Theorem of uniqueness

Of course, both statements can be proved from the theorem of uniqueness for the attraction field. In addition, it is appropriate to comment a linear function reaches its maximum at terminal points of the interval. The same behavior is observed in the case of harmonic functions, which cannot have their extreme inside the volume. Otherwise, the average value of the function at some point will not be equal to its value at this point, and, correspondingly, the Laplacian would differ from zero. At the same time, saddle points may exist. 

The procedure of determination of these conditions, based on the use of Gauss s formula, is called the theorem of uniqueness. 

In essence, the theorem of uniqueness formulates the steps that have to be undertaken to find the attraction field and its potential. These steps are... 

At the same time it is worth to notice that in modern numerical methods of a solution of boundary value problems, based on replacement of differential equations by finite difference, these steps are performed simultaneously. In accordance with the theorem of uniqueness, the field inside the volume V is defined by a distribution of masses inside this volume and boundary conditions, and correspondingly it is natural to derive an equation establishing this link. With this purpose in mind we will again proceed from Gauss s theorem,... 

As we already know a determination of the function G q, p) satisfying all these conditions represents a solution of the boundary value problem and in accordance with the theorem of uniqueness these conditions uniquely define the function G q, p). In general, a solution of this problem is a complicated task, but there are exceptions, including the important case of the plane surface Sq, when it is very simple to find the Green s function. Let us introduce the point s, which is the mirror reflection of the point p with respect to the plane of the earth s surface, Fig. 1.10, and consider the function G (p, q,. s) equal to... 

As follows from Chapter 1, we have formulated an external Dirichlet s boundary value problem, which uniquely defines the attraction field. In this light it is proper to notice the following. In accordance with the theorem of uniqueness its conditions do not require any assumptions about the distribution of density inside of the earth or the mechanism of surface forces between the elementary volumes. In particular, these forces may not satisfy the condition of hydrostatic equilibrium. 

Thus, we have demonstrated that the potential C/g is a solution of Laplace s equation and satisfies the boundary condition at the surface of the ellipsoid of rotation and at infinity. In other words, we have solved the Dirichlet s boundary value problem and, in accordance with the theorem of uniqueness, there is only one function satisfying all these conditions. 

This equation is a second-order linear partial-differential equation with a rich mathematical literature [1]. For a large class of initial and boundary conditions, the solution has theorems of uniqueness and existence as well as theorems for its maximum and minimum values.1... 

Tikhonov s theorem of uniqueness for a one-dimensional (1-D) model (Tikhonov... 

Weidelt s theorem of uniqueness for a two-dimensional model with an electrical conductivity described by an analytic function (Weidelt, 1978) ... 

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. 

Classical methods of mathematical physics are employed at the first stage. Numerous physical problems lead to mathematical models having no advanced methods for solving them. Quite often in practice, the user is forced to. solve such nonlinear problems of mathematical physics for which even the theorems of existence and uniqueness have not yet been proven and some relevant issues are still open. 

The ground-state electronic density p(r) is uniquely related to the external potential Vext(r) as stated by the fundamental theorems of DFT [1,2,8]. At zero field, the external potential of an atom is due to its nuclei and vext(r) = —Ze2/r where Z is the nuclear charge. It is shifted by the quantity V... 

THE FUNDAMENTAL THEOREM OF ARITHMETIC states that every whole number greater than 1 is the product of prime factors. Furthermore, these prime factors are unique, and there is exactly one set of prime factors. 

M. Rosina, Some theorems on uniqueness and reconstruction of higher-order density matrices, in Many-Electron Densities and Reduced Density Matrices (J. Cioslowski, ed.), Kluwer Academic/ Plenum Pubhshers, New York, 2000, pp. 19—32. 

According to a theorem of Friedrichs,375 to each closed quadratic form J there is uniquely determined a self-adjoint operator R such that 1) 5)(/Z)C3)(./) 2) J[J,g = (Hf, gj for every / 2)(i ) and g 6 3)(V) 3) R is bounded below with the same lower bound as J. R is called the self-adjoint operator belonging to the quadratic form J. 

Such a definition can, evidently, be extended to any number of routes. It is clear that if A(1), A(2), A<3) are routes of a given reaction, then any linear combination of these routes will also be a route of the reaction (i.e., will produce the cancellation of intermediates). Obviously, any number of such combinations can be formed. Speaking in terms of linear algebra, the reaction routes form a vector space. If, in a set of reaction routes, none can be represented as a linear combination of others, then the routes of this set are linearly independent. A set of linearly independent reaction routes such that any route of the reaction is a linear combination of these routes of the set will be called the basis of routes. It follows from the theorems of linear algebra that although the basis of routes can be chosen in different ways, the number of basis routes for a given reaction mechanism is determined uniquely, being the dimension of the space of the routes. Any set of routes is a basis if the routes of the set are linearly independent and if their number is equal to the dimension of the space of routes. 

Usually statements of problems on chemical equilibrium include the initial amounts of several species, but this doesn t really indicate the number of components. The initial amounts of all species can be used to calculate the initial amounts of components. The choice of components is arbitrary because /xA or fiB could have been eliminated from the fundamental equation at chemical equilibrium, rather than fiAB. However, the number C of components is unique. Note that in equation 3.3-2 the components have the chemical potentials of species. This is an example of the theorems of Beattie and Oppenheim (1979) that (1) the chemical potential of a component of a phase is independent of the choice of components, and (2) the chemical potential of a constituent of a phase when considered to be a species is equal to its chemical potential when considered to be a component. The amount of a component in a species can be negative. 

Therefore, the two boundary conditions can be specified at the same boundary, and it is not necessary to specify them at different locations. In fact, the fundamental theorem of linear ordinary differential equations guarantees that a unique solution exists when both conditions are specified at the same location. 

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. 

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... 

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. 

A more mathematical approach would invoke a theorem of differential equations, which says that a second order partial differential equation that is as nice as the one we have here, with two initial conditions of the form we just used, must have a unique solution. The branch of mathematics you would have to study to learn this theorem is called partial differential equations sometimes, or if the professor plans to give you the most general version, the area of study might be called differential operators on manifolds. Mathematically, this type of theorem makes the claim for our model of nature that the physical explanation is attempting to make for nature herself. Then we would again invoke the theory of Fourier series to tell us that the sines and cosines are good enough to do the job. 

More generally, the Hohenberg-Kohn theorem of SDFT states that in the presence of a magnetic field B r) that couples only to the electron spin (via the familiar Zeeman term), the ground-state wave function and all ground-state observables are unique functionals of n and m or, equivalently, of n- and. In the particular field-free case, the SDFT HK theorem still holds and continues to be useful, e.g., for systems with spontaneous polarization. Almost the entire... 

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]. 

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. 

---