Origin time

Since the integral is over time t, the resulting transform no longer depends on t, but instead is a function of the variable s which is introduced in the operand. Hence, the Laplace transform maps the function X(f) from the time domain into the s-domain. For this reason we will use the symbol when referring to Lap X t). To some extent, the variable s can be compared with the one which appears in the Fourier transform of periodic functions of time t (Section 40.3). While the Fourier domain can be associated with frequency, there is no obvious physical analogy for the Laplace domain. The Laplace transform plays an important role in the study of linear systems that often arise in mechanical, electrical and chemical kinetic systems. In particular, their interest lies in the transformation of linear differential equations with respect to time t into equations that only involve simple functions of s, such as polynomials, rational functions, etc. The latter are solved easily and the results can be transformed back to the original time domain. 

Original time scale (seconds) Time scale with origin shifted to the centre k ... 

Approximate Origin Time of the Major Multi-Cellular Plant and Animal Groups ... 

The system of coupled differential equations is equivalent to the original time-dependent Schrodinger equation, and no approximation has been made. If the perturbation AH is weak, the coefficients ck may be expanded in powers of A as... 

Apart from some special drift processes that we will treat separately, the noise in the measurements is expected to be the result of random processes much faster than the changes in the useful signal itself. Fourier transform spectral methods exploit this difference in frequency for separating the two components by considering a frequency-domain representation of the signal instead of its original time domain representation. 

Processing in general transforms an original time domain signal s(t) with the aid of some processing function(s) into a manipulated frequency domain signal S"(f). Manipulations can be performed either in the time domain ( s(t) => s (t) ) prior to the Fourier transformation, or in the frequency domain (S (f) S"(f)) after the Fourier... 

A most convenient way to solve the differential equations describing a mass transport problem is the Laplace transform method. Applications of this method to many different cases can be found in several modern and classical textbooks [21—23, 53, 73]. In addition, the fact that electrochemical relationships in the so-called Laplace domain are much simpler than in the original time domain has been employed as an expedient for the analysis of experimental data or even as the basic principle for a new technique. The latter aspect, especially, will be explained in the present section. 

The case p < 1 corresponds to slowing down the rate of articulation by means of a time-scale expansion, while the case p > 1 corresponds to speeding up the rate of articulation by means of a time-scale compression. Events which take place at a time t 0 according to the new time scale will have occurred at p 1 t.(] in the original time scale. 

Finally, based on the channel assignment a phase correction is introduced in each channel, making the phase of the modified filter output yp (n) at time n = pnpQ equal to the phase at the time instant n = np in the original time scale. Denoting the phase correction for each cluster by <]), the modified channel signal becomes... 

From the experimental results in Figs. 1 and 2, the first-order reaction trend lines could not be traced back to the origin (time at 0 and a glucose fraction at 1) on extrapolation. This phenomenon can be observed in most previously reported data. It indicates that the reversible reactions are much faster than irreversible glucose decomposition and equilibrium is attained at early phase of the reaction. Conner et al. (9) suggested an... 

Turning now to the slow dynamics, we consider the same limit e —> 0 in the original time scale t. On defining... 

We now multiply Equation (3.10) by e and consider the limit of an infinitely high recycle flow rate (i.e., e —> 0) in the original time scale t. In this limit, we obtain the constraints in Equation (3.12), or equivalently, the linearly independent constraints... 

The process of Fourier transformation converts the raw data (e.g. a time series) to two frequency domain spectra, one which is called a real spectrum and die other imaginary (diis terminology conies from complex numbers). The true spectrum is represented only by half the transformed data as indicated in Figure 3.18. Hence if there are 1000 datapoints in the original time series, 500 will correspond to the real transform and 500 to the imaginary transform. 

Theoretically, it is possible to conceive of multiplying the original time series by increasingly positive exponentials until peaks are one datapoint wide. Clearly there is a flaw in our argument, as otherwise it would be possible to obtain indefinitely narrow peaks and so achieve any desired resolution. 

This is equivalent to having the full intensity for a fraction of time a of the original time. 

Teleseismic travel time tomographic images are determined from the inversion of relative travel time residuals so as to minimize bias resulting from errors in the earthquake location and origin time and the effects of 3D structures outside the Earth volume of interest. A second reason for using relative arrival times is that they can be determined more precisely than absolute arrival times in the presence of noise. The relative travel time is given by... 

There are numerous variables that can affect trace element determinations before the analysis of the sample is undertaken and these require careful control. Guidelines giving details of sample collection procedures, and procedures for limitation of contamination in a range of sample types, are available for essential and toxic trace elements. Age, sex, ethnic origin, time of sampling in relation to food intake, time of day and year, history of medication, tobacco usage, and other factors should be recorded when establishing reference intervals from healthy control populations. 

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