1. LA DIGNIDAD COMO CONCEPTO
1.4. La Dignidad humana en el Estado Social de Derecho en Colombia
In mathematics, a sequence is an ordered list contain- ing an ordered set of terms (or elements). The number of terms n is called the length of the sequence, which may be also infinite, so that the sequences can be finite (converging to some limit) or infinite (e.g. all odd positive integers 1, 3, 5, ...). The terms of the sequence can be inte- ger numbers (forming integer sequences), real numbers, etc.
A recurrence relation is an equation that defines a sequence recursively: each term of the sequence is de- fined as a function of the preceding terms:
yn+1 = f(yn-k), 0 k n1 (2)
The linear recurrence relation of the k+1 order is
yn+1 = c0 + c1 yn + c2 yn-1 + c3 yn-2 + ... + ck+1 yn-k ,
0 k n1 (3) which is equivalent to another form of equation
y(n+1) = c0 + c1 y(n) + c2 y(n1) + c3 y(n2) + ... + ck+1 y(nk) ,
0 k n1 (4) where only the first powers of previous terms y are used and the coefficients ci are constants (in non-linear recur-
rence relations the y terms may have another power, may be also in denominator, and the coefficients may depend on n). If c0 = 0 the recurrence relation is called homogene-
ous otherwise it is called non-homogeneous. Some of the coefficients in eqs. (3) and (4) may be zero, the order of recurrence relation is given by the last non-zero term y(n-
k), i.e. by the value of k+1.
Originally, recurrent relations were applicable to discrete functions of arguments expressed e.g. by integer values of n. However, the recurrent relations can also be used for approximation of some properties dependent on the arguments x expressed by rational numbers. In this case equidistant argument values must be used:
y(x + x) = c0 + c1 y(x) + c2 y(x x) + c3 y(x - 2x) + ... +
ck+1 y(x kx) (5)
where the increment x is constant.
In mathematics there are several well known exam- ples of using integer arguments, e.g. in recursive definition of factorial:
(a) 0! = 1; (b) n! = n (n − 1)! where n 1 (6) or in recursive definition of power:
(a) a0 = 1; (b) an = an−1 a where n 1 (7)
In both examples there is (a) a basis, where the func- tion is explicitly evaluated for one or more values of its argument, (b) a recursive step, stating how to compute the function from its previous values. Another often cited ex- ample, originated in middle ages, is the Fibonacci se- quence 0, 1, 1, 2, 3, 5, 8, 13, ... , defined as a sequence whose two first terms are F0 = 0, F1 = 1 and each subse-
quent term is the sum of the two previous ones:
Fn+1 = Fn + Fn−1 for n 1 (8)
This is the basis for calculation of golden section ratio frequently used in art and architecture but also for unidimensional optimization in mathematics. Formally, it is the linear homogeneous second order recurrence rela- tion.
The most well-known integral argument in chemistry is the number of carbon atoms in molecules of organic compound homologues. This allows recurrence equations to be effectively used to approximate diverse properties of homologues. Zenkevich applied this principle for calcula- tion of boiling points, critical temperature, viscosity, parti- tion coefficient, chromatographic retention index, refrac- tive index, density surface tension, critical pressure, ioni- zation potential, permittivity and dipole moment (ref.12 and
further papers cited therein).
Recurrent algorithm for estimation of boiling points at the reduced pressure13 is an example of using non-
discrete arguments and eq. (5) in chemistry. Further utili- zation of this type of recurrences will be described in fur- ther text on voltammetry.
