Augmented paths

ND indices are molecular descriptors calculated from augmented matrices P, P, and P where the first two columns contain (1) the square root of the bond vertex degree 8 and (2) the equilibrium electronegativity of atoms [Nie, Dai et al, 2005]. ND indices are the leading eigenvalues of the symmetrized augmented path matrices Mi, M2, and M3 ... 

Referring to Fig. 9, the effect of the shear is to catalyze the reaction, presumably through suppression of the interfacial barrier by stretching the flow. The latter is believed to reduce the diffusion path, promoting the reaction rate, and hence the rate of increase in the viscosity. A similar effect is produced with temperature as a parameter, which also augments the reaction rate. The modified reaction rate constant in case of any external stimulus or perturbation acting on the system may be computed from the scalar K, where ... 

Observe that we have in this procedure worked out some of the steps previously left to the THEOREM PROVER, The previous procedure involves having the progranmer select a set of inductive assertions and critical points, and then feed this into the computer parts a VERIFICATION CONDITION GENERATOR and a THEOREM PROVER. In this alternative construction we still need inductive assertions as the nature of the Rule of Iteration for WHILE statements shows. Now the inductive assertions are fed directly into the THEOREM PROVER which las been augmented by the special axioms and rules D0,D1,D2,D3 and D4 in addition to all of the usual arithmetic axioms, rules of inference, rules for handling identities and special axioms for the primitives in question (such as the factorial axioms in our example). In effect the THEOREM PROVER works backwards from the output condition and the various inductive assertions using DO - D3 to find what amounts to path verification conditions -... 

It does not seem clear which procedure is easier to implement or more efficient (if and when it works ). Both can be augmented to handle other constructions in any particular programming language. The first procedure is augmented by changing the PATH VERIFICATION CONDITION GENERATOR and the second by adding additional axioms or rules to the THEOREM PROVER. 

Suppose in a recursion augmented scheme P, a procedure F is called and then, before this call is completed (and END F reached), F is called again. Suppose further that every test encountered during these recursive calls of F always yields the same answer (i.e., each test T either always gives amswer TRUE or else always gives answer FALSE). Then the sane path must be followed during the... 

Thorough investigations with dimethyl diazomalonate and catalysts of the type (RO)3P CuX have revealed that the efficiency of competing reaction paths, the synjanti or EjZ selectivity in cyclopropane formation as well as the cis/trans ratio of carbene dimers depend not only on catalyst concentration and temperature but also on the nature of R58) and of the halide anion X 57 6". Furthermore, the cyclopropane yield can be augmented in many cases at the expense of carbene dimer... 

With the augmentation of the net flow, the disjoint P-V paths are determined step by step. And, finally, by (8), we can decide whether G is Kekulean or not. 

Amid all these developments, value creation has remained a key objective, although the paths to it have, of course, been altered and augmented by the shifts in the business environment. 

The atoms of the vanadium group metals have five valence electrons. In vanadium (Z — 23) and niobium (columbium, Z = 41), these valence electrons lie beyond ra re-gas cores, whereas in tantalum (Z = 73), they lie beyond the xenon core which has been augmented by fourteen 4/ electrons. The +5 oxidation state is characteristic of this family for niobium and tantalum it is the only oxidation state of importance. Oxidation is often regarded as removal of five valence electrons, followed by coordination of the pentapositive ion (which cannot exist for appreciable time in chemical systems) to basic groups which are present (H2O, OH, Cl, etc.). Although such a description almost certainly has very little resemblance to the actual path of oxidation of these metals, it is clerically convenient and may be used if not taken literally. In the same way, the lower oxidation states of vanadium may be considered vanadium atoms with the two 4s electrons removed, and with additional removal of one or two 3d electrons. 

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