Triangular form

Let us suppose that we can convert the n x coefficient matrix in equation system (6.5) into an upper triangular form as... 

Solution of Ax = b hy Using LU Factorization Suppose that the indicated system is compatible and that A = LU (the case PA = LU is similarly handled and amounts to rearranging the equations). Let z = Ux. Then Ax = LUx = h imphes that Lz = h. Thus to solve Ax = h we first solve Lz = h for z and then solve Ux = z for x. This procedure does not require that A be invertible and can be used to determine all solutions of a compatible system Ax = h. Note that the systems Lz = h and Ux = z are both in triangular forms and thus can be easily solved. 

The macrocyclic borate ring is sandwiched by two smaller (V0)60i2 rings of triangular form via six axial B-(/X3-0)-V bonds, of which three are directed to one side of the macrocycle and three to the other one. On average, ten of the... 

Next, with the second row in C2 kept intact, the upper triangular form is achieved by operating on the third row ... 

Here we have introduced N(a>) as the Fourier transform of the noise n(x). Frieden assumed a simple triangular form for t(co) and constant noise-to-signal ratio for all frequencies. Goldman (1953) provides an appropriate discussion of ensemble averages and the like. 

The next pivot will be [A]4 3 = 4.0, thereby interchanging rows 3 an 4. To eliminate xj from equation 3 we need the single multiplier -2.0/4.0 = -0.5, and obtain the matrix in the desired upper triangular form ... 

In Figure 15.1, some examples of different disciplines and methods relevant to organic research are placed in relation to two methodological dimensions. One is the complexity of the research world described above. The other is the degree of involvement of the researcher or the research unit in its research world. The triangular form illustrates that in simple and well-controlled research worlds, the detached and involved phases of research can be employed in close cooperation, whereas detached, descriptive, historical methods are widely different from involved, interactive, developmental action research methods though they both work with complex research worlds. 

Figure 15.1 Some research disciplines and methods of relevance for organic research ordered according to two methodological dimensions the complexity of the research world (spanning from causal over adaptive to self-reflexive entities) and the degree of involvement of the researcher (adapted from Alroe and Kristensen 2002). The triangular form shows that simple research worlds allow the researcher to intervene and withdraw from the research world in closely connected processes, whereas the involved and detached stances are widely different in relation to complex research worlds.

Very few of these compounds have been characterized by X-ray crystallographic methods. As with alkylperoxo complexes, they can have a linear or a bridged structure, but no triangular form... 

Magnetic Susceptibility of TiNi has been previously observed [39] to be temperature independent and interpreted as due to Pauli spin susceptibility. This categorizes the magnetic property as one that is insensitive to the atomic arrangement. The magnetic susceptibility has the constant values, 2.1 x 10 6 (emu/g) below the Ms and 3.0 x 10"6 (emu/g) above the As temperature. Between these two temperatures a plot of the data has a triangular form but as predicted, no difference is observed between those obtained from complete and incomplete cycles. 

Each set of values /ajg for normal mode p forms a N x N matrix, where N is the number of nuclei. A diagonal term Jaa p represents the contribution which atom a makes to Jp, and the sum Ja/3 p + the contribution due to the coupled motion of the pair of nuclei a and / . The graphical representation as full and empty circles, depending on the sign, in a matrix, in upper triangular form, maps the way nuclear motion creates Raman and ROA intensity in the vibrating molecule [42],... 

The model equations result in an upper-triangular form for the matrix A of Equation 18.6, which may be consequently exploited to construct the general solution. Starting with the equation for i = n and proceeding backwards, the general solution may be determined implicitly as... 

From the rare gas isolation studies of trisilver (8,9), it is thought that the Ag,0 structural isomer has an equilateral triangular form (trapped in a 60° trisubstitutional vacancy site) with an E electronic ground state. Its spectroscopy is best described in terms of a vibronically active, pseudo rotating triatomic, as calculated for Li, (23) and observed for K, (24). 

The first point is that A is normally read from disk in triangular form, and must be transcribed to full square form, a process which is thoroughly vectorisable. The amount of store allowed for each column of the square form of A would normally be N words. However in the case that N is divisable by 8, we allocate N+1 words, which prevents subsequent store conflicts. A similar store allocation algorithm is applied to Q, whose dimension is in general N rows by M columns (M[Pg.26]

Corollary. Let S be any solvable matrix group over an algebraically closed field. Then S has a normal subgroup of finite index which can be put in triangular form. 

As we have seen previously, there is a subtle equilibrium between the trinuclear Ag3 and the dinuclear Ag complex whereas this equihbrium was not observed in the mixed dinuclear AuAg complex. Similarily, the mixed trinuclear AuAg complex apparently exists only in the open form shown in Figure 3.5 and not in the triangular form shown in Figure 3.4 [16]. 

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