Arithmetics

Perl knows basic arithmetic. Symbols known as operators are responsible for the various arithmetic operations  

The following example does a little math and prints out the result  

The account begins with binary arithmetic, moves on to on-off (flip-flop) electronic switches, then to serial and parallel processing, and finally to computers/transputers. 

In the next column, there are one seven (not one ten, as with decimal 17) and two sevens (from the 24), making a total of three sevens the total number of sevens is then 3 + 1 (carried from the previous column) = 4, and this is written down. Thus, the total obtained from the addition of 17 and 24 in the two systems is 41 in one and 44 in the other Which is correct In fact, we have cheated a little because 17 (decimal) written in the heptimal system should be two sevens and a 

Addition of two decimal numbers. Note the carryover of one ten from the rightmost column into the left one. 

Addition in the heptimal system. Now. one seven is carried from the rightmost column into the leftmost one. 

The above results appear most confusing, but only if the bases are mixed. Working with one base all the time (e.g., decimal), there is no confusion. 

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Crude petroleum is fractionated into around fifty cuts having a very narrow distillation intervals which allows them to be considered as ficticious pure hydrocarbons whose boiling points are equal to the arithmetic average of the initial and final boiling points, = (T, + Ty)/2, the other physical characteristics being average properties measured for each cut. 

V.L. Druskin and L.A. Knizhnerman Krylov subspace approximation of eigen-pairs and matrix functions in exact and computer arithmetics. Num. Lin. Alg. Appl., 2 (1995) 205-217... 

For examples of different types of similarity measures, see Table 6-2. The Tanimoto similarity measure is monotonic with that of Dice (alias Sorensen, Czekanowski), which uses an arithmetic-mean normaJizer, and gives double weight to the present matches. Russell/Rao (Table 6-2) add the matching absences to the nor-malizer in Tanimoto the cosine similarity measure [19] (alias Ochiai) uses a geometric mean normalizer. 

The resulting similarity measures are overlap-like Sa b = J Pxi ) / B(r) Coulomblike, etc. The Carbo similarity coefficient is obtained after geometric-mean normalization Sa,b/ /Sa,a Sb,b (cosine), while the Hodgkin-Richards similarity coefficient uses arithmetic-mean normalization Sa,b/0-5 (Saa+ b b) (Dice). The Cioslowski [18] similarity measure NOEL - Number of Overlapping Electrons (Eq. (10)) - uses reduced first-order density matrices (one-matrices) rather than density functions to characterize A and B. No normalization is necessary, since NOEL has a direct interpretation, at the Hartree-Fodt level of theory. 

As Eq. (29) shows, corresponds to the arithmetic mean and to the geometric mean of the homodimeric parameters AA and BB. 

Arithmetical operations on complex numbers are performed much as for vectors. Thus, if a j hi and y = c + di, then ... 

This is similar in spirit to the arithmetic-mean rule but with each individual r,) being weighted according to the square of its value. The well depth in this function starts with a formula proposed by Slater and Kirkwood for the Cg coefficient of the dispersion series expansion ... 

The second generator is an arithmetic sequence method that generates random number using the following mathematical operation ... 

To estimate the computational time required in a Gaussian elimination procedure we need to evaluate the number of arithmetic operations during the forward reduction and back substitution processes. Obviously multiplication and division take much longer time than addition and subtraction and hence the total time required for the latter operations, especially in large systems of equations, is relatively small and can be ignored. Let us consider a system of simultaneous algebraic equations, the representative calculation for forward reduction at stage is expressed as... 

When we report the result of a measurement a , there are two things a person reading the report wants to know the magnitude (size) of the measurement and the reliability of the measurement (its scatter ). If measuring errors are random, as they very frequently are, the magnitude is best expressed as the arithmetic mean p of N repeated tr ials xi... 

If very many measurements are made of the same variable a , they will not all give the same result indeed, if the measuring device is sufficiently sensitive, the surprising fact emerges that no two measurements are exactly the same. Many measurements of the same variable give a distribution of results Xi clustered about their arithmetic mean p. In practical work, the assumption is almost always made that the distribution is random and that the distribution is Gaussian (see below). 

