Three-chain cumulant

Cumulative double bonds are those present in a chain in which at least three contiguous carbon atoms are joined by double bonds non-cumulative double bonds comprise every other arrangement of two or more double bonds in a single structure. The generic name cumulene is given to compounds containing three or more cumulative double bonds. 

The only, but essential, use we make here of Eq. (11.19) is to extract the properties of the functions W u) etc. Note that by construction these functions are independent of (L, M), of momenta, and of chain lengths. Writing down Eq. (11.19) for three different choices of (L,M) we therefore have a system of three equations linear in W, 1/V, t/, which allows us to express these functions in terms of renormalized cumulants, Three important results immediately follow. 

Cumulative increase in contaminant body burdens up three or more trophic levels in a food chain. Biomagnification occurs when the intake of a contaminant exceeds the capacity of an organism to excrete and/or metabolize the contaminant in question. Volume 2(10). 

Figure 12-3 Plots of the degree of polymerization versus the cumulative number of synthetic steps for various repetitive syntheses (a) conventional linear solid-phase synthesis (b) nonlinear straight-chain sequence synthesis (c) dendrimer synthesis (branching multiplicity of three) (d) double exponential den-drimer synthesis (branching multiplicity of three). In all cases, the degree of polymerization is defined as the total number of monomer units per polymer molecule.

Ligands with chains of carbon atoms that have cumulated (consecutive) double bonds, designated cumulenylidene ligands, are also known. Such metallacumulene complexes have drawn interest because of possible applications as 1-dimensional molecular wires and for nse in nanoscale optical devices. In recent years, complexes with 2- and 3-carbon chains have also been developed as effective catalysts. Metallacumulene complexes with carbon chains containing three or more atoms have recently been reviewed. ... 

Next, focus on products handled by the supply chain and verify if the supply process matches product characteristics. If all SKUs are sorted in order of decreasing sales (i.e., from the highest to lowest sales levels) and the cumulative sales are plotted vs. the corresponding ranking of products, the data usually generates a Pareto distribution. Products can thus be divided into three categories A products that represent 20% of the products but 80% of the sales volume, B products that represent 30% of the products and 15% of the sales volume, C products that represent 50% of the products and 5% of the sales volume. 

Figure 8. Diagonal portion of the Green function for the randomly tagged chain at high dilution. The universal curves labeled 3R, 2 and 2C are, respectively, the three-body rational self-consistent approximation, the two body self consistent approximation and the two-particle cumulant. Taken from Figure 2 of reference [13].

Two-Particle Cumulant Approximation. The preceding discussion on the three particle Fade and the other approximations is not limited to the case of ideal chain statistics, but it is only in this... 

In Figure 8 we have plotted Equation (62) as well as the two body self-consistent approximation and the three-body rational self consistent approximation for this problem. The three-body rational approximation is believed to describe excitation transport on an ideal chain quite accurately [12]. Although the agreement between curves 3R and 2C is not as good as was observed for the case of randomly distributed chromophores in an infinite volume [13], the two body cumulant is bounded by the two-body and three- ody curves. Thus, it should give a quantitative approximation to G (t). It should be obvious that combination of Equation (57) for C with Equation (62) should lead to a straightforward determination of from measurement of the transient fluorescence anisotropy. 

---