Scale transformation

The oxidation of organic compounds by manganese dioxide has recently been reviewed. It is of limited application for the introduction of double bonds, but the advantages of mildness and simple workup make it attractive for some laboratory-scale transformations. Manganese dioxide is similar to chloranil in that it will oxidize A -3-ketones to A -dienones in refluxing benzene. Unfortunately, this reaction does not normally go to completion, and the separation of product from starting material is difficult. However, Sondheimer found that A -3-alcohols are converted into A -3-ketones, and in this instance separation is easier, but conversions are only 30%. (cf. Harrison s report that manganese dioxide in DMF or pyridine at room temperature very slowly converts A -3-alcohols to A -3-ketones.)... 

When applied to the relaxation time of a polymer, dimensional analysis of Eq. (22) shows that the following scaling transformation should be written for tr ... 

In a similar fashion, the scaling transformation of zz as given from Eq. (35) is the following ... 

The classical dynamics of the FPC is governed by the Hamiltonian (1) for F = 0 and is regular as evident from the Poincare surface of section in Fig. 1(a) (D. Wintgen et.al., 1992 P. Schlagheck, 1992), where position and momentum of the outer electron are represented by a point each time when the inner electron collides with the nucleus. Due to the homogeneity of the Hamiltonian (1), the dynamics remain invariant under scaling transformations (P. Schlagheck et.al., 2003 J. Madronero, 2004)... 

If A is a diagonal matrix, it corresponds to a scaling transformation. Each coordinate of the vector x is scaled by the corresponding diagonal element. 

The key to all this is that the scale of measurement of most (if not all) variables is arbitrary. Although we are most familiar with a linear scale of measurement, there is nothing which makes this the correct scale on its own, as opposed to a logarithmic scale [familiar logarithmic measurements are that of pH values, or earthquake intensity (Richter scale)]. Transforming a set of data (converting X to X ) is really as simple as changing a scale of measurement. 

The word fractal was coined by Mandelbrot in his fundamental book.1 It is from the Latin adjective fractus which means broken and it is used to describe objects that are too irregular to fit into a traditional geometrical setting. The most representative property of fractal is its invariant shape under self-similar or self-affine scaling. In other words, fractal is a shape made of parts similar to the whole in some way.61 If the objects are invariant under isotropic scale transformations, they are self-similar fractals. In contrast, the real objects in nature are generally invariant under anisotropic transformations. In this case, they are self-affine fractals. Self-affine fractals have a broader sense than self-similar fractals. The distinction between the self-similarity and the selfaffinity is important to characterize the real surface in terms of the surface fractal dimension. 

One must note that q and p are coordinates and momenta up to a scale transformation. A more general form of the Hamiltonian // is... 

Ludena, E. V., L6pez-Boada Local-Scaling Transformation Version of Density Functional Theory Generation of Density Functionals. 180, 169-224 (1996). 

Local-Scaling Transformation Version of Density Functional Theory ... 

Local-scaling transformations, or point transformations, are generalizations of the well-known scaling transformations. The latter have been widely used in many domains of the physical sciences. Scaling transformations carry a vector into /( ) = Xr, where k is just a constant. In the case of local-scaling transformations, A is a function (i.e., k = A(r)). Notice that the transformed vector/(r) 6 conserves the same direction as the original one and is given by /(r) = k(f)r. In terms of the operator/associated with this transformations, we can relate F and J(F) by ... 

In order to see that they correspond to the general transformations studied by Moser [58], consider the effect of applying a local-scaling transformation, denoted by the operator f, to each of the coordinates appearing in the wavefunc-tion i(Fi,. ..,F v) 6 ifjv. Hence, the resulting wavefunction 2(Fi,...,Fjv) e is given by ... 

The full-fledged introduction of local-scaling transformations into density functional theory took place in the works of Kryachko, Petkov and Stoitsov [28-30, 32, 34], and of Kryachko and Ludena [1, 20, 31, 33, 35-37],... 

Let us now show that when we apply a local-scaling transformation to a closed contour Si(r C) of p, (defined by pi( ) — C = 0), we obtain a closed... 

Fig. 3. Schematic representation of the transformation of a closed density contour curve of pj (dark contour on the left-hand side) into that of p2 (dark contour right) by local-scaling transformations...

Let us consider now the application of local-scaling transformations to sets of single-particle functions or orbitals. As it was shown in Sect. 2.1, a set of plane waves gives rise to the transformed orbitals described by Eq. (2). In particular, the application of this transformation to one-dimensional plane-waves leads to Harriman s equidensity orbitals [27], which are given by ... 

We illustrate here a specific example of the application of local-scaling transformations to atomic orbitals [111]. Consider the i is(r) and / 2s( ) orbitals of the Raffenetti type for the beryllium atom [71] ... 

The density associated with the Hartree-Fock-Raffenetti wavefunction is denoted by puVif)- We take this to be the initial density in our local-scaling transformation, i.e., pi r) = puVir)- We take as the final density, that associated with the 650-term Cl wavefunction of Esquivel and Bunge [73], which we call P2ir) = pair). These two densities are practically about the same, as can be seen clearly in Fig. 4, where we have also plotted their difference. The transformed radial orbitals are given by ... 

Fig. 5. Graph of the iocai-scaling transformation function/(r) and related quantities...

Table I. Selected values of the Raffenetti-Hartree-Fock orbitals Isg and 2shf for Be, of their locally-scaled transformed functions IsJ, and 2sgr and of their differenees di, = Ish, - Isgj,) and = (2siif — 2sgp). [Reproduced with permission from Table I Ludeiia et al. [Ill]]...

---