Decimation method

It is also possible to consider a polymer chain as being made of discrete links embedded in ad-dimensional space, and one can try to perform direct renormalization on polymers. This decimation method consists in representing a polymer of N links as a succession of sequences made of n links, and in considering each... 

This computation program, which looks straightforward, was described by de Gennes in 1977,10,11 but it was not easy to put it into practice. However, in 1978, Gabay and Garel12 succeeded in obtaining, with this method, the first term in the expansion of v as a function of e = 4 — d (d = space dimension) namely v = 1/2(1 + s/8 +. ..). However, this result was already known, and at the present time (1988), no algorithm exists which could be used to proceed much further with the decimation method.13... 

However, the principles and the techniques of renormalization theory are not directly related to the existence of fields. They apply whenever one deals with a critical system, i.e. whenever one has to describe large-scale phenomena which depend only globally on the chemical microstructure. Thus, because an ensemble of long polymers in a solution constitutes a critical system, renormalization principles and renormalization techniques must be directly applicable to their study. Actually, this idea appeared quite naturally. It led to the decimation method which has been described previously and which lacks efficiency. However, the same idea can be applied in a much better way. This direct renormalization method (des Cloizeaux 1980)37,38 consists in adapting to polymers methods which had been successful in field theory.39 In other words, the aim is to bypass the Laplace de Gennes transformation (see Chapter 11). This method applies to semi-dilute solutions as well as to dilute solutions. 

Let us be more precise. We consider a real chain with only two-body interaction, but we assume that, in order to construct a more realistic model, we perform a finite number of renormalizations of the chain by using the decimation method. We see immediately that the two-body interactions of the initial chain generate three-body interactions (and also four-body interactions, and so on). When we discussed this kind of renormalization, we even said that this effect was a drawback of the method. Here, this effect preserves us from difficulties it shows that two-body interactions can, in a simple way, generate three-body interactions (but the existence of a short-range cut-off has to be taken into account). 

Density is generally measured at 15°C using a hydrometer in accordance with the NF T 60-101 method it is expressed in kg/1 with an error of 0.0002 to 0.0005 according to which category of hydrometer is utilized. However, in practice only three decimal places are usually retained. 

A lustrous metal has the heat capacities as a function of temperature shown in Table 1-4 where the integers are temperatures and the floating point numbers (numbers with decimal points) are heat capacities. Print the curve of Cp vs. T and Cp/T vs. T and determine the entropy of the metal at 298 K assuming no phase changes over the interval [0, 298]. Use as many of the methods described above as feasible. If you do not have a plotting program, draw the curves by hand. Scan a table of standard entropy values and decide what the metal might he. 

The schlieren microscope is able to detect refractive index variations to six decimal places. Any small difference in optical path (index difference, film thickness, etc) is very precisely detected by the schlieren microscope, especially in the Dodd modification. It is, in effect, a darkfield method. The specimen is illuminated with light in a portion of the illuminating cone and that direct light is masked in the conjugate back focal plane of the objective (Fig. 3). The only light to pass through this plane is refracted, reflected, or diffracted by the specimen. 

It is rarely that any greater degree of accuracy than that attained in the above methods is required. If, however, scientific accuracy is necessary in specific gravity determinations, the usual standard of comparison, at whatever temperature the determination may be made, is water at its maximum density temperature, namely 4°. If no correction is made for the weight of the air contained in the bottle or tube, accuracy to the fourth place of decimals is ensured by reducing the observed weighings to vacuum weighings. This can be done by the equation—... 

These methods may be called analytical, by contrast with another class of iterations that might be called arithmetic, since they exploit the fact that the number representation is finite and digital. The familiar Homer s method is an example. The first step is to establish that a root lies between a certain pair of consecutive integers. Next, if the representation is decimal, f(x) is evaluated at consecutive tenths to determine the pair of consecutive tenths between which the root lies. This is repeated for the hundredths, thousandths, etc., to as many places as may be desired and justified. 

Recently, Teymour and coworkers developed an interesting computational technique called the digital encoding for copolymerization compositional modeling [20,21], Their method uses symbolic binary arithmetic to represent the architecture of a copolymer chain. Here, each binary number describes the exact monomer sequence on a specific polymer chain, and its decimal equivalent is a unique identifier for this chain. Teymour et al. claim that the... 

The volume, electron population, and charge of the whole molecule can also be obtained by the same method. Table 6.1 gives the atomic volumes, populations, and charges of the atoms in methanal. We see that the electron populations of the atoms and the corresponding charges are additive to four decimal places. 

These 10 sets of problems will familiarize you with arithmetic operations involving decimals (which are a really special kind of fraction). You use decimals every day, in dealing with money, for example. Units of measurement, such as populations, kilometers, inches, or miles are also often expressed in decimals. In this section you will get practice in working with mixed decimals, or numbers that have digits on both sides of a decimal point, and the important tool of rounding, the method for estimating decimals. 

The exact mass of an ion (4 to 6 decimal points) reliably defines its elemental and isotopic composition, while the method is called high resolution mass spectrometry. The measurements are conducted manually or automatically (computerized). Manual measurements are based on the parallel acquisition of the peak of interest with the closest peak of an ion with the known composition. Any compound with an intense ion peak with m/z value in the region +10% may serve as a marker. The most widespread markers are perfluorokerosene, perfluorotributylamine, and other polyfluorinated compounds. The use of these compounds is based on their volatility, as well as on the fact that fluorine is a monoisotopic element. In the spectra of these compounds intense ion peaks randomly cover all the range between m/z 19 and M+. ... 

There can be no doubt that instrumental methods of analysis have revolutionized analytical chemistry, in terms of increased sensitivity, more rapid throughput, multielement capability, computerized calibration, and data handling, etc. There is a cost, too, of course - increased capital expenditure, increased instrumental complexity, and, above all, the current tendency to believe implicitly the output of a computer. Just because a machine gives an analysis to 12 places of decimals doesn t mean that it is true (see Chapter 13) ... 

To use the equation method, recall this important information when translating words into math is means equals and of means multiply. Also, for equations, change the percent to a decimal by dividing by one hundred... 

Equation Method Write the equation, substituting in an equal sign for is, the multiplication sign for of, and change the percent to a decimal. 

I can almost hear some of the musings of the unbelievers After all, this study was done by psychiatrists. Who knows, maybe they absent-mindedly moved the decimal points. In any case, the new treatment method did not make it into mainstream American psychiatry, much less general medical practice. Nor did the good news that physostigmine was an effective antidote for atropine delirium. (Incidentally, I met Forrer s colleague. Dr. Miller, in 1981, 30 years after their first publications about atropine coma therapy. He was invited to... 

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