Monotonic sequences

Thus we can reformulate the question Which is the first nonvanishing term in the monotone sequence, Ria 2a - ... 

Wolery, T. J. and L. J. Walters, Jr., 1975, Calculation of equilibrium distributions of chemical species in aqueous solutions by means of monotone sequences. Mathematical Geology 7,99-115. 

I, Hg, H, Pb, Bi, Po, and At. However, due to the automatism in some cases, the values of the fitted parameters do not follow the intuitive monotonous series along the rows of the Periodic Table. E.g. the parameters for phosphorus fall out from the monotonous sequence along the row. More recent analysis indicates that this happens due to the fact that experimental values for phosphorus compounds used in the parametrization turned out to be wrong by 140 kcal/mole. 

This alternate form of the equations produces a faster convergence as shown in an example given by Wigley (41) and also converges more rapidly than Newton-Raphson. EQTemploys an additional control on the continued fraction method which generates monotone sequences (43,44). Its chief virtues are strict error bounds and increased stability with respect to a range of analyses of aqueous solutions used as input. 

Iteration and convergence method explicit equations Monotone sequences and secant method Newton- Raphson Free ion molali-ties by difference Newton- Raphson conti nued fraction Newton- Raphson Newton-Raphson conti nued fraction conti nued fraction for anions only conti nued fraction conti nued fraction conti nued fraction brute force... 

Walters, L. J., Jr. and Wolery, T. J., A monotone-sequences algorithm and FORTRAN IV program for calculation of equilibrium distributions of chemical species. Comp. Geosci. i, 57-63 (1975). 

While the evidence (fossils, sediment type, stratification) supports a lacustrine environment for all the deposits, the monotonous sequence at Condor is unique. At Condor, a stratified lake system similar to that proposed by Smith and Lee (17) for density stratification in the Piceance Creek Basin may have developed. 

A sequence is called increasing (decreasing) if a +i > a a +i < a ) and is called nondecreasing (nonincreasing) if a +i > aSa + n)- A seqnence is called strictly monotone if it is either increasing or decreasing. A seqnence is bonnded if there is a nnmber M such that a < M. A bounded strictly monotone sequence converges. 

Ball et al. associated the staircase profile of the (KC1)32 monotonic sequences with an efficient structure-seeker and contrasted this picture... 

Figure 1.1. Monotonic sequences for Ar,9 27). The structures correspond to the highlighted sequences and N is the number of atoms participating in each rearrangement (Section III.A).

Here, we use monotonic sequence basin (MSB s) to produce a more coarse-grained picture of the PES. Disconnectivity graphs that only include the minimum at the bottom of each MSB can be produced by excluding all the minima directly connected to a lower-energy minimum. In the resulting... 

Figure 1.9. Disconnectivity graphs for (a) LJ38 and (b) LJ55. Only the branches corresponding to monotonic sequence basin bottoms are shown. Energy is in units of e.

Many of the trends described in this section can be seen in the representative monotonic sequences in Figure 1.16. Relaxation to the global minimum is hindered when the range of the potential is short. 

Figure 1.20. A representative monotonic sequence fo the Cgo potential energy surface, showing the energies of minima and the transition states that connect them as a function of the Stone-Wales stack.

All the techniques are iterative and, except for the simplest chemical systems, require a computer. The methods include optimization by steepest descent (White et al., 1958 Boynton, 1960) and gradient,descent (White, 1967), tback substitution (Kharaka and Barnes, 1973 Truesdell and Jones, 1974), and progressive narrowing of the range of the values allowed for each variable. (.the monotone sequence method Wolery and Walters, 1975). 

The situation changes to a monotonous sequence of stabilities on going to the less solvating solvent 95 % aqueous methanol. From the consideration of solvent transfer coefficients of the anions only one would expect the chloride complex to experience... 

However, maybe all elements react rapidly with water to form a colorless basic solution and hydrogen gas Well, this is not true. The noble gases do not. They only dissolve in water, without any accompanying chemical reaction. They seem to be for the water structure just inert balls of increasing size. No wonder then that (at 293 K) one obtains the following monotonic sequence of their solubilities 8.61 (He)< 10.5 (Ne)< 33.6 (Ar)< 59.4 (Kr)< 108.1 (Xe) < 230 (Rn) cm /kg. 

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