Poisson formula

In our assumptions, system (27) has the finite number of roots (by Lemma 14.2 in Bykov et al., 1998), so that the product in Equation (26) is well defined. We can interpret formula (26) as a corollary of Poisson formula for the classic resultant of homogeneous system of forms (i.e. the Macaulay (or Classic) resultant, see Gel fand et al., 1994). Moreover, the product Res(R) in Equation (26) is a polynomial of R-variable and it is a rational function of kinetic parameters fg and Tg (see a book by Bykov et al., 1998, Chapter 14). It is the same as the classic resultant (which is an irreducible polynomial (Macaulay, 1916 van der Waerden, 1971) up to constant in R multiplier. In many cases, finding resultant allows to solve the system (21) for all variables. ... 

Occasionally, measured band progressions are theoretically analyzed by applying the Poisson formula on the intensity distribution. According to this, the relative band intensity of the member n in the progression is... 

Firstly, the Poisson formula (4.26) is employed to provide the extended useful integral formulation... 

The following Poisson formula can be used to coimect volume charge with the potential ... 

Correspondingly, the probability AP(f(f) to form no nuclei within Au can be expressed by means of the Poisson formula (see Chapter 3, equation (3.1)) ... 

The shot noise process is defined in terms of the Poisson process by means of the formula... 

The Poisson distribution is actually a special case of the binomial distribution, a fact that is only of mild peripheral interest here, as we will not be using that fact. The formula for the Poisson distribution is... 

In the practical matter of performing the summations indicated for the various formulas that must be evaluated, the question arises as to how many terms need to be included this question is analogous to the need to decide the limits of integration that was implicit in evaluating the analogous expressions for the Normal Distribution. In the case of the Poisson distribution this is one decision that is actually easier to make. The reason is... 

We can model this as a Poisson process and use the simple formula ... 

Kimura (1983) developed a curve fitting formula to this graph, which is similar to the simple Poisson equation but with an extra term ... 

In the experiment, the transmission intensities for the excited and the dark sample are determined by the number of x-ray photons (/t) recorded on the detector behind the sample, and we typically accumulate for several pump-probe shots. In the absence of external noise sources the accuracy of such a measurement is governed by the shot noise distribution, which is given by Poisson statistics of the transmitted pulse intensity. Indeed, we have demonstrated that we can suppress the majority of electronic noise in experiment, which validates this rather idealistic treatment [13,14]. Applying the error propagation formula to eq. (1) then delivers the experimental noise of the measurement, and we can thus calculate the signal-to-noise ratio S/N as a function of the input parameters. Most important is hereby the sample concentration nsam at the chosen sample thickness d. Via the occasionally very different absorption cross sections in the optical (pump) and the x-ray (probe) domains it will determine the fraction of excited state species as a function of laser fluence. 

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