Cumulant density, properties

The cumulants of the density matrices, for short density cumulants, have a few nice properties [17]. We note in particular the Hermiticity, for example. 

In a more complex situation than that of two electrons occupying each its orbital one can expect much more sophisticated interconnections between the total spin and two-electron densities than those demonstrated above. The general statement follows from the theorem given in [72] which states that no one-electron density can depend on the permutation symmetry properties and thus on the total spin of the wave function. For that reason the difference between states of different total spin is concentrated in the cumulant. If there is no cumulant there is no chance to describe this difference. This explains to some extent the failure of almost 40 years of attempts to squeeze the TMCs into the semiempirical HFR theory by extending the variety of the two-electron integrals included in the parameterization. 

The electro-active surface of the porous carbon electrode for EDLCs is accessible only through the cumulative resistance of the electrolyte inside the pore. Therefore, the porous structure of the porous carbon becomes one of the most important factors influencing the energy/power densities. Fractal analysis has proven to be useful to describe the geometric and structural properties of rough surfaces and pore surfaces.56 66... 

The properties of dienes depend upon the arrangement of the double bonds and the other substituents. If the double bonds are widely separated(as in isolated dienes),the compound behaves like a simple alkene. If the compound is conjugated, there is potential delocalization of the tt electrons. Conjugated double bonds are, in general, more stable than isolated double bonds because of the resonance effect. Cumulated double bonds (allenes) are more unstable than isolated double bonds because of their geometry and electron density. 

Subsequently we show some important properties of the moment generating function. This include the number and weight average of the degree of polymerization and other averages, as well as the distribution function. There is a potential confusing situation in nomenclature. In polymer chemistry, the term cumulative distribution is used for what is called in mathematics the cumulative distribution function or simply distribution function. Further, polymer chemistry uses the term differential distribution for the mathematical term probability density function. 

In general, particle properties vary within a particle system and the particle size X should be regarded as a distributed parameter. Size distributions can be depicted by the cumulative function Q, the density function qr, or the transformed density function q ... 

Fractionating sizing techniques combine a size-related classification process with the measurement of particle quantities. The classification may yield a physical separation of difl erently sized particles (e.g. by sieving, cf. Fig. 2.1) or it may successively deplete the disperse system of the coarsest or finest particles (e.g. in a sedimentation column). Accordingly, the measured quantities, which can be absolute amounts or concentration values, represent either a density or a cumulative function of the size distribution. Anyhow, the type of quantification (e.g. weighing) determines the type of quantity of the measured size distribution (e.g. mass), whereas the classification defines the probed particle property. The classification process should be ideal (i.e. with maximum selectivity) and well-defined by a monotone correlation between the particle property (e.g. settling velocity) and the... 

Physical properties covered include purity, freezing point, vapour pressure, liquid density, vapour density, refractive index, rate of change of boiling point with pressure, latent heat of fusion, latent heat of evaporation, critical values, compressibility, viscosity, heat content, surface tension, and solubility. The 456 tables cover 434 aliphatic compounds and 22 miscellaneous compounds and elements. There is a cumulative index to the three volumes. Searching the Chemical Literature. 

From a statistical mechanics viewpoint, molecular systems share several properties with Ising spin systems if density fluctuations are substituted with spin flips. Polymer melts and blends without long-range interactions typically belong to the 3d-Ising universality class [92, 93], and critical points can be determined with techniques that were originally derived for spin systems [94]. To this extent we can calculate second- or fourth-order cumulants [94] ... 

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