Delta function representation

If we interpret the weights and abscissas as a delta-function representation of the NDF ... 

For the moment estimates, we have seen that the composition PDF, /, (delta functions (i.e., the empirical PDF in (6.210)). However, it should be intuitively apparent that this representation is unsatisfactory for understanding the behavior of fyiir) as a function of fj. In practice, the delta-function representation is replaced by a histogram using finite-sized bins in composition space (see Fig. 6.5). The histogram h, (k) for the /ctli cell in composition space is defined by... 

Instead of solving evolution equations for the moments, the evolution of the weights and nodes in the quadrature approximation can be directly tracked (Marchisio Fox, 2005 McGraw Wright, 2003). The evolution equations for weights and nodes can be derived by formally substituting the delta-function representation of the NDF into Eq. (7.3). If the weights and nodes are continuous functions of time, this procedure yields... 

It should be recognized that in this relation the function G is the plane Gaussian kernel [see, for example, the form (6.28)], and T and S are the one-velocity plane transport kernels. The explicit forms of these functions are not required for this calculation if desired for other purposes, these may be obtained from the integral relation (7.191) by introducing a suitable delta-function representation of the source [see Sec. 5.2dj. 

If the Fourier integral representation of the delta function is introduced and the siun over all possible final-state vibration-rotation states Xf is carried out, the total rate Rj propriate to this non BO case can be expressed as ... 

The Fourier representation of the Dirac delta function leads then to the result... 

The delta-function addendnm removes the divergences from these matrix elements and allows their representation npon bases. When the pwc basis fnnctions are... 

This relation may be obtained by the same derivation as that leading to equation (B.28), using the integral representation (C.7) for the three-dimensional Dirac delta function. 

The inverse Fourier transform then gives an integral representation of the delta function... 

The partition function, Z(4>y), cannot be calculated exactly. It could be rewritten using the integral representation of the functional Dirac delta function and evaluated within the saddle place approximation. The calculations lead to the following expression [36,126,128] ... 

The obvious disadvantage of this simple LG model is the necessity to cut off the infinite expansion (26) at some order, while no rigorous justification of doing that can be found. In addition, evaluation of the vertex function for all possible zero combinations of the reciprocal wave vectors becomes very awkward for low symmetries. Instead of evaluating the partition function in the saddle point, the minimization of the free energy can be done within the self-consistent field theory (SCFT) [38 -1]. Using the integral representation of the delta functionals, the total partition function, Z [Eq. (22)], can be written as... 

There appears to be some confusion on this point in hie literature. In an Eulerian PDF code, the notional particles do not represent fluid particles, rather they are a discrete representation of the composition PDF (e.g., a histogram). Thus, the number of notional particles needed in a grid cell is solely determined by the statistical properties of the PDF. For example, if the PDF is a delta function, then only one particle is required to represent it. Note, however, that the problem of determining the number of particles needed in each grid cell for a particular flow is non-trivial (Pfilzner et al. 1999). 

For further derivations we will need the matrix element of the Dirac delta function, 8(r, , ). Using the following representation of the delta function,... 

Thirumalai et al. [7] noted that one can use a Gaussian prelimit representation of the delta function to do the job ... 

An analytical integration of an integrodifferential equation under a singular time boundary is always a complicated matter. The treatment of the method, based on a representation of the delta functional as a Fourier transform, and working in the complex plane, would be out of place in this report. It can be found in detail in Ref. 7) where also the solution obtained is discussed. It is shown that this solution is especially simple if the elution curves show a positive skewness, i.e. if they are tailed on the right-hand-side of their maximum (this is always true in PDC and GPC). A renormalization of the found concentration profile and a recalculation of the coordinates (z, t) to the elution volumina (V, V) then yield the spreading function of the considered column (Greschner 7))... 

In the vicinity of the maximum the energy-loss function (3.18) is of Lorentz form. With y- 0 it transforms into a delta function. In order to see this, let us use the representation of a delta function for a nonnegative variable (see the Mathematical Appendix A in Ref. 99) ... 

In terms of heat transfer this can be physically interpreted that, if the infinite heat source is at the boundary (the infinite character is given by the delta function), then for a smooth surface only half of the delta function must be included. The general integral representation of Poisson s equation becomes,... 

The integral representation of the delta-function in (2.56), which was used to derive (4.7), requires integration from —oo to +oo. 

Dirichlet function, which is an approximation of Delta function, S x). Various approximate representations of Dirac delta function are provided in Van der Pol Bremmer (1959) on pp 61-62. This clearly shows that we recover the applied boundary condition at y = 0. Therefore, the delta function is totally supported by the point at infinity in the wave number space (which is nothing but the circular arc of Fig. 2.20 i.e. the essential singularity of the kernel of the contour integral). 

A useful representation of the delta function is given by using the representation theorem together with (3.25). 

Before discussing the use of molecular dynamics to compute this quantity, one notes that by replacing the delta function by its integral representation ... 

Mathematically the bandlimited data can be introduced by filtering the theoretical signals. We consider, for example, the effect of this filtering on a delta-function. There is a well-known integral representation for a delta-function ... 

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