Matrix admissible

In general for a set of nonlinear equations the necessary condition for determinancy is that there exists at least one admissible set of output variables for the equations (C7, S4). We can think of an output variable as that variable for which a given equation is solved either by an iteration process or by an elimination process. The set of all such assigned pairs of variables and equations is called the output set. Clearly an admissible output set must satisfy the conditions (i) each equation has exactly one output variable, (ii) each variable appears as the output variable of exactly one equation, (iii) each output variable must occur in the assigned equations in such a manner that it can be solved for uniquely. Such an output set (circled entities) is illustrated in terms of an occurrence matrix in Fig. 4. Algorithms for finding output sets have been published by Steward (S4) and Gupta et al. (G8). 

A matrix B is called admissible (with respect to B) if it can be obtained by fixing the free parameters of B at some particular value. The symbol ( ) denotes matrices with fixed elements (matrix in the usual sense). 

The maximal rank of an (m x g) matrix having no specified structure is equal to min (m, g). The inclusion of the structure into the problem makes it possible for matrices to have less than full rank, independent of the values of the free parameters, as was shown by Schields and Pearson (1976). Therefore, a structured matrix B has full generic rank if, and only if, there exists an admissible matrix B with full rank. 

Separability can be exploited even with admission of relativistic effects, by using the standard density matrix formalism with a simple extension to admit 4-component Dirac spin-orbitals this opens up the possibility of performing ab initio calculations, with extensive d, on systems containing heavy atoms. 

Since with the fixed value of vector Pbr and the lack of constraint (62) the admissible region of solutions is a polyhedron, F reaches its minimum at one of its vertices. With the rank of matrix A equal to m-1 and n unknowns the reference solution contains no less than n- m-1) zero components, which equals the number of chords of the system of independent loops of the network graph. In this case the graph tree is a polyhedron vertex and the optimal variant should be among the set of trees of a redundant scheme. 

Drugs placed on the surface of hair must have some mechanism for entry into the hair matrix. If a solid is placed on the hair surface, most of it can be readily removed. However, after the deposition of the solid drugs onto the hair surface, hair may be bathed at some point in an aqueous media, be that sweat, sebum, normal hygienic solutions, or during the hair analysis procedure. Alternatively, an individual might come in contact with a solution of a drug by transfer of sweat from another individual. It is for these reasons that we and others studied solution phase transfer as the vector for admission and incorporation of drugs into hair. 

Because the vector m is constrained by the mass conservation requirement = const, the space of possible m values has — 1 dimensions. If the number of independent chemical reactions, R, is less than - 1, then some vectors m are not accessible at some assigned M this, as will be seen, has important consequences in the consideration of heterogeneous chemical equilibria. Now consider the special case where R = N — 1, so that indeed all admissible m s are accessible. Because the kernel of a contains only the zero vector, there exists an M X N matrix A such that... 

The set of components that can (but generally need not) participate in the admissible reactions is a priori fixed by the given mixture, along with their (possibly only conventional) chemical formulae, thus also the species formula masses are given. According to (C.20), let be the number of atoms of element in. If there are H elements E, present in the formulae Q then the H X K matrix A of elements is the atom matrix of the set of species C j, for arbitrarily given orders of the indices. 

We have thus completed the classification. The group I of columns represents the observable variables, the group II the unobservable ones. In this manner, the canonical format of the extended matrix enables one to identify uniquely the following invariants of the given linear system (independent of the admissible equivalent transformations). 

It is not difficult to show that if the condition (8.1.12) is fulfilled then y and rankB = 2 in the whole admissible region, in particular on iM. In the linearized system, we thus obtain the degree of redundancy equal to 1. On the other hand, so long as > 0, we obtain again the condition (8.1.14) and this condition satisfied, we have m >0 and m > 0. Further, because the two columns of B are linearly independent and because on as shown above, with the third column of (8.5.3) the rank of the matrix equals also 2, the m3-column of the Jacobi matrix is a linear combination of the columns of the new matrix B (8.5.6). According to (7.1.27), at points of Mthe. measured variable m3 can be qualified as nonredundant, in accord with the tentative qualification before formula (8.1.16). But observe that at points zi M, the matrix (8.5.3) is of rank 3, thus in the linearized system, again by (7.1.27), m3 will be qualified as redundant. If... 

We suppose that the state vector z can take its values in some N-dimensional interval Vet/ where ll is the admissible region (8.5.8). The interval can be assessed as some neighbourhood of a vector Zq e fSf. A first information can be obtained in the same manner as above, in the linear case. Taking different Zq e we can examine the behaviour of the Jacobi matrix Dg(Zo) on fW (restricted to t thus on r U). [We can also, in the case of balance models, start from different values of the independent parameters representing the degrees of freedom and determining Zq e fW see Sections 8.2 and 8.3. But such procedure may be rather tedious.] In the reconciliation, however, also the behaviour of Dg(z) in a neighbourhood of the solution manifold is relevant. 

The theoretical concepts of redundancy/nonredundancy are more tricky see Chapter 8, Section 8.5, finally Subsection 8.5.4. In practice, assuming the covariance matrix F diagonal, the concepts can be replaced by those of redundancy (adjustability)/nonadjustability. So if, in a series of measurements, a variable Xh is found nonadjustable (10.3.14) then it is, almost certainly, nonadjustable in the whole admissible region. Otherwise, at some points a variable can be also almost nonadjustable see Remark to Section 10.5 below. 

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