Density superposition

Time-crosslink density superposition. Work of Plazek (6) and Chasset and Thirion (3, 4) on cured rubbers suggests that there is one universal relaxation function in the terminal region, independent of the crosslink density. Their results indicate that the molar mass between crosslinks might be considered as a reducing variable. However, these findings were obtained from compliance measurements on natural rubber vulcanizates,... 

Time-crosslink density superposition is inapplicable to our data. 

This point Is further demonstrated in Figure 6 where we have applied arbitrary vertical and horizontal shifts to the log D(t) curves for the MAPO crossllnked polybutadiene networks at the various crosslink densities. Superposition of all three curves along the entire time domain Is clearly impossible. The short time parts, however, superposed very well. 

Ebbels T, Keun H, Beckonert O, et al. (2003). Toxicity classification from metabonomic data using a density superposition approach CLOUDS . Analyt. Chim. Acta. 490 109-122. 

Figure 1. The charge-density difference (bonding charge density) between NiaX and the superposition of neutral Ni and X atomic charge densities on the (001) planes for (a) X = A1 and (b) X = Si. The solid (dotted) contours denote contours of increased (decreased) density as atoms are brought together to form the NiaX (X = Al, Si) crystal. Contours start from 4.0 X 10 e/(a.u.) cind increase successively by a factor of root 2.

Intensity at a Point of Superposition (1.17) The measurable physical parameter of an optical wave is its energy density or intensity. If two fields are superimposed, the measured intensity is given by the sum of the individual intensities plus aterm which describes the long term correlation of the field amplifudes. Long ferm means time scales which are large compared to the inverse of the mean frequency uj/2Tt (about 10 s) the time scale is set by the time resolution of the detector. This is why the held product term is expressed in the form of an ergodic mean ). An interferometer produces superimposed helds, the correlation of which carries the desired information about the astronomical source. We will discuss exactly how this happens in the following sections. 

Fig. 3 Probability density for the Fig. 4 Probability density for the vibrational eigenstates (/>o, and (j)2 optimal superpositions and tp ...

In order to generate a set of calculated structure factors Fc(Q) from a set of coordinates, it is necessary to introduce a model for the time variation of the electron density. The usual assumptions in macromolecular crystallography include harmonic isotropic motion of the atoms and in addition, the molecular scattering factor is expressed as a superposition of atomic scattering factors. With these assumptions the calculated structure factor (equation III.2) is given by.27... 

Fig. 8.7 Superposition of density-of-states for B-N bonding BN>. states with corresponding metal states, reflected by their valence states from alkaline-earth (as Ca), lanthanide (as La), and 3d-metal (as Ni), and corresponding block schemes...

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

At the beginning we assumed that masses are absent inside the volume V and, correspondingly. Equation (1.107) describes the potential of the attraction field, caused by masses located outside V. Now suppose that there are also masses in this volume and their distribution is characterized by the density Then, applying the principle of superposition, we obtain... 

In the previous chapters our attention was paid mainly to the study of the figure of the earth and with this purpose in mind we represented the total field as a sum of the normal and secondary fields. In this chapter, we will discuss a completely different application of the gravity method, related to exploration geophysics, in which the gravitational field is measured in order to study lateral changes of the density near the earth s surface. By analogy, we also represent the gravitational field as a superposition of two fields ... 

Thus, while the relation between the partial current densities and potential is exponential, in the region of low polarization a linear relation is obtained between polarization and the net CD, owing to a superposition of the currents of forward and reverse process. At A = 10 mV, the error introduced by the approximation above will be between 1 and 20%, depending on the relative values of a and p it becomes even smaller with decreasing polarization. Hence we can by convention consider the interval of polarization values between -10 and 10 mV as that of low polarization where the linear relation (6.6) is valid. 

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