Exponential Multiplied By Time

If you remember from Chap. 6, repeated roots of the characteristic equation yielded time functions that contained an exponential multiplied by time. 

Equation (9.17) can be generalized for a repeated root of nth order to give 

The impulse function is an infinitely high spike that has zero width and an area of one (see Fig. 9.1a). It is a function that cannot occur in any real system, but it is a useful mathematical function that will be used in several spots in this book. 

One way to define is to call it the derivative of the unit step function, as sketched in Fig. 9.1b. 

Now the unit step function can be expressed as a limit of the first-order exponential step response as the time constant goes to zero. 

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D. EXPONENTIAL MULTIPLIED BY TIME. In the Laplace domain we found that repeated roots l/(s -I- a) occur when we have the exponential multiplied by time. We can guess that similar repeated roots should occur in the z domain. Let us consider a very general function ... 

Exponential / 7.2.5 Exponential Multiplied by Time / 7.2.6 Impulse (Dirac Delta Function 8 d) Inversion of Laplace Transforms Transfer Functions... 

The deformation functions, however, must also describe density accumulation in the bond regions, which in the one-center formalism is represented by the atom-centered terms. They must be more diffuse, with a different radial dependence. Since the electron density is a sum over the products of atomic orbitals, an argument can be made for using a radial dependence derived from the atomic orbital functions. The radial dependence is based on that of hydrogenic orbitals, which are valid for the one-electron atom. They have Slater-type radial functions, equal to exponentials multiplied by r1 times a polynomial of degree n — l — 1 in the radial coordinate r. As an example, the 2s and 2p hydrogenic orbitals are given by... 

The coefficient is merely a decimal number written in the ordinary way. That coefficient is multiplied by the exponential part made up of the base (10) and the exponent. (Ten is the only base which will be used with numbers in exponential form in the general chemistry course.) The exponent tells how many times the base is multiplied by the coefficient ... 

Exponential expressions are in the form xa. The number x is multiplied by itself a times. 

The complementary solution consists of oscillating sinusoidal terms multiplied by an exponential. Thus the solution is oscillatory or underdamped for ( < 1. Note that as long as the damping coefficient is positive (C > 0), the exponential term will decay to zero as time goes to infinity. Therefore the amplitude of the oscillations will decrease to zero. This is sketched in Fig. 6.6. 

According to spectral-kinetic parameters, the optimal conditions of luminescence excitation and detection, so called selection window (SW) parameters, were calculated in the following way. At optimal for the useful component excitation, the liuninescence spectra, decay time and intensity were determined for this mineral and for the host rock. After that, on the personal computer was calculated the proportion between useful and background signals for the full spectral region for each 50 ns after laser impulse. For calculation the spectral band was simulated by the normal distribution and the decay curve by the mono-exponential function. The useful intensity was multiplied by the weight coefficient, which corresponds to the concentration at which this component must be detected. 

The exponential term can be thought of as a Green function, with the time dependence always implicit. Thus an excitation at x causes a response at x whose phase is delayed by the distance between them multiplied by the real part of kp (this corresponds approximately to 2 /Ar), and whose amplitude is decreased exponentially by the distance between them multiplied by the imaginary part of kp (this corresponds to the decay associated with the propagation of the leaky Rayleigh wave). The magnitude x — x is used because... 

Solutions of Eq. 25-46 for nonsteady-state conditions are difficult to obtain analytically, yet numerical procedures are straightforward. For the case of slow reactions, more precisely for Da 1, the solution can be approximated by multiplying the time-dependent solution for a conservative substance with the exponential factor exp(- t). For instance, for the pulse input Eq. 25-20 is modified into ... 

The half-life is the time required for the number of nuclei present to decrease by a factor of 2. The number of decays that occur in a radioactive sample in a given amount of time is called the activity A of the sample. The activity is equal to the number of nuclei present, N, multiplied by the probability of decay per nucleus, A, that is, A = N. Therefore, the activity will also decrease exponentially with time,... 

This expansion is valid to second order with respect to St. This is a convenient and practical method for computing the propagation of a wave packet. The computation consists of multiplying X t)) by three exponential operators. In the first step, the wave packet at time t in the coordinate representation is simply multiplied by the first exponential operator, because this operator is also expressed in coordinate space. In the second step, the wave packet is transformed into momentum space by a fast Fourier transform. The result is then multiplied by the middle exponential function containing the kinetic energy operator. In the third step, the wave packet is transformed back into coordinate space and multiplied by the remaining exponential operator, which again contains the potential. 

Result of multiplying the time series in Figure 3.23 by a positive exponential (original signal is dotted line)... 

Simply to multiply correlations, such as have just been derived, by a suitable exp (—at) is obviously improper as it gives incorrect behaviour at small times. It may be empirically useful to allow such a correlation to run down nearly to its y(oo) and then switch in the exponential decay with an arbitrary smoothing at the transition. One way to do this while preserring the initial slope is to multiply by exp[- aifit + e — 1)], but this spoils the initial curvature. In any case, the switching in of the exponential ought to arise naturally from the analysis of the problem. Libration in a two-well potential. For simplidty we consider two equivalent equilibrium positions separated by tt radians. The similar potential wells are taken to be harmonic to a height much greater than kT, and separated by equal potential barriers on either side. 

Equations (2)-(5) represent a complete recipe for calculating the response of the spin system to a m.p. sequence. The spectrum is obtained, as in the real experiment, by a discrete Fourier transformation of the time series after it had been multiplied by a suitable filter function, for example, a decaying exponential. 

Fig. 45. Time/temperature scaling of spin lattice relaxation curves in a saturation-recovery experiment. At each temperature the raw amplitude data arc first normalized by the amplitude of the fully relaxed signal so that all values fall between zero (saturation at short times) and 1 (lull recovery at long times). The time points t are multiplied by the temperature at which the relaxation curve was obtained. If the relaxation is governed by the Korringa process, the scaled points fall on a temperature-independent curve, even if the relaxation is not simply exponential, as in the cases shown here. The sample is Pt/Ti02 of dispersion 0.60 (determined by electron microscopy) at several hydrogen coverages (calculated from the dispersion) 0.1, 0.5, and 1.0 monolayers. The squares in c show data at 110 K for another Pt/TiOi catalyst of dispersion 0.36.

Exponential notation is an alternative way of expressing numbers in the form fl ( a to the power ), where a is multiplied by itself n times. The number a is called the base and the number n the exponent (or power or index). The exponent need not be a whole number, and it can be negative if the number being expressed is less than 1. See Table 39.2 for other mathematical relationships involving exponents. 

In the exponential type potentials, the distance is multiplied by a constant and used as the argument for the exponential. Computationally it takes significantly longer time (typical factor of 5) to perform mathematical operations like taking the square root and calculating exponential functions, than to do simple multiply and add. The Lennard-Jones potential has the advantage that the distance itself is not needed, only R raised to... 

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