Equilibrium shapes of flat knots

We study the equilibrium shapes of prime and composite knots confined to two dimensions. Using scaling arguments we show that, due to self-avoiding effects, the topological details of prime knots are localized on a small portion of the larger ring polymer. Within this region, the original knot configuration can assume a hierarchy of contracted shapes, the dominating one given by just one small loop. This hierarchy is investigated in detail for the flat trefoil knot, and corroborated by Monte Carlo simulations.     

Apex exponents for polymer-probe interactions

We consider self-avoiding polymers attached to the tip of an impenetrable probe. The scaling exponents gamma(1) and gamma(2), characterizing the number of configurations for the attachment of the polymer by one end, or at its midpoint, vary continuously with the tip's angle. These apex exponents are calculated analytically by epsilon expansion, and numerically by simulations in three dimensions. We find that when the polymer can move through the attachment point, it typically slides to one end; the apex exponents quantify the entropic barrier to threading the eye of the probe.     

Integrated system modeling and optimization via quasilinearization

System modeling and system optimization are two coupled and strongly related concepts in the modern approach to large-scale systems. Yet, they have been treated as two separate problems in the literature. The identification of system parameters, often referred to as system modeling, is essential in order to obtain an optimal control policy. This work considers the two problems jointly and provides a computational methodology in tackling the integrated problem formulation. This is done by viewing one of the objective functions in the bicriterion problem formulation as a constraint. A computational strategy such as quasilinearization is employed for the solution of the integrated problem. An example problem is introduced, and numerical results using an IBM 360/91 digital computer are presented.The authors are very grateful to Professor C. T. Leondes for his invaluable assistance, guidance, and comments. This research was supported in part by the Air Force Office of Scientific Research, Grant No. 699-67, and in part by the National Science Foundation, Grant No. GK-4086.