Delta function multiplication

Table 1. Basic Quantities in Analyses of CW Laser Scattering for Probability Density Function. In Eq. 1 within the table, F(J) is the photon count distribution obtained over a large number of consecutive short periods. For example, F(3) expresses the fraction of periods during which three photons are detected. The PDF, P(x), characterizes the statistical behavior of a fluctuating concentration. Eq. 1 describes the relationship between Fj and P(x) provided that the effects of dead time and detector imperfections such as multiple pulsing can be neglected. In order to simplify notation, the concentration is expressed in terms of the equivalent average number of counts per period, x. The normalized factorial moments and zero moments of the PDF can be shown to be equal by substitution of Eq.l into Eq.2. The relationship between central and zero moments is established by expansion of (x-a)m in Eq.(4). The trial PDF [Eq.(5)] is composed of a sum of k discrete concentration components of amplitude Ak at density xk. [The functions 5 (x-xk) are delta functions.]...

This, as we saw in Chapter 3, is a Dirac delta function, which always sums to zero except when (a s) becomes integral, that is, when a s is a multiple of X. Because a is invariant, only when o and k articulate certain relationships that result in specific directions for s will that be true. When s takes on those unique directions, then waves scattered by all N points have relative phases of zero and thus constructively interfere. The amplitudes of all of the N scattered waves then add arithmetically. 

This is a nonlinear Langevin equation of the first order. It contains a multiplicative noise term. The noise A(t) may be represented, according to van Kampen [58], by a random sequence of delta functions. Thus each delta function jump in A(t) causes a jump in (r). Hence, the value of at the time the delta function arrives is indeterminate and consequently so is g at this time also. A problem arises, as the equation does not indicate which value of one should substitute in g whether the value of before the jump, the value after or a mean of both. 

The full multiple spawning (FMS) method has been developed as a genuine quantum mechanical method based on semiclassical considerations. The FMS method can be seen as an extension of semiclassical methods that brings back quantum character to the nuclear motion. Indeed, the nuclear wave function is not reduced to a product of delta functions centered on the nuclear positions but retains a minimum uncertainty relationship. The nuclear wave function is expressed as a sum of Born-Oppenheimer states ... 

It follows that the probability density cp, 6 t) of the trantit time 9 at time t can be expressed as a multiple average of a delta function 5 taken over... 

A multiple regression analysis is performed on a matrix constructed from a series of chemical shifts values associated with parameters which have the properties of a delta function, i.e. parameters which are equal to 1 or O according as whether a substituent is present or not in the considered position. This treatment, which has been very successful in spectroscopy, leads to additive substituent incre-... 

Note since a and p are essentially the Dirac delta functions in the spin coordinate CO, the process of integration reduces here to scalar multiplication of vectors.]... 

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,... 

According to Eq. (7.115), the desired response is obtained by convolution of the basic response (Rsp delta function ) with the respective excitation function. The delta function is now just the derivative of the step function of interest to us, so that instead of Rsp delta function it is also possible to use the time derivative of the response to the step function, that we already know (5/ Rsp step function ) . Since the Laplace transformation ( ) converts the convolution into a multiplication it is more concise to write... 

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