Superposition, assumption

Double-layer interaction potentials (t/di) are evaluated with Gregory s (28) constant-charge, linear-superposition assumption (LSA) expression... 

Kirkwood derived an analogous equation that also relates two- and tlnee-particle correlation fiinctions but an approximation is necessary to uncouple them. The superposition approximation mentioned earlier is one such approximation, but unfortunately it is not very accurate. It is equivalent to the assumption that the potential of average force of tlnee or more particles is pairwise additive, which is not the case even if the total potential is pair decomposable. The YBG equation for n = 1, however, is a convenient starting point for perturbation theories of inliomogeneous fluids in an external field. 

It is apparent therefore that the Superposition Principle is a convenient method of analysing complex stress systems. However, it should not be forgotten that the principle is based on the assumption of linear viscoelasticity which is quite inapplicable at the higher stress levels and the accuracy of the predictions will reflect the accuracy with which the equation for modulus (equation (2.33)) fits the experimental creep data for the material. In Examples (2.13) and (2.14) a simple equation for modulus was selected in order to illustrate the method of solution. More accurate predictions could have been made if the modulus equation for the combined Maxwell/Kelvin model or the Standard Linear Solid had been used. 

The method of superposition of configurations is essentially based on the assumption that the basic orbitals form a complete set. The most popular basis used so far in the literature is certainly formed by the hydrogen-like functions, which set contains a discrete and a continuous part. The discrete subset corresponds physically to the bound states of an electron around a proton, whereas the continuous part corresponds to a free electron scattered by a proton, or classically to the elliptic and hyperbolic orbits, respectively, in a central-field problem. 

The first assumption involved in using the Boltzmann superposition principle is that elongation is proportional to stress, that is, compliance is independent of stress. The second assumption is that the elongation created by a given load is independent of the elongation caused by any previous load. Therefore, deformation resulting from a complex loading history is obtained as the sum of the deformations that can be attributed to each separate load. 

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... 

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. 

The non-covalently bound BPDEs to DNA formed initially appear to be intercalation complexes (1 6,52-55) Meehan et al. (1 6) report that the BPDE intercalates into DNA on a millisecond time scale while the BPDE alkylates DNA on a time scale of minutes. Most of the BPDE is hydrolyzed to tetrols (53-56). Geacintov et al. (5l ) have shown with linear dichroism spectral measurements that the disappearance of intercalated BPDE l(+) is directly proportional to the rate of appearance of covalent adducts. These results suggest that either there may be a competition between the physically non-covalently bound BPDE l(+) and an externally bound adduct or as suggested by the mechanism in the present paper, an intercalative covalent step followed by a relaxation of the DNA to yield an externally bound adduct. Their results for the BPDE i(-) exhibit both intercalative and externally bound adducts. The linear dichroism measurements do not distinguish between physically bound and covalent bound forms which are intercalative in nature. Hence the assumption that a superposition of internal and external sites occurs for this isomer. 

It remains to comment on the fact that, contrary to expectation, the integral relation for G involves only g<2) and no higher correlation functions. This arises because in writing Eq. (172) the implicit assumption is made that the probability of finding a second partner, say / , to a given defect, a, is the same as it would be if the defect a did not already have a partner / . In fact g<3,( a/ S ) has been replaced by g(2,( a/J ). This is even stronger than the Kirkwood superposition approximation33... 

The formula for specialized distribution functions makes no such assumptions and hence involves g<3). It also involves g(4), g(5),.. . since correction is made for the possibility of a defect having two, three,. . . other partners simultaneously. Using Eq. (171) and the superposition approximation one finds that for the sodium chloride type lattice... 

As shown in Fig. 5.4, the flow domain can be denoted by 2 with inlet streams at Ain boundaries denoted by 3 2, (/el,..., Ain). In many scalar mixing problems, the initial conditions in the flow domain are uniform, i.e., cc(x, 0) = 40). Likewise, the scalar values at the inlet streams are often constant so that cc(x e 3 2, t) = c(f for all / e 1,..., Nm. Under these assumptions,38 the principle of linear superposition leads to the following relationship ... 

There are many types of deformation and forces that can be applied to material. One of the foundations of viscoelastic theory is the Boltzmann Superposition Principle. This principle is based on the assumption that the effects of a series of applied stresses acting on a sample results in a strain which is related to the sum of the stresses. The same argument applies to the application of a strain. For example we could apply an instantaneous stress to a body and maintain that stress constant. For a viscoelastic material the strain will increase with time. The ratio of the strain to the stress defines the compliance of the body ... 

Our next assumption is that we can use the superposition of base states to predict the probabilities of experimental outcomes. For example, consider the energy observable. Suppose we have a finite linear combination... 

From the beginnings, attempts to model the line shapes of collision-induced absorption spectra were based on the assumption that the various rotational lines of induced spectra, Figs. 3.10 through 3.14, are superpositions of scaled and shifted line profiles, g(C)(v), °f a small number of different, e.g., overlap- and quadrupole-induced, types [313, 404],... 

I. The theory of molecular dislocations used to describe deformation and relaxation is based on the assumption of a distribution of the thermal vibration energy similar to that applicable to gas molecules. In general we consider the superposition of thermal motion in one direction to be given by the geometric position of two possible conformations. In this single dimension the phase space elements are dx (space coordinates) and dp (momentum coordinates). The sum of states... 

The superposition of the calculated transitions corresponding to all the tautomers (Figure 2) reproduces satisfactorily the entire experimental spectrum. The low-intensity transition observed experimentally in the range 350-450 nm is predicted as forbidden in ZINDO/S calculations. The comparison between these calculations allowed transitions and the experimental absorption spectrum (see Section 13.14.3.1.1 (i)) leads to the assumption that the TtJt band located between 250 and 330 nm belongs principally to tautomers 4 and 7 (thione form). 

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