Universal functions definition

In recent years, a dependable dipole function for He-Ar, last column of Table 4.3, has been obtained [278] which we compare with the universal dipole function mentioned [23], Fig. 4.5. The He-Ar interaction potential is one of the better known functions [13] and suggests Rmj = 6.518 bohr. Both functions were normalized to unity at the separation R = 5 bohr in the figure. The comparison shows that at small separations the logarithmic slope of the most dependable dipole function is roughly one half that of the universal p, and p diverges rapidly from p(R) for R — o. Similar discrepancies have been noted for other rare gas systems (Ne-Ar, Ne-Kr, and Ar-Kr [152]). Even if for these other systems the dipole function is not as well known as it is for He-Ar, it seems safe to say that for the rare gas mixtures mentioned the induced dipole function is definitely not identical with the universal function at the distances characteristic of the spectroscopic interactions the universal dipole function is not consistent with some well established facts and data. We note that the ratio of // (/ ) and the He-Ar potential is indeed reasonably constant over the range of separations considered (not shown in the figure). 

There is no universally accepted definition. Some definitions of the H bond are based on a single operational criterion (such as an infrared spectral shift), others on a listing of suitable functional groups, and still others on a mathematical or physical picture of H bond formation. We will use the term H bond to refer to an interaction that fits the following operational definition ... 

Functionally, the effective dwell time (EDT) at 90% level can be defined as the time it taken by the forcetime curve to traverse the 90% of the peak height. Likewise, the effective contact time (ECT) is the time between points at 10% of the peak height. The shape of the force curve depends, as we know, on the deformation mechanisms of the powder and therefore, all other variables being equal, EDT and ECT will be different for brittle and plastic materials. Although somewhat useful for material characterization, such variables should not be confused with the universally accepted definitions of contact time (time of contact) and dwell time (time of immobility). 

The functional food concept constitutes a worldwide trend. That it is viewed very differently in the USA, Europe and Asia is clearly evident from the fact that there is no universally accepted definition of the term functional food . The following are some examples of the international definitions of functional food . 

As seen in fii, x is, as a first approximation, a universal function of B and x it is evident, therefore, that for a definite reduced temperature and reduced pressure, the right-hand side of equation (20.2fi) has the same value for all gases. It is consequently, possible to construct a generalized diagram giving the (approximate) value of (II — H)/T for any gas with the reduced pressure as coordinate and the reduced temperature as parameter. Such a diagram is represented in Eig. 14. ... 

If the ground state is degenerate, several of the degenerate ground-state wave functions may produce the same density, so that a unique functional T[n] does not exist, but by definition these wave functions all yield the same energy, so that the functional Ev[n] continues to exist and to be minimized by no. A universal functional F[n can also still... 

To date, a number of national authorities, academic bodies, and the industry have proposed definitions for functional foods. Although the term functional food has already been defined several times, there is no universally accepted definition for this emerging food category. Definition ranges from the very simple to the more complex (Sir6 et al., 2008) foods that, by virtue of physiologically active components, provide health benefits beyond basic nutrition (International Life Sciences Institute, 1999) food similar in appearance to a conventional food, consumed as part of the usual... 

Finally, the shear rate as deflned by Equation 2.36b is clearly the appropriate argument for the viscosity function only for one-dimensional flows like the one used here. We need a quantity that reduces to dvx/dy for the one-dimensional flow but is properly invariant to the way in which we choose to deflne our coordinate system. The appropriate function, which follows directly from the principles of matrix algebra, is one half the second invariant of the rate of deformation, which is usually denoted Ud- Ud is shown in Table 2.6, where it is identical to the dissipation function O divided by r] for the special case of Newtonian fluids. (It is important to keep in mind that the function /) in Table 2.6 is the proper form for the dissipation only for a Newtonian fluid, whereas IId is a universally valid definition that depends only on the velocity field.) For an arbitrary flow field, then, the power-law and Carreau-Yasuda equations would be written, respectively. 

PESs determined by electronic structure calculations are by definition discrete. Search for the discovery of an underlying universal functional representation is one of the Holy Grails of molecular spectroscopy Perhaps it started in the 1930s, and the analysis Dunham provided is basic to much of the work in which potential energy curves are approximated by polynomials. Dunham showed that, if a potential energy curve of a diatomic molecule can be expressed as a power series... 

There is no universally accepted definition of probable maximum loss (PML) for purposes of earthquake risk analysis, but it is often understood to mean the loss with 90 % nonexceedance probability given shaking with 10 % exceedance probability in 50 years. For a single asset, PML can be calculated from the seismic vulnerability function by inverting the conditional distribution... 

Figure 5.2b shows that the data collapse implied by Equation 5.8 works remarkably well for a subregion containing 15 grains. Similar data collapse was observed for the other components of the force moment tensor and for m as small as 4. For smaller m, there is more of a spread in the data however, there is remarkably fast convergence as a function of m to a universal functional form, which can be interpreted as the thermodynamic entropy G(L/t/) for kl = 11,22,12. The solid line in Figure 5.2b is the functional form deduced from the assumed equation of state and the definition of of dc (Equation 5.3)... 

It has been proposed recently [210] to use the Minkowski functionals to define the scaling length l for the 2D systems as l( = (lEuier(t)/L2)-1 2 and Is = E(t) 1, where L2 is a volume of the 2D system. A similar scaling length could be defined for the symmetric 3D systems which possess the bicontinuous interface that is, l% = (—XEuierW/ ) an(J h - E(x) l. However, this definition cannot be applied for the asymmetric blends where the change of the Euler characteristic is not universal. 

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