Scaling assumption

This is Widom s scaling assumption. It predicts a scaled equation of state, like the law of corresponding states, that has been verified for fluids and magnets [102]. 

Table 6. Relationship between the fractal dimension dp the exponent for the molar mass dependence of the second virial coefficient and the expected exponent m for the osmotic modulus when the scaling assumptions of Eqs. (93)-(96) are made. The experimental data were derived from the exponents for the second virial coefficient...

Different critical exponents are not independent. The relationships between them can be obtained by employing the scaling assumptions. In particular, one can assume that the average number of i clusters (i.e., clusters containing j sites) per site can be represented at arbitrary and i as (72)... 

Rammal and Toulouse [140] introduce a scaling assumption for mode frequency on a fractal. Changing the length scale of the object from L —> L/b, the frequency changes as... 

Family et al. [15] extended the scaling assumption for the particle-size distribution of irreversible aggregation to cases involving also reversible aggregation. Using part of their derivation, let S(k,t) be the mean particle size defined by... 

McHorney, C.A. Ware, J.E., Jr. Lu, J.F. Sherbourne, C.D, The MOS 36-item short-form health survey (SF-36) III. Tests of data quality, scaling assumptions, and reliability across diverse patient groups. Med. Care 1994, 32, 40-66. 

Furthermore, the following scaling assumptions have been made [122] ... 

In the semidilute regime, the scaling assumption is that the correlation length decreases as a power law in concentration ... 

The dynamic scaling assumption relates the island density... 

Table A2.5.1 shows the Ising exponents for two and three dimensions, as well as the classical exponents. The uncertainties are those reported by Guida and Zinn-Justin [22]. These exponent values satisfy the equalities (as they must, considering the scaling assumption) which are here reprised as functions of P and y ...

This separation-of-time-scales assumption allows the electrons to be described by electronic wavefunctions that smoothly ride the molecule s atomic framework. These electronic functions are found by solving a Schrodinger equation whose Hamiltonian contains the kinetic energy of the electrons, the Coulomb... 

It is remarkable to check how a is approximately constant for different values of Z, a result which agrees with the scaling assumptions. 

In principle, the adsorption length D depends also on the ionic strength through /c b As discussed below, the scaling assumption (Eq. 59) is only valid as long as K and D are not of the same order of magnitude. Otherwise, h should be a function of both kx... 

We find numerical solutions for the PE profile equations in various cases. These numerical solutions agree well with simple scaling assumptions describing the adsorption of PEs. Scaling expressions for the amount of adsorbed polymer T and the width D of the adsorbed layer, as a function of the fractional charge / and the salt concentration Cgait> are obtained for two cases constant fs and constant a. 

This scaling assumption implies an asymptotic expansion of the scaling function of the form... 

Table 8.2 provides key parameters for the AHTR-VT, AHTR-IT, S-PRISM, and GT-MHR reactors used here as the basis for cost scaling assumptions. There are two major assumptions and a number of additional assumptions used in the cost comparisons that are summarized in the following sections. Detailed comparisons at a subsystem level are given in Table 8.3. 

The plasma physics basis for CT reactor projections is derived from MHD stability theory and experimental observations of macrostability, and from postulated transport scaling or scaling of related devices (tokamak and diffuse, toroidal pinch). A central issue, on which different reactor projections can be based, centers on the observed macrostability of certain CT s (reversed field theta-pinch Toroidal apparent contradiction with MHD stability theory. Although the conflict may be resolved by inclusion of finite-ion-larmor radius (FLR) effects in the theory, scaling assumptions to reactor conditions may or may not invoke FLR. 

Effect of Time-Scale Assumptions on Process Models and Their Reconciliation... 

Thermoreversible gelation has been studied in o-xylene for syndiotactic poly(methyl methacrylate) based block copolymers. The dynamic properties of solutions and gels have been analyzed and discussed on the basis of scaling assumptions. At the gel point, where the loss angle tan 5c=G 7G is independent of the probing frequency, the samples obey the typical power law G ((o) G (co) o... 

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