Scale symmetry

A relatively recent type of space-time symmetry has been introduced to explain the results of certain high-cncrgy scattering experiments. This is scale symmetry and it pertains to the rescaling or dilation of the space-time coordinates of a system without changing the physics of the system. Other symmetries, such as chirality, are more of an abstract nature, but aid the theorist in an effort to bring order into the vast array of possible elementary particle reactions. 

Wintgen (1987). This section consists of three parts. In part (a) we derive the trace formula for the one-dimensional heUum atom, a system with an odd-even symmetry. In part (b) we use the classical scaUng properties of the one-dimensional helium atom to apply the scaled energy technique. In part (c) we generalize the technique to apply to autonomous systems without scaling symmetries. 

It is not necessary for the classical Hamiltonian of a system to possess intrinsic scaling symmetries in order to apply the technique of scaled... 

Umberger, D.K., Mayer-Kress, G. and Jen, E. (1986). Hausdorff dimensions for sets with broken scaling symmetry, in Dimensions and Entropies in Chaotic Systems, ed. G. Mayer-Kress (Springer, New York). 

Fractal models for soil structure and rock fractures are becoming increasingly popular (e.g., Sahimi, 1993 Baveye et al., 1998). The primary appeal of these models is their ability to parsimoniously parameterize complex structures. Scale symmetry or scale invariance, in which an object is at least statistically the same after magnification, is a fundamental property of fractals and can also be observed in numerous natural phenomena. Thus, it is logical that some investigators have examined theoretical transport in known prefractals. 

In the years after the discovery of aperiodic crystals (incommensurate modulated, intergrowth and quasicrystals) crystallography was for me a very rich and open field of research, but not mysterious. Even the surprising combination in snow crystals of sixfold circular rotations with hyperbolic rotations [1], leading to hexagrammal scaling symmetry, fitted into the whole because the atomic positions in ice are invariant with respect to both types of crystallographic rotations [2]. 

Scaled, symmetry-corrected value from this work [FlF/6-31G(d)]. 

Scaled, symmetry-corrected value for structure I, computed at the MP2(full)/6-31G(d) level (this work). " From the experimental Af7/°m (Ref. 26) and 5°m( -C4H8) and GB( -C4H8) = —172.5 1.1 kcalmoP (Ref. 107) corrected with the computed values for structures I and II. This value pertains to the formation of an equilibrating mixture of isomers I and II in the gas phase. 

From the DPA onset of tricyclopropylcarbinol and the computed Af/F ,[(c-C3H5)3CH] (this work). Scaled, symmetry-corrected HF/6-31G(d) value from this work. 

Erom the same source, for the homologous chloride anion exchange, together with fH° (tert-C4H9CI) = 42.99 0.50kcalmor (Ref. 26) and Af/rm(l-AdCl) = -42.45 0.60 kcal mol . "From the DPA and FT ICR equilibrium study of the same process with the entropy corrections (scaled, symmetry-corrected HF/6-3 lG(d) values) from Ref. 64. 

Scaled, symmetry-corrected HF/6-31G(d) value (this work). 

Hohenstein, E. G., Rarrish, R. M., Sherrill, C. D., Turney, J. M., and Schaefer, H. F. [2011b]. Large-scale symmetry-adapted perturbation theory computations via density fitting and Laplace transformation techniques Investigating the fundamental forces of DNA-intercalator interactions,/ Chem. Phys. 135, p. 174107, doiilO.1063/1.3656681. 

Surface Alignment, Length-scales, Symmetry and Microscopic Interactions... 

Spinodal decomposition of liquids or solids gives a distribution of the phases that has scale symmetry and these composites are fractals within a range of sizes. 

Diffusion-limited aggregation of particles also yields fractal agglomerates that have scale symmetry within wide cutoff limits. 

On the macroscopic scale, symmetry can be extremely pleasing to the human eye. It is exploited by architects in the design of great buildings, with symmetry in both the overall structure and in its component elements (such as around doorways, windows, etc. see Figure 2.2(a)). Symmetry is also to be found in the natural world where, for example, the different kinds of symmetry present in flowers can be used in their identification (Figure 2.2(b)). 

The same model is shown to fit the electronic structure of all atoms when described in dimensionless units. The scaling symmetry observed here obeys the symmetry law of Hatiy quoted by Janner [26] ... 

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