Linear-superposition assumption

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

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 ... 

To convert an optical signal into a concentration prediction, a linear relationship between the raw signal and the concentration is not necessary. Beer s law for absorption spectroscopy, for instance, models transmitted light as a decaying exponential function of concentration. In the case of Raman spectroscopy of biofluids, however, the measured signal often obeys two convenient linearity conditions without any need for preprocessing. The first condition is that any measured spectrum S of a sample from a certain population (say, of blood samples from a hospital) is a linear superposition of a finite number of pure basis spectra Pi that characterize that population. One of these basis spectra is presumably the pure spectrum Pa of the chemical of interest, A. The second linearity assumption is that the amount of Pa present in the net spectrum S is linearly proportional to the concentration ca of that chemical. In formulaic terms, the assumptions take the mathematical form... 

We note that s and v are the responses of the playback system, including mechanical components and amplification/ equalization circuitry, to the recorded audio and noise signals, respectively. The assumption of a linear system allows the overall response x to be written as the linear superposition of individual responses to signal and noise components. 

It must be pointed out that the formula given in Figure 12 is extremely simplified because it has been derived in analogy to the model illustrated in Figure 9 under the assumptions of a linear superposition of deformation components and the action of an instantaneous constant load. Because of differences between these idealized assumptions and the actual experimental conditions (e.g. continuous loading instead of instantaneous loading) the formula should be considered only as a first rough approximation. 

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 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. 

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... 

Firstly, it has been shown that there may be many experimental problems in a direct determination of the experimental fimction. In shear, damping functions obtained from step strain and from step strain rate experiments do not match each other. This poses an important question on the validity of the separability assumption in the short time rai e. Significant departures from this factorization have already been observed in the case of narrow polystyrene fractions by Takahashi et al. [54]. These authors found that time-strain superposition of the linear and nonlinear relaxation moduli was only possible above a cert2un characteristic time. It is interesting to note that this is predicted by the Doi-Edwards theory [10] and according to this theory, this phenomena is attributed to an additional decrease of the modulus connected to a tube contraction process and time-strain separability may hold after this equilibration process has been completed. Other examples of non-separability were also reported by Einaga et al. [55] and more recently by Venerus et al. [56] for solutions. 

The other aspect of the symmetry basis for AOM, and we now return to the more general case of Eq. (3) where the ligands are not necessarily linearly hgating, is that expressed by assumption 111 (p. 71). This assumption makes AOM an additivity model or a superposition model (76). This was discussed thoroughly in a previous paper (70) where it was shown that the general matrix element of AOM within an I basis contains the rotation ) matrices Ddimensional irreducible representation of the three-dimensional rotation group. We shall not pursue this general case here. 

The superposition principle, which forms the basis of all multiple-dose models in this section, is true only as long as all elimination processes follow first-order (linear) elimination kinetics. Since the assumption of first-order elimination kinetics has already been made for all the previous single-dose models that are being combined by superposition, the application of the superposition principle does not add any new model assumptions. 

The assumption of constant C therefore, permits separation of the momentum equation from its dependence on the energy equation and results in an energy equation that is linear in I so that general solutions can be obtained from a superposition of individual solutions. 

Nonuniform Surface Temperature. Nonuniform surface temperatures affect the convective heat transfer in a turbulent boundary layer similarly as in a laminar case except that the turbulent boundary layer responds in shorter downstream distances The heat transfer to surfaces with arbitrary temperature variations is obtained by superposition of solutions for convective heating to a uniform-temperature surface preceded by a surface at the recovery temperature of the fluid (Eq. 6.65). For the superposition to be valid, it is necessary that the energy equation be linear in T or i, which imposes restrictions on the types of fluid property variations that are permitted. In the turbulent boundary layer, it is generally required that the fluid properties remain constant however, under the assumption that boundary layer velocity distributions are expressible in terms of the local stream function rather than y for ideal gases, the energy equation is also linear in T [%]. 

Use the shear creep data in Figure 4.4, together with the method of time-temperature superposition, to estimate the shear creep compliance for linear polyethylene at 20°C and a creep time 10 s. Ust the assumptions that you make in this long extrapx>lation of the creep data. 

The pseudo-elastic design method may be used for components submitted to intermittent loading, provided that the intervals during which the material is unloaded are suffident to allow virtually complete recovery. Some manufacturers provide recovery data that enable the validity of this assumption to be tested. Altemativefy, the Boltzmann superposition prind-ple may be used to determine whether the assumption gives a satisfactory q>proximatk>n (see Oiapter 4). If not, or if die 1 is varying in a more complex manner, a more complete anafysis of deflection behaviour based upon the Boltzmaim prindple may be necessary. Linearity can be assumed for strains up n> about 0.005. 

Since both problems considered in this section are linear, a superposition of the two generalized forces, i.e., simultaneous application of the pressure gradient and the electric field, leads to relationship (1) with the electro-osmotic tensors (of, a, p, and K), This is true only under the assumption that the ion distribution is slightly distorted, by appliction of either VP or E. The condition imposed on the latter quantity is obviously... 

The problem is that this superposition principle is the basis of many theories, for instance electromagnetic and gravitational field theories. The string theory has a large part of its elanental assumptions based on it too. One of the most used properties of the wave function in qnantum mechanics is precisely its linearity. 

Boltzmann Superposition and the Constitutive Law for Linear Viscoelasticity. The underlying assumption of the Boltzmann superposition principle is that responses to loads or deformations applied to a material at different times are linearly additive. This set of assumptions leads to the constitutive laws of linear viscoelasticity theory which can be considered as a linear response theory. For discussion purposes, consider a Maxwell material that is subjected to a two-step deformation history. The history is such that a deformation yi = Ayi... 

The present theory of calorimetry is a result of the authors own work. Its essential feature is the simultaneous application of the relationship and notions specific to heat transfer theory and control theory. The present theory has been used to develop a classification of calorimeters, to discuss selected methods of determining thermal effects and thermokinetics, and to describe the processes proceeding in calorimeters of various types. Calorimeters have been assumed to constitute linear systems. This assumption allowed the principle of superposition to be used to analyze several constraints acting simultaneously in and on the calorimeter. 

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