Piecewise Linear Reasoner

The Piecewise Linear Reasoner (PLR) [5] takes parameterized ordinary differential equations and produces maps with the global description of dynamic systems. Despite its present limitations, PLR is a typical example of a new approach that attempts to endow the computer with large amounts of analytical knowledge (dynamics of nonlinear systems, differential topology, asymptotic analysis, etc.) so that it can complement and expand the capabilities of numerical simulations. 

We describe the system playback process by a service curve s(f), It is reasonable to use the system maximal chunk ID at any time f as s(f), and then playback rate is r t) =ds f)/dt. For a channel with playback rate variations, the playback rate vs. time should be a piecewise linear function. 

A fully quantitative treatment of the above intuitive ideas is difficult at the present time, for two reasons in the thermal case the death rate depends exponentially on the state variable and in the chemical case one deals with a birth and death process with highly nonlinear transition probabilities whose time-dependent behavior remains poorly known, despite recent significant progress [2,5] In a preceding paper [ ] we circumvented this difficulty in the thermal case by adopting an idealized piecewise linear representation of the transition rates, which captures their essential features while allowing a rather exhaustive analytic treatment. Here we present an alternative description using the full form of the transition rates, and the more limited aim we fix to ourselves is to determine the critical time beyond which transient bimodality is expected to occur. 

The resorting to polynomials of higher orders leads to success only in those instances where the shape can reasonably be represented by polynomial approximation. Other strategies include piecewise fitting of linear functions or the use of appropriate transformations with the aim of retaining... 

The technical conditions on f are quite reasonable if a physical situation has a discontinuity, we might look for solutions with discontinuities in the function f and its derivatives. In this case, we might have to consider, e.g., piecewise-defined combinations of smooth solutions to the differential equation. These solutions might not be linear combinations of spherical harmonics. 

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