General Remarks about Transformation

The common tool for obtaining the spectrum in the frequency domain from a time function is the Fourier transformation. This exhibits some limitation like a frequency vector, which is multiple integers of the fundamental frequency. Moreover, it requires periodic signals. Practically, periodic signals should be used in their entire length  

It is often worth tuning the transformation algorithm to get what is wanted. For calculation of random lines within the Fourier spectrum (equidistant frequencies), discrete Fourier transformation or sparse Fourier transformation is a good choice (Gbrtzel algorithm). 

For assessing a small window within the spectrum but with high-frequency resolution, z-transformation (Chirp-z-transformation) can be used. A popular algorithm is the Bluestein algorithm. The latter solutions are especially helpful for single shot signals. The continuous analogy is the Laplace transformation. 

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Characteristics and implementation of the treatments depend on the expected results and on the properties of the material considered a variety of processes are employed. In ferrous alloys, in steels, a eutectoid transformation plays a prominent role, and aspects described by time-temperature-transformation diagrams and martensite formation are of relevant interest. See a short presentation of these points in 5.10.4.5. Titanium alloys are an example of the formation of structures in which two phases may be present in comparable quantities. A few remarks about a and (3 Ti alloys and the relevant heat treatments have been made in 5.6.4.1.1. More generally, for the various metals, the existence of different crystal forms, their transformation temperatures, and the extension of solid-solution ranges with other metals are preliminary points in the definition of convenient heat treatments and of their effects. In the evaluation and planning of the treatments, due consideration must be given to the heating and/or cooling rate and to the diffusion processes (in pure metals and in alloys). 

We conclude our discussion of polymorphism with some brief general remarks. The physical factors most commonly involved in bringing about an enantiotropic polymorphic transformation are changes... 

Olefin metathesis (or x-metathesis) reactions are one of the most unusual transformations in chemistry and also one of the most commercially important The general reaction, which is given by Equation (19.15), involves the exchange of substituents between two different alkenes. What is remarkable about this reaction is that it involves the breakage of a C=C double bond under relatively mild conditions using an organometallic catalyst... 

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