Nonlinear Scale invariant Formalism and its Application to Some Differential Equations

Abstract

Scale

Scale Invariant Limit and Emergence of Complexity: Applications to Traffic Flow

“…Indeed, exploiting scale invariance, (1) may be rewritten as 7Where and so that as in such a manner that goes to zero at a much slower rate. However, denotes an ultrametric valuation defined over a class of infinitesimals those are assumed to reside in [1,2,4] (see below for more details). We simply note here that the scale δ may be identified with an accuracy level beyond which ie, elements of , are practically invisible (undetectable) relative to the real variable .…”

“…The theory of infinitesimals and the corresponding concept of an ultrametric valuation is being developed in reference [1,2,3,4].…”

“…Classically, of course, the answer is trivial: moves toward 0 linearly with uniform rate 1. Previously, we have mostly discussed non classical but analytic consequences of such a variation [1,2,3,4]. Here, we present a few applications of new non classical results to some simple problems in traffic flow [6,7].…”

“…Such a solution has been constructed under various setting previously (see [1] and references therein). Further, the scale invariance itself can be interpreted as to offer a mechanism of giving away with the (integrated) forcing term to appear explicitly in (4). The precise nature of the forcing term can therefore be considered as of secondary importance; the primary information with regard to a motion is formation of discontinuities and singularities of various kinds on the solution curve as the motion is unfolded.…”

“…In the following two sections, we present the precise analytical formalism leading to the promised scale invariance along with associated non classical consequences [1,2,3,4,5]. In Sec.4, we initiate in a more systematic way applications of the formalism to traffic flow [6,7] which we consider as a prototype for an intelligent system.…”