Jordan basis

Gauss-Jordan Basis for computing inverse can solve multiple sets of equations. Less efficient for a single set of equations. 

Remark. The Lyapunov function is a universal tool of stability theory. Typically, a proof concerning stability consists of either constructing a Lyapunov function, or proving its existence. Moreover, its applicability is not limited to critical equilibria for example, in our analysis of studying the structurally stable equilibria (Theorem 2.4), we have implicitly shown that the norm of a vector in a Jordan basis is a Lyapunov function. 

Jordan oompared the use of plane wave and oonventional Gaussian basis orbitals within density funotional oaloulations in ... 

Iron(III)-catalyzed autoxidation of ascorbic acid has received considerably less attention than the comparable reactions with copper species. Anaerobic studies confirmed that Fe(III) can easily oxidize ascorbic acid to dehydroascorbic acid. Xu and Jordan reported two-stage kinetics for this system in the presence of an excess of the metal ion, and suggested the fast formation of iron(III) ascorbate complexes which undergo reversible electron transfer steps (21). However, Bansch and coworkers did not find spectral evidence for the formation of ascorbate complexes in excess ascorbic acid (22). On the basis of a combined pH, temperature and pressure dependence study these authors confirmed that the oxidation by Fe(H20)g+ proceeds via an outer-sphere mechanism, while the reaction with Fe(H20)50H2+ is substitution-controlled and follows an inner-sphere electron transfer path. To some extent, these results may contradict with the model proposed by Taqui Khan and Martell (6), because the oxidation by the metal ion may take place before the ternary oxygen complex is actually formed in Eq. (17). 

There is, in fact, no evidence supporting the existence of a cap containing many Tbe GTE molecules, even at very high tubulin concentrations where rapid polymerization is favored. Over an extraordinarily large range of tubulin concentration, O Brien and Erickson observed no accumulation of GTP beyond that anticipated on the basis of the boundary stabilization model of Karr et This was confirmed independently by Jordan and Wilson. ... 

Of course, it is dangerous to exclude the possibility of aluminium polymerization on the basis of calculations. Reality is invariably more complicated than the simplified models put into computers. However, in view of the uncertainties surrounding existing systems, and the doubts thrown by calculations, any well-defined aluminium alkyl claimed to be active should at least be checked, as an isolated complex, for its propensity to olefin insertion vs. chain transfer, e.g,. using the Al-i-butyl/ethene experiment reported by Jordan [15], as explained above. 

Colchicum ritchii from southern Jordan afforded, on extraction of the bulbs and removal of the alkaloids, three nonnitrogenous compounds named colchicone (7), 3-demethylcolchicone (8), and comigerone (9) (12). Because of their close chemical relationship with colchicinoids they are discussed here and are listed in Table I. The structures of these ketones were elucidated mainly on the basis of H-NMR and MS analysis, as well... 

As shown previously analogous equations can be derived in a statistical framework both for localized fermions in a specific pairing mode and/or for bosons subject to a quantum transport environment [7]. The second interconnection regarding the relevance of the basis f is related to the fact that a transformation of form (20) connects canonical Jordan blocks to convenient complex symmetric forms. This will not be explicitly discussed and analysed here except pointing out the possible relationship between temperature scales and Jordan block formation by thermal correlations (see e.g. [7-9,14], for more details). 

If possible, bring 6i into the basis set 6 as the pivotal variable for row i and column i, by applying the modified Gauss-Jordan method described at the end of Section E.3 to the full A-matrix, including the last row and column. The requirements for such a move are LBAS(i) must be zero An must be positive (to ensure a descent of 5) Di and all resulting Dj must exceed ADTOL and every parameter must remain within the permitted region ofEq. (6.4-2). 

The transformed A-matrix quadratic expansion of S in terms of the set Be of current basis parameters. The submatrix A is the inverse of the current basis matrix. The basis determinant Aee is the product of the pivots An that were used in steps of Type 1, divided by the pivots A that were used in steps of Type 3. The use of strictly positive pivots for the Gauss-Jordan transformations ensures a positive definite submatrix A , hence a positive determinant Aee throughout the minimization. 

Jordan and Smith proposed formation of CO2 anion radical by one electron transfer to CO2 molecule as an initial step in CO2 reduction on the basis of their polarographic study, ... 

The exact foundation of modem quantum mechanics was laid in 1925-1926, by Wemer Heisenberg, Max Bom and Pascual Jordan, and by the Austrian physicist Erwin Schrodinger. The latter s formulation in terms of wave functions has, in particular, proved well suited for the description of atoms, molecules and solids, and their interaction with light. Thus, the Schrodinger equation supplies the basis for our understanding of the chemical bond and for the description of, say, atomic and molecular spectra. 

Since most spectroscopic investigations have been concerned with the Ha line, we shall examine in detail the structure to be expected for this line on the basis of the new theory, and review the work done after Heisenberg and Jordan had given the new interpretation to Sommerfeld s formula. 

Kim et al. [2] were apparently the first to apply ab initio methods to (H20)e, carrying out SCF optimizations of the hexamer as early as 1988. However, as it became possible to carry out MP2 calculations with suitable flexible basis sets on water clusters, it became clear that inclusion of electron correlation effects is crucial for describing the bonding in these species [5,11,12,20,23,67,68]. The first MP2 level calculations on (H20)6 were by Tsai and Jordan [12]. Subsequently, Kim and coworkers [69,70], Xantheas et al. [71],... 

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