Combinatorics,graph theory,and computing

Journal of Algebraic Combinatorics -     

Flow auctions

Flow goods (like electricity) are sold through auctions in a dynamic framework. An important design question is the frequency of such auctions. We use a simple dynamic auction model in continuous time to answer this question. We focus on the relationship between the persistency of bidders’ valuations and the optimal choice of frequency. If the seller focuses on the equilibrium in which bidders follow a repeated static Nash strategy, then the frequency of auctions should typically increase when persistency declines. However, accounting for the fact that bidders can follow different equilibria that are collusive in nature, the comparative statics are reversed, forcing the seller to reduce the frequency when bidders’ valuations are less persistent. The argument builds on the fact that high frequency auctions are more conducive to collusion among bidders.     

The continuous grey pattern problem

A new location problem is formulated and solved. It is the continuous version of the grey pattern problem which is a special case of the Quadratic Assignment Problem. The problem is a minimization of a convex function subject to non-convex constraints and has infinitely many optimal solutions. We propose several mathematical programming formulations that are suitable for a multi-start heuristic algorithm. In addition to solving these formulations by the Solver in Excel and Mathematica, a special Nelder–Mead algorithm is proposed. This special algorithm provided the best results. One suggested modification may improve the performance of the Nelder–Mead algorithm for other optimization problems as well.     

Numerical-Asymptotic Expansion Matching for Computing a Viscous Flow Around a Sharp Expansion Corner

A computational technique which is based on a numerical-asymptotic expansion matching for computing the local singular behavior of a viscous flow around a sharp right-angle expansion corner is presented. Moffatt's (1964) asymptotic solution is extended and a matching with a time-marching finite-difference scheme of the Navier--Stokes equations is formulated. Local mesh refinement around the corner is required to meet the validity of the asymptotic solution. Flows in an expanding channel with expansion ratio D/d=3 at various Reynolds numbers 1≤Re≤700 are simulated. The results are compared with those from a standard finite-difference scheme that uses second-order forward/backward differences near the corner. It is found that the results of the standard scheme converge toward those of the present technique as the level of local refinement near the corner is increased. The time-dependent parameters of the first two terms of the asymptotic solution at the steady-state solution are also described for various cases of Re and D/d. It is demonstrated that the present method enhances the accuracy of the simulations and requires less refinements near the corners to achieve converged numerical results. Received 14 August 2000 and accepted 25 October 2001     

The influence of pressure on the capillary flow of poly(methyl methacrylate)

Summary Pressure effects in the capillary flow of a single sample of poly(methyl methacrylate),M _v = 1.33 · 10⁵, were evaluated. The length/diameter ratios of the different capillaries used varied from 4 to 100. The tests were made with an Instron rheometer in the range 160–250 °C. The pressure-viscosity model, derived from the free volume-WLF equation, was used to pressure correct the capillary data. The corrected data agreed well with data obtained at atmospheric pressure using aWeissenberg rheogoniometer. A derived expression to calculate an increase in the flow activation energy with increasing stress predicts the observed increase in activation energy.

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Zusammenfassung Druckeffekte bei der Kapillarströmung einer einzelnen Polymethylmethacrylat-ProbeM _v = 1,33 · 10⁵ werden abgeschätzt. Die Längen/Durchmesserverhältnisse der verschiedenen verwendeten Kapillaren variierten zwischen 4 und 100. Im Temperaturbereich zwischen 160 und 250 °C wurden die Versuche mit einem Instron-Rheometer durchgeführt. Das Druck-Viskositätsmodell, das aus der WLF-Gleichung abgeleitet war, wurde zur Druckkorrektur der Kapillardaten verwendet. Die korrigierten Werte zeigten eine gute Übereinstimmung mit den Werten, die bei Atmosphärendruck mit Hilfe einesWeissenberg-Rheogoniometers gewonnen worden waren. Ein abgeleiteter Ausdruck zur Berechnung des Anstiegs der Strömungs-Aktivierungsenergie mit steigender Spannung sagt den beobachteten Anstieg in der Aktivierungsenergie voraus.

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On leave from Montecatini Edison, Milan (Italy).     

A variational problem determined by probability measures

ABSTRACT

An optimization problem of maximizing an integral of a function over a family of probability measures is considered. The problem is a generalization of a well-studied variational problem in mathematical economics, concerning optimal allocations. The specific generalization that we examine arises also in the limit of singularly perturbed optimal control problems. We examine the mathematical problem and allude to the singular perturbation motivation.     

NMR-assisted computational studies of peptidomimetic inhibitors bound in the hydrophobic pocket of HIV-1 glycoprotein 41

Due to the inherently flexible nature of a protein–protein interaction surface, it is difficult both to inhibit the association with a small molecule, and to predict how it might bind to the surface. In this study, we have examined small molecules that mediate the interaction between a WWI motif on the C-helix of HIV-1 glycoprotein-41 (gp41) and a deep hydrophobic pocket contained in the interior N-helical trimer. Association between these two components of gp41 leads to virus–cell and cell–cell fusion, which could be abrogated in the presence of an inhibitor that binds tightly in the pocket. We have studied a comprehensive combinatorial library of α-helical peptidomimetics, and found that compounds with strongly hydrophobic side chains had the highest affinity. Computational docking studies produced multiple possible binding modes due to the flexibility of both the binding site and the peptidomimetic compounds. We applied a transferred paramagnetic relaxation enhancement experiment to two selected members of the library, and showed that addition of a few experimental constraints enabled definitive identification of unique binding poses. Computational docking results were extremely sensitive to side chain conformations, and slight variations could preclude observation of the experimentally validated poses. Different receptor structures were required for docking simulations to sample the correct pose for the two compounds. The study demonstrated the sensitivity of predicted poses to receptor structure and indicated the importance of experimental verification when docking to a malleable protein–protein interaction surface.     

Calculation of a conductive-laminar thermo-convection regime of a Newtonian fluid in a rectangular cavity with isothermal vertical boundaries

A method of calculating a free-convection flow of a viscous incompressible medium in a rectangular cavity is proposed. The lateral boundaries of the cavity have the same constant temperature, the top and bottomboundaries are thermally insulated, and the half-width to height ratio is much smaller than unity. A decomposition of the flow domain into zones with ascending and descending flows made it possible to formulate and solve analytically a coupled parabolic system of linear equations based on the classic Oberbeck-Boussinesq assumptions.