Over-determination

It is also possible to satisfy this relation and have the structure be underdetermined and locally over determined. Examples of two-dimensional structures are shown in Table 1. 

Here the pair-force fj (r, r -) is unknown, so a model pair-force fij(r , rj, p, P2 pm) is chosen, which depends linearly upon m unknown parameters p, p2 - Pm- Consequently, the set of Eq. (8-2) is a system of linear equations with m unknowns p, P2 - - Pm- The system (8-2) can be solved using the singular value decomposition (SVD) method if n > m (over-determined system), and the resulting solution will be unique in a least squares sense. If m > n, more equations from later snapshots along the MD trajectory should be added to the current set so that the number of equations is greater than the number of unknowns. Mathematically, n = qN > m where q is the number of MD snapshots used to generate the system of equations. 

For each studied system, the range of distances at which pair-forces are modeled as cubic splines is given, as well as mesh sizes and the number of resulting unknowns, the number of configurations included in a set to generate an over-determined system of equations, and the number of sets for which the least squared solution is averaged. 

It is important to stress that for this to work, the independently known matrix A of absorptivity coefficients needs to be square, i.e. it has previously been determined at as many wavelengths as there are chemical species. Often complete spectra are available with information at many more wavelengths. It would, of course, not be reasonable to simply ignore this additional information. However, if the number of wavelengths exceeds the number of chemical species, the corresponding system of equations will be over determined, i.e. there are more equations than unknowns. Consequently, A will no longer be a square matrix and equation (2.22) does not apply since the inverse is only defined for square matrices. In Chapter 4.2, we introduce a technique called linear regression that copes exactly with these cases in order to find the best possible solution. 

Almost all crystals suitable for X-ray powder diffraction can be studied by electron diffraction. Several of the most demanding problems with powder diffraction are overcome by electron diffraction. There is no problem of overlapping reflections in electron diffraction and all diffraction spots can be unambiguously indexed. There is no problem of underdetermination (less data than unknown parameters) for electron diffraction since 10-100 times more reflections than parameters can be obtained by ED, whereas in X-ray powder diffraction the over-determination is close to one. 

We note in passing that many more spectra will be accumulated compared to the observable species present. This represents an over-determined problem, one of many qualities present in real physical ill-posed problems. 

The goal of the analysis is to determine the source contributions m. by inverting Eq. (1). Usually there are more equations than unknowns, that is, more measured elemental concentrations p. than source contributions m.. The values of m. for this over-determined system can be el tlmated using a leait squares method in which the following assumptions are made ... 

In a previous work [33] we suggest an effective approach to study of conditional symmetry of the nonlinear Dirac equation based on its Lie symmetry. We have observed that all the Poincare-invariant ansatzes for the Dirac field i(x) can be represented in the unified form by introducing several arbitrary elements (functions) ( ), ( ),..., ( ). As a result, we get an ansatz for the field /(x) that reduces the nonlinear Dirac equation to system of ordinary differential equations, provided functions ,( ) satisfy some compatible over-determined system of nonlinear partial differential equations. After integrating it, we have obtained a number of new ansatzes that cannot in principle be obtained within the framework of the classical Lie approach. 

If several ( ) charged species i equilibrate across the phase boundary, the set of Eqns. (4.116) has to be solved simultaneously for i = 1,2,..This does not lead to an over-determination of Atpb but ensures that the chemical potentials of the electroneutral combinations of the ions (= neutral components of the system) are constant across the interface. The electric structure (space charge) of interfaces will be discussed later. 

Ratanathanawongs and Crouch [19] have described an on-line post-column reaction based on air-segmented continuous flow for the determination of phenol in natural waters by high performance liquid chromatography. The reaction used was the coupling of diazotised sulphanilic acid with the phenol to form high coloured azo dyes. The detection limit for phenol was 17pg L 1 which represents a 16-fold improvement over determination of phenol with ultraviolet detection. 

Treating the positional parameters and the elements of the thermal displacement tensor as variables, a best fit of the observed to the calculated structure factors in a least squares sense is determined. As this is a non-linear procedure, it is essential to over determine the problem. For a routine structure... 

Generally speaking the classical alkaline hydrolysis reaction followed by back-titration could be used to determine steroid esters. The results for 21-acyloxycortico-steroids (where R = H, OH) are however quite unsatisfactory. The reason behind the unacceptable result is the auto-oxidation of the a-ketol side-chain in an alkaline medium. Because of the formation of acidic products in side reactions, there is an over consumption of base in the titration and a corresponding over determination of the amount of steroid. 

Fully ab initio variational calculations, using a wavefunction consisting of VB structures and with optimization of both orbitals and structure coefficients, may be carried out by a variety of methods these range from the direct spin-free approach, in which only the permutation symmetry of the wavefunction is used (as in the pre-Slater era), to methods in which the structures (with spin factors included) cure expanded over determinants of spin-orbitals. 

Another factor to be taken into account is the degree of over determination, or the ratio between the number of observations and the number of variable parameters in the least-squares problem. The number of observations depends on many factors, such as the X-ray wavelength, crystal quality and size, X-ray flux, temperature and experimental details like counting time, crystal alignment and detector characteristics. The number of parameters is likewise not fixed by the size of the asymmetric unit only and can be manipulated in many ways, like adding parameters to describe complicated modes of atomic displacements from their equilibrium positions. Estimated standard deviations on derived bond parameters are obtained from the least-squares covariance matrix as a measure of internal consistency. These quantities do not relate to the absolute values of bond lengths or angles since no physical factors feature in their derivation. 

The conunonly used methods for solving the phase problem required for stmcture solutions are the direct methods and Patterson maps. Direct methods use relationships between phases such as triplets (0 = 4>h + 4>k + -h-k The probabihty of 0 >= 0 increases with the magnitude of the product of the normalized stmcture factors of the three reflections involved. Once these triplets associated with high certainty are identified based on diffraction intensities, they are used to assign new phases based on a set of known phases. Since the number of phase relationships is large the problem is over determined. Another approach is based on the Sayre equation, which is derived based on the relationship between the electron density and its square ... 

A convenient way to derive the explicit expression for the time course of the reaction in the general three component system [Eq. (3)] is first to reduce the matrix K [Eq. (10)] to a 2 X 2 matrix. This is possible because the constraint given by the law of conservation of mass tells us that the system is over determined with regard to amounts, i.e., only two of the three amounts Ui, U2, and as need be specified to determine the system. First, let us shift the origin of the natural coordinate system to the equi-... 

In this paper, an inverse problem for galvanic corrosion in two-dimensional Laplace s equation was studied. The considered problem deals with experimental measurements on electric potential, where due to lack of data, numerical integration is impossible. The problem is reduced to the determination of unknown complex coefficients of approximating functions, which are related to the known potential and unknown current density. By employing continuity of those functions along subdomain interfaces and using condition equations for known data leads to over-determined system of linear algebraic equations which are subjected to experimental errors. Reconstruction of current density is unique. The reconstruction contains one free additive parameter which does not affect current density. The method is useful in situations where limited data on electric potential are provided. 

Equation (9.56) can also be derived by considering the least squares problem as an over-determined system of equations. That is, we would like to find 0 such that... 

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