Decaying exponential function

An ideal plug flow reactor, for example, has no spread in residence time because the fluid flows like a plug through the reactor (Westerterp etal., 1995). For an ideal continuously stirred reactor, however, the RTD function becomes a decaying exponential function with a wide spread of possible residence times for the fluid elements. 

To convert an optical signal into a concentration prediction, a linear relationship between the raw signal and the concentration is not necessary. Beer s law for absorption spectroscopy, for instance, models transmitted light as a decaying exponential function of concentration. In the case of Raman spectroscopy of biofluids, however, the measured signal often obeys two convenient linearity conditions without any need for preprocessing. The first condition is that any measured spectrum S of a sample from a certain population (say, of blood samples from a hospital) is a linear superposition of a finite number of pure basis spectra Pi that characterize that population. One of these basis spectra is presumably the pure spectrum Pa of the chemical of interest, A. The second linearity assumption is that the amount of Pa present in the net spectrum S is linearly proportional to the concentration ca of that chemical. In formulaic terms, the assumptions take the mathematical form... 

The signal-to-noise ratio (S/N) can be improved by multiplying the FID by a decaying exponential function, exp(—nt LB), which attenuates the noise at the end of the FID but broadens peaks in the transformed spectrum by the amount LB (Hz). 

The autocorrelation function for a polydisperse system represents the weighted sum of decaying exponential functions, each of which corresponds to a different particle diameter. For such a system ... 

The solution of the differential equation (2.165) for the time function G(t+) is the decaying exponential function... 

As suggested above, the noise amplitude in a spectrum can be attenuated by de-emphasising the latter part of the FID, and the most common procedure for achieving this is to multiply the raw data by a decaying exponential function (Fig. 3.34a). This process is therefore also referred to as exponential multiplication. Because this forces the tail of the FID towards zero, it is also suitable for the aptodisation of truncated data sets. However, it also... 

Example Consider 10,000 power supplies in the field with a failure rate of 10% every year. That means in 2005 if we had 10,000 working units, in 2006 we would have 10,000 x 0.9 = 9000 units. In 2007 we would have 9000 x 0.9 = 8100 units left. In 2008 we would have 7290 units left, in 2009, 6561 units, and so on. If we plot these points —10,000, 9000, 8100, 7290, 6561, and so on, versus time, we will get the well-known decaying exponential function (see Figure 7-1). 

On the other hand, for go > (4coao) % i e., for y > 1, the correlation function is simply the sum of two monotonically decaying exponential functions. Thus, the structure of a bicontinuous microemulsion is characterized by /wo length scales the domain size of oil or water domains and the correlation length... 

Microemuisions exist for values of the parameter y [in and see Eqs. (33) and (34)] less than 1 and greater than —1, with more negative values associated with more structure. As can be seen, the correlation function is an exponentially decaying oscillatory function of the separation r. On the other hand, for values of y > 1, the Fourier transform is simply a sum of two monotonically decaying exponential functions, and the liquid is unstructured. It is this difference in bulk behavior that proves crucial to the interfacial wetting behavior. 

The Buckhingam potential is similar to LJ with a more physical description of the repulsion term given in terms of a decaying exponential function ... 

If the density function F(A ) takes the form of decaying exponential function... 

If there are small perturbations about the equilibrium state the system, as discussed in Section 4.9, relaxes to equilibrium (see Problem 7.4). The displacement variables [X] — Zq, etc., are linear combinations of decaying exponential functions. Direct negative feedback, due to reverse reactions like 2X X + A, keep [X] and [Y] under control no matter what the initial conditions. 

If the displacements are small the quadratic terms can be dropped. Using the matrix method of Chapter 5 and assuming that f and rj are linear combinations of decaying exponential functions e the decay constants X are solutions of the determinantal equation... 

Fig. 5.1 The boundary condition represented by a series of step functions and by a series of decaying exponential functions (Cai and Weitsman 1994)...

Whitney and Drzal [87] presented an analytical model to predict the stresses in a system consisting of a broken fiber surrounded by an unbounded matrix. The model (Fig. 8) is based on the superposition of the solutions to two axisymmetric problems, an exact far-field solution and an approximate transient solution. The approximate solution is based on the knowledge of the basic stress distribution near the end of the broken fiber, represented by a decaying exponential function multiplied by a polynomial. Equilibrium equations and the boundary conditions of classical theory of elasticity are exactly satisfied throughout the fiber and matrix, while compatibility of displacements is only partially satisfied. The far-field solution away from the broken fiber end satisfies all the equations of elasticity. The model also includes the effects of expansional, hygrothermal strains and considers orthotropic fibers of the transversely isotropic class. 

In the simplest case of particles of uniform size, G St) consists, after subtraction of the long-time baseline I(t)), of a decaying exponential function. 

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