Examples of application of recurrence relations in evaluation of physicochemical constants and in chromatography
An illustrative example of advantageous use of re- current relations is the dependence of boiling points of a homologous series of compounds vs. number of carbon atoms in the molecule. Fig. 1 shows such a typical depend- ence for perfluoro-n-alkanes from CF4 up to C9F20. The
corresponding boiling point data are available in Beilsteins Handbuch der organischen Chemie and are collected also in Ref.7. In contrast to the dependence in Fig. 1, the corre-
sponding recurrence relation (Fig. 2) is perfectly linear and exhibits excellent correlation. In constructing the recur-
Fig. 1. Typical non-linear dependence of boiling points (in °C)
vs. number of carbon atoms n for perfluoro-n-alkanes CnF2n+2
Fig. 2. Linear dependence of the 1st order non-homogeneous recurrence relation Tb(n+1) = c0 + c1Tb(n) for boiling points (in °C) for perfluoro-n-alkanes
rence relation the boiling point data on vertical axis are just shifted by one row with respect to the data corre- sponding to horizontal axis. It is evident that the calculated linear recurrence dependence allows not only a very accu- rate interpolation, important when some experimental point within the dependence seems to be dubious, but, what may be even more important, it provides a very good way for extrapolation here the accuracy of the extrapola- tion result depends on the accuracy of individual points employed in this dependence.
An important relationship in chromatography relates the selected retention parameter of the homologous series to the number of carbon atoms in the molecule, n. Depend- ence of retention time tR on n is non-linear and the same is
valid for the net retention time corrected for the dead time
t0 as demonstrated in Fig. 3 constructed for the homolo-
gous serie of n-alkanes (C5 to C13). Surprisingly, the rele-
vant first order non-homogeneous recurrence relation of the type tR(n+1) = c0 + c1tR(n) provides a perfect linear
plot with an excellent coefficient of determination. Such a perfect linearity is achieved regardless of using the not corrected or the net values of retention time, however, the not corrected values are more recommended since any error in t0 has an adverse effect on the calculation accu-
racy.
There exists another possibility to plot the net reten- tion time in logarithmic form14, log(t
R t0) vs. n, which
equivalent21 to the use of Kovats retention indices:
log(tR t0) = b0 + b1 n (9)
We have proved a very good linearity of this depend- ence for the investigated n-alkane series: b0 = 1.9753, b1 =
0.2605, R2 = 0.9998. However, Zenkevich in his more
detailed study14 found that for extrapolation the recurrence
relation of the type tR(n+1) = c0 + c1tR(n) is much more
accurate than the relationship (9); the extrapolation error
of eq. (9) was quite large even for extrapolating by 2 or 3 carbon atoms.
Accurate calculation of voltammetric current by infinite series solution and extrapolation by recurrent relations
Infinite series corresponding to reversible electrode reduction in linear scan voltammetry was given by Nichol- son and Shain15 and later on it was converted for the oxi-
dation process to the form16
where I* and E* denote dimensionless current and poten-
tial, resp., and j is summation index. Only few terms of this series are needed for an accurate current I* calculation
if potential E* is sufficiently negative (corresponding to
the real potential values smaller than formal redox poten- tial of the investigated electroactive compound). However, for the values E* 0 the series (10) is divergent and a good
way how to overcome this problem is the use of some transformation of the infinite series into a converging se- quence. Generally, several ways of such a transformation were described e.g. by Wimp18 as well as in our papers16,17.
With regard to the use of recurrences it is worth to demon- strate epsilon algorithm, which performs two-dimensional non-linear recurrence equations:
= 0, = sj , j 0 (11)
= + 1 / , k > 0 (12) The calculation scheme is demonstrated in Fig. 1. As Fig. 3. Non-linear dependence of the net retention time (in
min) upon number of carbon atoms n in alkanes CnH2n+2
Fig. 4. Linear dependence of the 1st order non-homogeneous recurrence relation tR(n+1) = c0 + c1tR(n) demonstrating line- arity even for the not corrected values of retention time (in min) for alkanes
(10)
* * ( 1) 0 1 j 1 ej E j I j
( ) 1j ( ) 0j ( 1) 1 j k ( ) 1 j k ( ( 1)j ( )j) k k indicated by eq. (12) it uses the non-linear third order ho- mogeneous recurrence relation. Partial sums of the series (10), which involve all terms up to index j, enter to the k = 0 column. The first value calculated by recurrence is , which represents the partial sum of the transformed and converging sequence. Further partial sums of this new sequence are the last terms in each row. The algorithm may use several hundreds of rows in difficult cases.
