Linear sizes scale

Dixon SL, Merz KM Jr (1996) Semiempirical molecular orbital calculations with linear system size scaling. J Chem Phys 104(17) 6643-6649... 

Mauri, F., G. Galli, and R. Car. 1993. Orbital formulation for electronic-structure calculations with linear system-size scaling. Phys. Rev. B 47, 9973. 

Li, X. P., R. W. Nunes, and D. Vanderbildt. 1993. Density Matrix Electronic Structure Method with Linear System-Size Scaling. Phys. Rev. B 48,10891. 

Before carrying out particle sizing on a sample, the microscope s ocular scale must be calibrated. This is normally done with a stage micrometer, which has a linear graduated scale. The micrometer is aligned with the eyepiece ocular to determine the length per ocular scale division. The ocular can then be used to read the diameters of particles on a slide. When sufficient particles are sized, the length-number mean can be calculated ... 

Phase transitions are not only characterized by atomic or molecular structural changes - they can also be characterized by significant modifications in the microstmcture and domains and at a much larger size scale. One notable example has been recently reported by Glazer et al. [110] using linear birefringence measurements in LiTaOs and LiTa cNbi c03 crystals at high temperature. 

For the long reaction times, Id Ao, this new scale dominates over all other dimensions of the process and thus we can consider a model where the whole reaction volume is divided into such alternating blocks with linear sizes Id, and each of them contains either A or B particles predominantly (Fig. 2.8). 

Disordered porous media have been adequately described by the fractal concept [154,216]. It was shown that if the pore space is determined by its fractal structure, the regular fractal model could be applied [154]. This implies that for the volume element of linear size A, the volume of the pore space is given in units of the characteristic pore size X by Vp = Gg(A/X)°r, where I), is the regular fractal dimension of the porous space, A coincides with the upper limit, and X coincides with the lower limit of the self-similarity. The constant G, is a geometric factor. Similarly, the volume of the whole sample is scaled as V Gg(A/X)d, where d is the Euclidean dimension (d = 3). Hence, the formula for the macroscopic porosity in terms of the regular fractal model can be derived from (65) and is given by... 

In fact, if one measures the total number of bonds (sites) on the infinite cluster at the percolation threshold (pc) in a (large) box of linear size L, then this number or the mass of the infinite cluster will be seen to scale with L as where die (< d) is called the fractal dimension of the infinite cluster at the percolation threshold. Similar measurements for the backbone (excluding the dangling ends of the infinite cluster) give the backbone mass scaling as, de < die, where dfi is called the backbone (fractal) dimension. In fact, die can be very easily related to the embedding Euclidean dimension d of the cluster by... 

However, several strands of evidence showed that male eyespan was more sensitive to environmental conditions (David et al 1998). There was a strong sex x density interaction, with male eyespan declining much more conspicuously than female eyespan over the same range of stresses (Fig. 2). The sex difference persisted when we examined relative eyespan size (dividing by body size). Finally, a linear model with body size and larval density as effects showed that male eyespan declined with larval density even when body size scaling had been accounted for, but that female eyespan did not. 

The convergence law of the results of the PR method is related to the corrections to the finite-size scaling. From Eq. (55) we expect that at the critical value of nuclear charge the correlation length is linear in N. In Fig. 9 we plot the correlation length of the finite pseudosystem (evaluated at the exact critical point Xc) as a function of the order N. The linear behavior shows that the asymptotic equation [Eq. (60)] for the correlation length holds for very low values of N [87]. 

This simple scaling results because a volume of size r contains n = rjb) monomers from that chain section. Equation (6.143) describes the minus one slope region of Fig. 6.31 on scales smaller than the size of the linear chains (with r < bN ). When r reaches the size scale of the linear chain sections bN j, Eq. (6.143) gives the expected result for the volume fraction of a linear chain inside its own pervaded volume in a melt, 4> A lOn scales larger than the linear chains, the fractal dimension V = 4- applies to chain sections of size r that are directly connected withinTlic volume P [see Eq. (6.142) and Fig. 6.31] ... 

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