Scale-invariance

J. Krug, M. Schimschak. J Phys (France) I 5 1065, 1995 J. Krug. In A. McKane, M. Droz, J. Vannimenus, D. Wolf, eds. Scale Invariance, Interfaces and Nonequilibrium Dynamics. New York Plenum Press 1995. 

Patterns of this third class in fact demonstrate a complex form of scale-invariance by their self-similarity, in the infinite time limit, different magnifications observed at the same resolution are indistinguishable. The pattern generated by rule R90, for example, matches that of the successive lines in Pascal s triangle ai t) is given by the coefficient of in the expansion of (1 - - xY modulo-tv/o (see figure 3.2). 

The mechanism of Self-organized criticality, a concept first introduced by Bak, Tang and Wiesenfeld [bak87a], may possibly provide a fundamental link between such temporal scale invariant phenomena and phenomena exhibiting a spatial scale invariance - familiar examples of which are given by fractal coastlines, mountain landscapes and cloud formations [mandel82],... 

A. Pocheau and D. Queiros-Conde 1996, Scale invariance of the wrinkling law of turbulent propagating interface. Physical Review Letters 76 (18) 3352-3355. 

Scaling is a very important operation in multivariate data analysis and we will treat the issues of scaling and normalisation in much more detail in Chapter 31. It should be noted that scaling has no impact (except when the log transform is used) on the correlation coefficient and that the Mahalanobis distance is also scale-invariant because the C matrix contains covariance (related to correlation) and variances (related to standard deviation). 

Power law relaxation is no guarantee for a gel point. It should be noted that, besides materials near LST, there exist materials which show the very simple power law relaxation behavior over quite extended time windows. Such behavior has been termed self-similar or scale invariant since it is the same at any time scale of observation (within the given time window). Self-similar relaxation has been associated with self-similar structures on the molecular and super-molecular level and, for suspensions and emulsions, on particulate level. Such self-similar relaxation is only found over a finite range of relaxation times, i.e. between a lower and an upper cut-off, and 2U. The exponent may adopt negative or positive values, however, with different consequences and... 

For comparatively high repetition-rates (period T < 5t) fluorescence decays could also overlap between adjacent pulses. Thanks to the scale-invariant properties of the exponentials, no error is introduced when the decay is a pure single-exponential. Conversely, the preexponential factors can be altered when multiple lifetime decays... 

Xiang S. J., Huang J. W., Yang R. (2006). Time-scale invariant audio watermarking based on the statistical features in time domain. Proceedings of the 8th Information Hiding Workshop, 2006. 

Since TD-DFT is applied to scattering problems in its QFD version, two important consequences of the nonlocal nature of the quantum potential are worth stressing in this regard. First, relevant quantum effects can be observed in regions where the classical interaction potential V becomes negligible, and more important, where p(r, t) 0. This happens because quantum particles respond to the shape of K, but not to its intensity, p(r, t). Notice that Q is scale-invariant under the multiplication of p(r, t) by a real constant. Second, quantum-mechanically the concept of asymptotic or free motion only holds locally. Following the classical definition for this motional regime,... 

Distance measures were already discussed in Section 2.4. The most widely used distance measure for cluster analysis is the Euclidean distance. The Manhattan distance would be less dominated by far outlying objects since it is based on absolute rather than squared differences. The Minkowski distance is a generalization of both measures, and it allows adjusting the power of the distances along the coordinates. All these distance measures are not scale invariant. This means that variables with higher scale will have more influence to the distance measure than variables with smaller scale. If this effect is not wanted, the variables need to be scaled to equal variance. 

S. Nicolay, E.-B. Brodie of Brodie, M. Touchon, Y. d Aubenton-Carafa, C. Thermes, and A. Arneodo, Erom scale invariance to deterministic chaos in DNA sequences towards a deterministic description of gene organization in the human genome. Physica A 342,270-280 (2004). 

In order to calculate the effects of CLF we have to ask how the fraction of monomers that is released through CLF at the chain ends grows with time. It has been recently shown that for Ktr the effect of reptation on escaping from the tube is negligible in comparison to CLF [90]. It is the first passage of a chain end that is assumed to relax the constraint of a tube segment on a chain. From the scale invariance of the Rouse equation (Eq. 3.7) an exact asymptotic result... 

S. Deffner, C. Jarzynski, and A. del Campo. Classical and quanmm shortcuts to adiabaticity for scale-invariant driving. Phys. Rev. X, 4(2) 021013-021031(2014). 

The scale-up procedure, then, is simple express the process using a complete set of dimensionless numbers, and try to match them at different scales. This dimensionless space in which the measurements are presented or measured will make the process scale invariant. 

The results in Figure 1 have been obtained by changing the rotational speed of the stirrer and the gas throughput, whereas the liquid properties and the characteristic length (stirrer diameter d) remained constant. But these results could have also been obtained by changing the stirrer diameter. It does not matter by which means a relevant number (here Q) is changed because it is dimensionless and therefore independent of scale ( scale-invariant ). This fact presents the ha-... 

Dimensional analysis is a method for producing dimensionless numbers that completely characterize the process. The analysis can be applied even when the equations governing the process are not known. According to the theory of models, two processes may be considered completely similar if they take place in similar geometrical space and if all the dimensionless numbers necessary to describe the process have the same numerical value [2], The scale-up procedure, then, is simple express the process using a complete set of dimensionless numbers, and try to match them at different scales. This dimensionless space in which the measurements are presented or measured will make the process scale invariant. 

The existence of the correlation length gives a proof to the hypothesis of the scale invariance [8, 9] in the vicinity of the critical point physical... 

Two methods appear to be very powerful for the study of critical phenomena field theory as a description of many-body systems, and cell methods grouping together sets of neighboring sites and describing them by an effective Hamiltonian. Both methods are based on the old idea that the relevant scale of critical phenomena is much larger than the interatomic distance and this leads to the notion of scale invariance and to the statistical applications of the renormalization group technique.93... 

In Chap. 6 we learned that in the excluded volume limit ftc > 0,n —> oo, the cluster expansion breaks down, simply because it orders according to powers of z = j3enef2 —> oo. To proceed, we need a new idea, going beyond perturbation theory. The new concept is known as the Renormalization Group (RG), which postulates, proves, and exploits the fascinating scale invariance property of the theory. 

The key aspects of the modern understanding of phase transitions and the development of renormalization group theory can be summarized as follows. First was the observation of power-law behavior and the realization that critical exponents were, to some extent, universal for all kinds of phase transitions. Then it became clear that theories that only treated the average value of the order parameter failed to account for the observed exponents. The recognition that power-law behavior could arise from functions that were homogeneous in the thermodynamic variables and the scale-invariant behavior of such functions... 

---