Asymptotic self-similarity

At present, we see only one route to complete solution of the problem for t as r, for example, up to t = 5r or lOr, we solve the partial differential equations with a given pressure law at the piston continuing the solution for t > r in asymptotic (self-similar) form, we determine the constants A and B from the condition that the two solutions coincide at the extreme point to which the calculation is carried. 

II. Nonideal Systems and the Hypothesis of Asymptotic Self-Similarity... 

The assumption of self-similarity often is applied, almost as a matter of course, when one uses inductive reasoning to predict the properties of a large system on the basis of studies performed on relatively small subsystems thereof. This holds true not only in the physical sciences but for other areas of research as well. For example, statisticians obviously are only able to interview very limited samples of human populations. To generalize such findings to an entire population it is assumed that testimony elicited from the small sample accurately reflects opinions held by all members. Thus, the (response distribution of the) population is assumed to be self-similar, on the scale set by the sample size. This assumption can be tested by interviewing several samples, each consisting of a different number of participants. If the distributions of opinions are identical for all samples beyond a certain threshold size, it then is concluded that results based on the larger samples accurately represent the population as a whole, that is, the distribution is asymptotically self-similar . 

II. NONIDEAL SYSTEMS AND THE HYPOTHESIS OF ASYMPTOTIC SELF-SIMILARITY... 

These three examples, as well as others, will be treated in greater detail below. Our intention here is simply to illustrate the practicability of separating the function representative of a physical quantity into an ideal part that easily can be evaluated and an excess part that almost never is susceptible to exact evaluation. The challenge, of course, is to devise a (necessarily approximate) method for estimating the excess part. It is in this context that the hypothesis of asymptotic self-similarity can be used to construct a nonperturbative calculational procedure. 

Kuhn s idea of the asymptotic self-similarity of a nonideal chain now can be expressed as two propositions (1) there exists a value g of the bare coupling parameter for which Eq. (5.156) reduces to the form... 

The situation changes dramatically when higher order terms are included in the initial polynomial approximation for the propagator amplitude [35]. Beginning with the initial, two-loop approximation that is quadratic in a, one finds that the dynamical system d(t a), a t a) exhibits a nontrivial, stable fixed point or attractor (t). This indicates that the corresponding approximation to the photon propagator amplitude is an asymptotically self-similar fractal object. 

Although these results fall short of fully resolving the issue of the high four-momentum behavior of the transverse photon propagator, they do establish that if this propagator is asymptotically self-similar it will exhibit a fractal-like asymptotic form... 

A typical property of fractals is that they are locally (asymptotically) self-similar of small-length scales. Fractals are shapes that look more or less the same on all or many... 

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