Box-normalisation

We use (6.22) to obtain from (6.21) the box-normalised wave-packet collision state at t = 0. 

The key quantity in the calculation of experimental observables is the collision amplitude (6.13). The box-normalised collision amplitude for the wave packet is given for t = 0 by using (6.26). 

Since the probability of the state is uniform over the whole box, the box normalisation is The properly-normalised wave-packet quantity that is independent of L is... 

We continue the formal development in terms of the box-normalised quantities. 

After some complex-number arithmetic we find for the box-normalised quantities... 

In the discrete notation for box-normalised channels the differential cross section i is defined as the transition rate (6.39) divided by the incident flux vL. ... 

In this section we first summarise the meaning of the notation for the channel and collision states with box normalisation and in the continuum limit L —> 00. We then define notation for the limit 6 —> 0-1- and write the corresponding integral equations. 

With box normalisation the channel states d>,) are countable. For discrete target states the index i stands for the internal quantum numbers n,j,m,, v of the target and projectile and the box quantum numbers nix,nty,ni characterising the relative motion. When L —> 00 the box quantum number set is replaced by the momentum continuum k,. The limiting procedure is summarised as follows... 

Note that this kind of normalisation, via the norm function, can only be performed column- (or row-) wise via a loop as seen in the Matlab box above. Calling norm with one matrix argument determines a different kind of normalisation coefficients. We refer to the Matlab help and function references for more detail. 

Figure 1. Normalised absorption at the Pt L edges boxes experimental data full lines calculated Lorentzian contribution, arctangent contribution and complete equation (1).

A full report of the defined integrals can be listed in a report dialog box, including the size or width of the integral region (in data points), the start and end rows and columns (number and ppm), the integral value (absolute, normalised) and the calculation mode used. 

Groupe d Etude International pour la Normalisation des Essais d Explo-sifs Secretary Der. Per Anders Persson, Swedish Detonic Research Foundation, Box 32058, S 12611 Stockholm, Sweden 87 199... 

The double limit e —> 0+, L —> oo must be taken in such a way that the whole system is inside the normalising box at all times, x is the length of the incident train of particles divided by their velocity v. We require... 

We must first find the density of final states, which we characterise in terms of the relative momentum /c . The permitted values of k in the normalisation box are given by (4.7). 

The differential cross section is independent of L, the size of the normalisation box. 

The first line of (29) is the zero-order polarisability and, of course, it approaches the well-known free-atom value 9/2Z4 as the box radius becomes infinitely large. The next two terms in (29) constitute the first-order correction, a to At first sight it might be thought that the e2s factor makes the final term in (29) a second-order quantity but, due to the presence of ex, this is not so. It is a result of the fact that the e2s term in the normalisation of i/Us is first order (see [31]). 

Climate dataset generation The climate dataset for simulation comprises of a synthetic wind speed, significant wave height and wave period time series. These are generated using a Multivariate Auto-Regressive (MAR) model, shown in Equation 1, normalised by the mean of the data p where is the simulated wind speed at time-step t, n is the number of variables, is a variable state vector, is a matrix of the MAR model coefficients and is a noise vector with mean zero and covariance matrix of the data, order p (Box and Jenkins, 1970). 

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