Transformation mathematical

In Laplace-transformation mathematics, the following symbols and variables are used ... 

Fourier Transformation. Mathematical process of converting the interference free induction decay into a spectrum. 

Since some adulteration of raw data occurs when they are transformed mathematically, by differentiation or taking logarithms or reciprocals or otherwise, it is better from a statistical point of view to change the rate equation to read in terms of total pressure, rather than to change the data to partial pressures or concentrations. Such a transformation is worked out for a... 

The final step in obtaining the spectrum by the FTIR method is turning back the data obtained as a result of the repetitive interference action of the moving mirror into an intensity wavelength line. It is here that Fourier Transform mathematics is utilized. It is the signal intensity that is stored in a digital representation of the Interferogram. This information is then Fourier transformed by the computer into the frequency spectrum. 

Additionally, with the inclusion of computers as part of an instrument, mathematical manipulation of data was possible. Not only could retention times be recorded automatically in chromatograms but areas under curves could also be calculated and data deconvoluted. In addition, computers made the development of Fourier transform instrumentation, of all kinds, practical. This type of instrument acquires data in one pass of the sample beam. The data are in what is termed the time domain, and application of the Fourier transform mathematical operation converts this data into the frequency domain, producing a frequency spectrum. The value of this methodology is that because it is rapid, multiple scans can be added together to reduce noise and interference, and the data are in a form that can easily be added to reports. 

This procedure is sometimes referred to as the Schvab-Zeldovich transformation. Mathematically, what has been accomplished is that the nonhomogeneous terms (m and H) have been eliminated and a homogeneous differential equation [Eq. (6.7)] has been obtained. 

Figure 3.4 Steps in X-ray structure determination. X-ray scattering by the crystal gives rise to a diffraction pattern. From the diffraction pattern, the molecular structure can be determined using Fourier transformation mathematical calculations. Source For diffraction photograph, Nicholls Ft. Double hehx photo not taken by Franklin, BioMedNet News and Comments, 2003. http //news.bmn.com/news/story day=030425 story=l caption name [accessed April 28, 2003].)...

Inverse Fourier Transform Mathematical transformation from the... 

In 1946, both Purcell and Bloch and their coworkers independently reported the first NMR spectra of paraffin and water, respectively. They were awarded the Nobel Prize for physics in 1952. Twenty years later, Ernst and Anderson applied Fourier-transform mathematics to this technique, increasing instrument sensitivity and spectral resolution and opening the door to many possible applications. Today, NMR analysis of compounds not only reveals chemical structure and conformation, but also molecular mobility and internal dynamics of systems. 

Although many measurements in medicine are not even approximately normally distributed, they can often be easily transformed mathematically... 

Fig. 7.1 Correlation between measurements. When two different properties are measured for a set of entities, one can plot one of them against the other, (a) If there is no relation between them at all, the points should be scattered with no evident pattern, (b) If the correlation is typical of quantities measured in biology a definite trend should be evident, (c) An excellent correlation is, in biology, typical of the agreement between two sets of measurements of the same thing, (d) A perfect correlation is most likely to arise from one set of measurements transformed mathematically to give the impression that two different things have been measured...

Fourier transformation Mathematical transformation of time domain functions into frequency domain. 

Sometimes raw signals are transformed mathematically, for example, in optical spectroscopy it is usual to convert transmittance to absorbance data using a logarithmic transformation. This changes the noise characteristics, often to a log-normal distribution, although the origins of the instrumental noise are still homoscedastic. 

Interferometers utilize the interference phenomenon to produce an interferogram at a specific field of view. The interferogram is subjected to Fourier transformation mathematics to create a measurement spectrum of a sample specimen. The basic components of a classical Michelson interferometer are demonstrated by Chandler (1951) and Steel (1983). 

Milling the surface may be necessary for a painted or dirty surface, but this can be done in a few seconds with hand held abraders. Although the reflection spectra obtained do not resemble absorption spectra, they can be transformed mathematically to give comparable results. Even without the transformation the data obtained is sufficient to set up a specific identification procedure that can recognise subtle differences in polymer structure and composition. MIR can identify and separate polymers of a similar type e.g. different form of polyamides or blends that only vary in proportion of polymers present [8]. A number of rapid identification methods are now marketed, based on MIR with sealed specular reflectance attachments. Both large scale and portable systems are available. The method can be automated and used in conjunction with a reference collection of spectra can visually display the most probable polymer type. The Bruker system P/ID 22 has been developed and tested in conjunction with the European Automobile Manufacturers Association... 

Infrared spectroscopy is a family of techniques that can be used to identify chemical bonds. When improved by Fourier transform mathematical techniques, the resulting test is known as FTIR. An FTIR scan can be used to identify compounds rather in the same way as fingerprints are used to identify humans an FTIR scan of the sample is compared to the FTIR scans of known compounds. If a positive match is found, the sample has been identified an example is shown in Figure 8.8. Not surprisingly, FTIR results are sometimes called fingerprints by analytical chemists. 

It is a property of Fourier transform mathematics that multiplication in one domain is equivalent to convolution in the other. (Convolution has already been introduced with regard to apodization in Section 2.3.) If we sample an analog interferogram at constant intervals of retardation, we have in effect multiplied the interferogram by a repetitive impulse function. The repetitive impulse function is in actuality an infinite series of Dirac delta functions spaced at an interval 1 jx. That is,... 

For decades, the subject of control theory has been taught using transfer functions, frequency-domain analysis, and Laplace transform mathematics. For linear systems (like those from the electromechanical areas from which these classical control techniques emerged) this approach is well suited. As an approach to the control of chemical processes, which are often characterized by nonlinearity and large doses of dead time, classical control techniques have some limitations. 

Fourier transform Mathematical process that can be used to analyze a function of time into the individual frequency components that it contains. 

Spin-warp technique Method of using gradient fields to encode position-dependent information into NMR signals, then using Eourier transform mathematics to decode the signal and produce an image. 

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