Every transformation algorithm has some limitations and in the studied case it is given by too large exponent when the summation index j is large. Therefore a linear third order non-homogeneous recurrence relation was used for calculation of dimensionless currents I* at large values
(1) 1
of potential E* ; in this case the recurrence relation was
used for extrapolation purposes. Extrapolation was per- formed using equation
I*(E* + E*) = c
0 + c1 I*(E*) + c2 I*( E* E*) + c3 I*( E*
2 E*) (13)
for E* = 0.1 dimensionless potential units (1 unit corre-
sponds to 25.69 mV). Extrapolation was made for currents in the potential region E* = 11.519.0. Extracted results
are summarized in Table I. Organization of columns in this spreadsheet table was made in such a way that the whole row (including the predicted current value) could be ob- tained by one vertical mouse movement, so that the whole table was prepared very quickly in two steps. In the first Fig. 5. Scheme of epsilon algorithm for first five calculation steps. Calculation starts in the j=2 row, the calculation element is first shifted right and then down – in the next row it starts again from the left side
Table I
Extrapolation values of dimensionless current I* calculated by the linear non-homogeneous third order recurrence relation
E* yn+1 ≡ I* yn yn-1 yn-2 yn+1 predicted yn+1 correct error
8.5 0.197510812 0.198797398 0.200111250 0.201453365 0.197510786 0.1975108117 -2.5346E-08 8.6 0.196250538 0.197510812 0.198797398 0.200111250 0.196250525 0.1962505382 -1.3141E-08 8.7 0.195015670 0.196250538 0.197510812 0.198797398 0.195015666 0.1950156697 -3.5445E-09 8.8 0.193805340 0.195015670 0.196250538 0.197510812 0.193805343 0.1938053397 3.7080E-09 8.9 0.192618721 0.193805340 0.195015670 0.196250538 0.192618730 0.1926187210 8.8794E-09 9.0 0.191455024 0.192618721 0.193805340 0.195015670 0.191455036 0.1914550238 1.2226E-08 … … … … 11.4 0.168847293 0.169627780 0.170419646 0.171223182 0.168847319 0.1688468274 2.5388E-08 11.5 0.168077938 0.168847293 0.169627780 0.170419646 0.168077938 0.1680779025 3.5469E-08 … … … … 13.0 0.157713077 0.158342575 0.158980087 0.159625771 0.157713077 0.1577041514 8.9251E-06 … … … … 19.0 0.130628391 0.130950113 0.131275230 0.131603781 0.130628391 0.1298896927 7.3870E-04
step, the predicted values I*(E* + E*) were calculated
from three previous currents I*(E*), I*( E* E*) and
I*(E* 2 E*) in the interval of E* = 8.5 to 11.4 and the
second and seventh column of the table were made equal (the correct yn+1 was set for yn+1). Then, in the second step,
to achieve a true extrapolation, the predicted yn+1 calcu-
lated in the sixth column was set for yn+1 in the second
column, which in further row was used as yn (column 3).
The correct I* values at very large E* were obtained by
means of Lerch transcendent function in Mathematica software17, which unfortunately cannot be used generally
for solving further electrochemical infinite series. As a result of very accurate correct I* values (calculated by
epsilon transformation as well as Lerch function) the cal- culation error after few extrapolation steps was very small but gradually slowly increased. After 76 extrapolations from E* = 11.4 to 19.0 the extrapolation error was only
7.4104, which is an excellent result, not achieved by any
of the previously investigated ways of extrapolation.
The support of this work by grants VVCE-0004-07 and VEGA projects 1/1005/09 and 1/0066/0 is highly ac- knowledged. J.M cordially thanks to Professor Igor G. Zenkevich (St. Peterburg State University) for providing a collection of his articles on recurrences.
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