Multiple counting

Fig. 11-7. Block diagram of multiple counting circuits used with spectrograph of Fig. 11-6. (Courtesy of Hall, Proceedings, Sixth Annual Conference on Industrial Applications of X-ray Analysis, Denver, Colo., 1957, page 297.)...

The LU decomposition is essentially a Gaussian elimination, arranged for maximum efficiency. The chief reason for doing an LU decomposition is that it takes fewer multiplications than would be needed to find an inverse. Also, once the LU decomposition has been found, it is possible to solve for multiple right-hand sides with little increase in work. The multiplication count for an n x n matrix and m right-hand sides is... 

No indications of spontaneous fission activities on the moon surface were obtained by neutron multiplicity counting of 3 kg of lunar rocks [45]. 

Stimulated by these studies, samples of primitive meteorites were inspected by neutron multiplicity counting. In the Allende meteorite available in large quantities, a weak fission activity at the 1014 g/g level was reported [71-73] but could not be chemically enriched [74-76]. 

One packaging (HDPE bottle with induction seal) Multiple packaging Multiple packaging (multiple count bottles, and multiple material blisters) Consumption test (for bottles) Light test (1200 K lux hours) Micro-testing at annual time points (to test the entire packaging operation and to track the changes over time)... 

Reduced density matrices, introduced by Husimi [9], yield corresponding probability densities for the presence of n n < N) particles simultaneously in selected volume elements dxj,... x thus, for n = 1, the probability/unit volume of finding an electron (no matter which) at Xj will be obtained by integrating fV p (x x) over the positions of all fV — 1 volume elements dx2,...dxjv and dividing by [N — 1) (to avoid multiple counting). The quantity so defined is the one-particle reduced density matrix more explicitly, it becomes... 

FIGURE 20.2 The type of devices used against U.S. military and civilian personnel in terrorist attacks during 2008.The majority of the 34,124 injirries sustained were due to explosives, explosives as part of improvised explosive devices (lEDs), and in vehicle bombs. NB Multiple counting occurred in events with multiple threat types. Source Center, U.S.N.C. (2009) 2008 Report on Terrorism, N.C. Center, Editor. 

In order to verify Eq. (3.52) we note that the dispersion energy according to expression (3.41) is derived from an infinite sum over all closed graphs drawn between molecules i, j, k,... as vertices. The vertex i yields the factor X,-, the line connecting i and J yields the factor T,-j. The factor 1/n in front of the sum of order n in Eq. (3.41) avoids multiple counting of graphs. It can be omitted if at the same time the initial point of each graph is fixed. 

Lunar rocks showed no indication of spontaneous fission activity in neutron multiplicity counting of a 3 kg sample [45]. 

Stimulated by these studies, samples of primitive meteorites were inspected by neutron multiplicity counting. In the Allende meteorite available in large... 

Accounting for the possibility that each fiber can cross the sphere in any position requires integration over the ranges Q < Si< R followed by division by n to avoid multiple counting of the same events. Finally we have ... 

The PqS are probability densities, e.g. Po(Osj,j) for a contact between segments Sj and both in molecule 1, and Poi ju2 kik2 hi2) simultaneous intermolecular contacts j j2 and fei,/c2 between chains 1 and 2, conditional upon an intermolecular contact already existing between segments and (2 of the two chains. The subscript zero signifies that these are taken as random flight distributions. The summations are to be understood as limited to avoid multiple counting of contacts. 

The multiple counting of lobe interactions for compressed dimer-shaped particles gives rise to anomalous behavior for the structural and mechanical properties, which do not approach those for disks in the 1 limit. We now describe a micromechanical model based on the area of overlap between compressed dimershaped particles that result in structural and vibrational properties that converge smoothly to those in the —> 1 limit at all values of the compression A( ). 

Figure 3.20 shows the vibrational density of states D(tu) (in the harmonic approximation) for static packings of dimer-shaped particles with and without multiple counting of the lobe interactions as a function of A(j) and With multiple counting, the maximum frequency in D(cu) can increase significantly above... 

Figure 3.21 shows the vibrational density of states D(o ) for static packings of bidisperse dimers as a function of compression A( ) on a logarithmic scale to focus on the low-frequency behavior. D(o ) contains a strong peak of mainly rotational modes at low frequency. With multiple counting of the lobe interactions in the total potential energy, the location of the low-frequency peak comin does... 

Figure 3.20 (Continued) The vibrational density of states D(a>) (in the harmonic approximation) for static packings of N = 48 bidisperse dimers at aspect ratio ct = 1 (solid line), 1.01 (dashed line), 1.1 (dot-dashed line), and 1.3 (dot-dot-dashed line) at compression A(() = 0.03 (c) Largest frequency cUmax at which D(tUn,ax) = 0.1 versus A(j) with (gray) and without (black) multiple counting of the interactions between lobes for aspect ratio ao = 1.0 (solid line), 1.01 (dashed line), 1.03 (dash-dash-dotted line), 1.1 (dash-dotted line), 1.3 (dot-dot-dashed line), and 1.5 (dotted line).

---