Orders of differences

FIGURE 16.5 Comparison of different ventilation unit combinations. The life cycle Is 20 years and the interest rate is A%. In sensitivity analysis the variation of the Interest rate did not change the order of different combinations signlBcantly. 

Cosmic rays < Y rays < X rays < nv rays < visible rays < i.r. rays < micro waves < radio waves and the increasing order of different energies are... 

In comparing glass from different panels one observed in general only about the same order of difference that exists between different fragments in the same panel which do not correlate very closely. Most of those differences probably represent the normal variation between batches of glass from the same workshop. 

An HPLC method for chlorogenic acids with lactones in six different commercial brands of roasted coffee was developed by Schrader et al. (143). Hydroxycinnamic acid derivatives, including mono- and di-caffeoylquinic acids, corresponding lactones, and feruloylquinic acids were extracted from coffee with methanol at 80°C for 1 h under reflux. An HPLC method using step-gradient elution with 2% aqueous acetic acid (eluent A) and ACN (eluent B) for a 75-min run time was developed. Determination was carried out by HPLC with UV detection at 324 nm, and further confirmation was conducted by HPLC-thermospray (TSP)-MS and HPLC-diode array detection. Elution order for mono-caffeoylquinic acid (CQA) was 3-CQA, 5-CQA, followed by 4-CQA, which was different from the usual elution order of mono-CQA (Fig. 17). These results indicate that it is currently not possible to predict the elution order of different reversed-phase packings due to the different selectivity (143). 

Although these hierarchic structures can be easily expressed with drawings, it is not easy to express them with plain text. Mathematicians encounter similar problems when writing mathematical expressions where ordering of different operations must be performed. Parentheses were invented to solve these problems, as they provide syntax to show the intended grouping of expression parts. 

First Order of Difference between Sequential Flexagrams)... 

We viewed the King Wen sequence as a continuum and intuited that the ordering principle related to a quality that connected the umelated pairs of hexagrams. We were led to compare the first order of difference, or degree of change, as one moves through the King Wen sequence. 

As if these synthetic symmetries were not enough, in addition we find that when the first order of difference of the King Wen sequence is graphed it appears random or unpredictable (fig. 18A). Elowever, when an image of the graph is rotated 180 degrees within the plane and superimposed upon itself, it is found to achieve closure at four adjacent points (fig. 18B). 

The point that we wish to make in this discussion is that the first order of difference among the hexagrams, as they are read in a linear sequence, was consciously and deliberately ordered in very ancient times. Why this was done we cannot yet say, but the fact that it was done validates our contention the these mathematical qualities of the / Ching were factors of which the neolithic Chinese were well aware. Whether they graphed the first order of difference, as we have done in figures 18A and B, is moot. Graphing is not necessary simple numerical quantification of the orders of difference gives the idea of the continuum of the sequence. Nevertheless,... 

Here X is a perturbation parameter which is introduced so as to define the order of different terms in the perturbation series but which is set equal to 1 in order to recover the physical situation. 

The different orders of differences are usually arranged in the form of a table of differences . To construct such a table, we. can begin with the first member of series of corresponding values of the two variables. Let the different values of one variable, say, x0, xv x2,... correspond with y0, ylt y2,... The differences between the dependent variables are denoted by the symbol u A, with a superscript to denote the order of difference, and a subscript to show the relation between it and the independent variable. Thus, in general symbols — ... 

We see that the numerical coefficients of the successive orders of differences follow the binomial law of page 36. This must also be true of yx + if n is a positive integer, consequently,... 

The correct value is 6 80094. The discrepancy is due to the fact that the order of difference above the third ought not to be neglected. But we can only get n - 1 orders of differences from n consecutive terms and equidistant terms values of a function. If more terms had been given we could have got a more exact result. 

The central difference formula of Stirling thus furnishes the same result as the ordinary difference formula of Newton. We get different results when the higher orders of differences are neglected. For instance, if we neglect differences of the second order in formulae (7) and (20), Stirling s formula would furnish more accurate results, because, in virtue of the substitution A1 = A1 - JA2 v we have really retained a portion of the second order of differences. If, therefore, we take the difference formula as far as the first, third, or some odd order of differences, we get the same results with the central and the ordinary difference formulae. One more term is required to get an odd order of differences when central differences are employed. Thus, five terms are required to get... 

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