use the neutron-scattering data more completely, what we can obtain is the mean square Fourier transform of the distribution functions for individual molecules and there is no uniqueness theorem for the problem of inverting that. Even if there was, knowing the distribution functions would not tell us the conformations. All we can do is to make models and see whether they will fit the scattering data within experimental error. If they don t, they are wrong. If they do, they are not necessarily right. You must call in all aids you can to limit the models to be tested. It is essential that they should pass tests of steric acceptability, as everyone who uses the corresponding trial-and-error method for X-ray crystal structure determination knows. 

Fortunately, the situation is different for inverse model (or inverse scattering) problems. There are many favorable situations when inverse geophysical problems happen to be unique. These situations are outlined by corresponding uniqueness theorems. For example, I list below some important uniqueness theorems of geophysics. 

Novikov (1938) proved the uniqueness theorem for a star-type body with the given homogeneous density distribution p(r) = pg. The theorem states that if it is known that the gravity field is generated by a star-type body with the a given constant density, the gravity inverse problem has a unique solution. In the case of spherical balls, this result is obvious according to formula (1.39), if pj = P2, then R — H-2 ... 

There are three famous uniqueness theorems for the electromagnetic inverse problem (Berdichevsky and Zhdanov, 1984) ... 

Note, also, that the proof of these theorems, including the 3-D case, can be obtained as a special case of a more general mathematical uniqueness theorem of inverse problems for general partial differential equations. We will outline this more... 

The inverse problem in this case is formulated as recovery of the unknown coefficient 7 of the elliptic operator from the known values of the field p(r, u>) in some domain or in the boundary of observations. In a number of brilliant mathematical papers the corresponding uniqueness theorems for this mathematical inverse problem have been formulated and proved The key result is that the unknown coefficient 7 (r) of an elliptic differential operator can be determined uniquely from the boundary measurements of the field, if 7 (r) is a real-analytical function, or a piecewise real-analytical function. In other words, from the physical point of view we assume that 7 (r) is a smooth function in an entire domain, or a piecewise smooth function. Note that this result corresponds well to Wcidelt s and Gusarov s uniqueness theorems for the magnetotelluric inverse problem. I would refer the readers for more details to the papers by Calderon (1980), Kohn and Vogelius (1984, 1985), Sylvester and Uhlmann, (1987), and Isakov (1993). 

Returning to the seismic wavefield inverse problem, we can assume that, based on general mathematical uniqueness theorems, the seismic inverse problem. 

Unfortunately, the number of uniqueness theorems for geophysical inverse problems is relatively small. These theorems cover very specific geophysical models. In practical situations we should rely on a more simple but important property of inverse problem solution. Following Hjelt (1992), we call this property practical uniqueness. It can be described using the following simple considerations. 

Sylvester, J., and G. Uhlmann, 1987, Global uniqueness theorem for an inverse boundary value problem Ann. Math., 125, 153-169. 

Thus we have proved the uniqueness theorem for the boundary-value electromagnetic problem. 

This completes the proof of the uniqueness theorem for the unbounded domain. 

Note that the solution of equation (10.62) and, correspondingly, equation (10.65), can be nonunique in a general case, since there may exist m (r) distributions that produce a zero external field, similar to the existence of the current distributions producing zero external field demonstrated in the beginning of this chapter. However, according to the uniqueness theorem by Gusarov (1981, see Chapter 1), the excess conductivity Aa (r) can be found uniquely in the class of piecewise-analytic functions. Therefore,... 

A similar approach can be used to prove the uniqueness theorem for the solution of the initial value problem in an unbounded domain. 

Chastened by this example, we state a theorem that provides sufficient conditions for existence and uniqueness of solutions to x = /(x). Existence and Uniqueness Theorem Consider the initial value problem X = fix), X(0) = Xg. ... 

For proofs of the existence and uniqueness theorem, see Borrelli and Coleman (1987), Lin and Segel(1988), or virtually any text on ordinary differential equations. 

There are various ways to extend the existence and uniqueness theorem. One can allow f to depend on time t, or on several variables x,..., x , One of the most useful generalizations will be discussed later in Section 6.2. 

Why is this the general solution First of all, it is a linear combination of solutions to X = Ax, and hence is itself a solution. Second, it satisfies the initial condition x(0) = Xq, and so by the existence and uniqueness theorem, it is the only solution. (See Section 6.2 for a general statement of the existence and uniqueness theorem.)... 

We have been a bit optimistic so far—at this stage, we have no guarantee that the general nonlinear system x = f(x) even has solutions Fortunately the existence and uniqueness theorem given in Section 2.5 can be generalized to two-dimen-... 

Existence and Uniqueness Theorem Consider the initial value problem X = f(x), x(0) = Xq. Suppose that f is continuous and that all its partial derivatives /dXj, J, J = 1,..., n, are continuous for x in some open connected set D c R". Then for x, e D, the initial value problem has a solution x(t) on some time interval... 

The existence and uniqueness theorem has an important corollary different trajectories never intersect. If two trajectories did intersect, then there would be two solutions starting from the same point (the crossing point), and this would violate the uniqueness part of the theorem. In more intuitive language, a trajectory can t move in two directions at once. 

As mentioned above, a set of experimental data does not necessarily correspond to a unique molecular structure. Moreover, even unphysical structures may be consistent with a set of experimental data. It is therefore necessary to carefully choose a set of constraints to limit the number of possible structures. The uniqueness theorem of statistical mechanics [30, 31] provides a guide to the number and type of constraints that should be appfied in the RMC method in order to get a unique structure [32]. For systems in which only two- and three-body forces are important, the uniqueness theorem states that a given set of pair correlation function and three-body correlation function determines aU the higher correlation functions. In other words, assuming that only two- and three-body forces are important, the RMC method must be implemented along with constraints that describe the three-body correlations [27]. 

Henderson, R.L. (1974). Uniqueness theorem for fluid pair correlation-functions. Phys. Lett., 49A, 197-8. 

Arimoto, S., Spivakovsky, M., Ohno, H., Zizler, P., Taylor, K.F., Yamabe, T. and Mezey, P.G. (2001) Structural analysis of certain linear operators representing chemical network systems via the existence and uniqueness theorems of spectral resolution. VI. Int. J. Quant. Chem., 84, 389-400. 

The modeling of stationarily operating enzyme electrodes is a special case of nonstationary modeling. In this context, stationary means the consideration of the behaviour after infinite time (steady state), i.e., solutions have to be found for which the time derivative has to be zero. For all models considered, the existence of a unique and stable mathematical solution is assumed. Since the respective partial differential equations are linear initial boundary problems, known existence and uniqueness theorems may be used (Kamke, 1956 Fife, 1979 Ozisik, 1980). 

The basic idea can be formulated as a formal theorem the Uniqueness Theorem... 

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