In so doing, we obtain the condition of maximum probability (or, more properly, minimum probable prediction error) for the entire distribution of events, that is, the most probable distribution. The minimization condition [condition (3-4)] requires that the sum of squares of the differences between p and all of the values xi be simultaneously as small as possible. We cannot change the xi, which are experimental measurements, so the problem becomes one of selecting the value of p that best satisfies condition (3-4). It is reasonable to suppose that p, subject to the minimization condition, will be the arithmetic mean, x = )/ > provided that... 

This method, because it involves minimizing the sum of squares of the deviations xi — p, is called the method of least squares. We have encountered the principle before in our discussion of the most probable velocity of an individual particle (atom or molecule), given a Gaussian distr ibution of particle velocities. It is ver y powerful, and we shall use it in a number of different settings to obtain the best approximation to a data set of scalars (arithmetic mean), the best approximation to a straight line, and the best approximation to parabolic and higher-order data sets of two or more dimensions. 

Clearly, proposing arbitrary candidates for p and selecting the one with the smallest value of xi — p) to find x is not very efficient, nor can it be readily generalized. This is especially so because, even with a data set of integral numbers, the arithmetic mean does not have to be an integer. 

The sum of squares of differences between points on the regression line yi at Xi and the arithmetic mean y is called SSR... 

Look up the experimental values of the first ionization potential for these atoms and calculate the average difference between experiment and the computed values. Depending on the source of your experimental data, the arithmetic mean difference should be within 0.010 hartrees. Serious departrues from this level of agreement may indicate that you have one or more of your spin multiplicities wrong. 

By the time j ou get to a ten step synthesis, the arithmetic demon ensures that the yield is down to a miserly 35%. And this with 90% yield in eaeh step So clearly a short synthesis is a good one. But we ean cheat the arithmetic demon by making our five steps convergent rather than linear. The convergent version is this ... 

The equation is sometimes written, = SRTI3000 t . The form preferred here is more correct because it makes no presupposition about the units used. Arithmetical adjustments will normally be necessary to give rate constants in the required units. 

If we assume that there are certain ideal val ues for bond angles bond distances and so on itfol lows that deviations from these ideal values will destabilize a particular structure and increase its po tential energy This increase in potential energy is re ferred to as the strain energy of the structure Other terms for this increase include steric energy and steric strain Arithmetically the total strain energy ( ) of an alkane or cycloalkane can be considered as... 

You have seen that measurements of heats of reaction such as heats of combustion can pro vide quantitative information concerning the relative stability of constitutional isomers (Section 2 18) and stereoisomers (Section 3 11) The box in Section 2 18 described how heats of reaction can be manipulated arithmetically to generate heats of formation (AH ) for many molecules The following material shows how two different sources of thermo chemical information heats of formation and bond dissociation energies (see Table 4 3) can reveal whether a particular reaction is exothermic or en dothermic and by how much... 

The simplest arithmetic ap proach subtracts the C—C (j bond energy of ethane (368 kj/mol 88 kcal/mol) from the C=C bond energy of ethylene (605 kJ/mol 144 5 kcal/mol) This gives a value of 237 kJ/mol (56 5 kcal/mol) for the tt bond energy... 

The most obvious feature of these C chemical shifts is that the closer the carbon is to the electronegative chlorine the more deshielded it is Peak assignments will not always be this easy but the correspondence with electronegativity is so pronounced that spec trum simulators are available that allow reliable prediction of chemical shifts from structural formulas These simulators are based on arithmetic formulas that combine experimentally derived chemical shift increments for the various structural units within a molecule... 

We have so far described the structure of DNA as an extended double helix The crys tallographic evidence that gave rise to this picture was obtained on a sample of DNA removed from the cell that contained it Within a cell—its native state—DNA almost always adopts some shape other than an extended chain We can understand why by doing a little arithmetic Each helix of B DNA makes a complete turn every 3 4 X 10 m and there are about 10 base parrs per turn A typical human DNA contains 10 base parrs Therefore... 

Raw data are collected observations that have not been organized numerically. An average is a value that is typical or representative of a set of data. Several averages can be defined, the most common being the arithmetic mean (or briefly, the mean), the median, the mode, and the geometric mean. 

The median of a set of numbers arranged in order of magnitude is the middle value or the arithmetic mean of the two middle values. The median allows inclusion of all data in a set without undue influence from outlying values it is preferable to the mean for small sets of data. 

Another measure of dispersion is the coefficient of variation, which is merely the standard deviation expressed as a fraction of the arithmetic mean, viz., s/x. It is useful mainly to show whether the relative or the absolute spread of values is constant as the values are changed. 

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