Numerical integration of ordinary differential equations on manifolds

Summary This paper is concerned with the problem of developing numerical integration algorithms for differential equations that, when viewed as equations in some Euclidean space, naturally evolve on some embedded submanifold. It is desired to construct algorithms whose iterates also evolve on the same manifold. These algorithms can therefore be viewed as integrating ordinary differential equations on manifolds. The basic method “decouples” the computation of flows on the submanifold from the numerical integration process. It is shown that two classes of single-step and multistep algorithms can be posed and analyzed theoretically, using the concept of “freezing” the coefficients of differential operators obtained from the defining vector field. Explicit third-order algorithms are derived, with additional equations augmenting those of their classical counterparts, obtained from “obstructions” defined by nonvanishing Lie brackets.     

Identifying several biased coins encountered by a hidden random walk

Suppose that attached to each site z ∈ ? is a coin with bias θ(z), and only finitely many of these coins have nonzero bias. Allow a simple random walker to generate observations by tossing, at each move, the coin attached to its current position. Then we can determine the biases {θ(z)}_z∈?, using only the outcomes of these coin tosses and no information about the path of the random walker, up to a shift and reflection of ?. This generalizes a result of Harris and Keane. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Fairness in Scheduling

On-line machine scheduling has been studied extensively, but the fundamental issue of fairness in scheduling is still mostly open. In this paper we explore the issue in settings where there are long-lived processes which should be repeatedly scheduled for various tasks throughout the lifetime of a system. For any such instance we develop a notion ofdesiredload of a process, which is a function of the tasks it participates in. Theunfairnessof a system is the maximum, taken over all processes, of the difference between the desired load and the actual load.An example of such a setting is thecarpool problemsuggested by Fagin and Williams [IBM Journal of Research and Development27(2) (1983), 133–139]. In this problem, a set ofnpeople form a carpool. On each day a subset of the people arrive and one of them is designated as the driver. A scheduling rule is required so that the driver will be determined in a “fair” way.We investigate this problem under various assumptions on the input distribution. We also show that the carpool problems can capture several other problems of fairness in scheduling.     

The speed of biased random walk on percolation clusters

 We consider biased random walk on supercritical percolation clusters in ℤ². We show that the random walk is transient and that there are two speed regimes: If the bias is large enough, the random walk has speed zero, while if the bias is small enough, the speed of the random walk is positive. Received: 20 November 2002 / Revised version: 17 January 2003 Published online: 15 April 2003 Research supported by Microsoft Research graduate fellowship. Research partially supported by the DFG under grant SPP 1033. Research partially supported by NSF grant #DMS-0104073 and by a Miller Professorship at UC Berkeley. Mathematics Subject Classification (2000): 60K37; 60K35; 60G50 Key words or phrases: Percolation – Random walk     

On the symmetric square: unstable local transfer

Summary We prove the fundamental lemma for spherical functions with respect to the natural (induction) lifting fromPGL(2) toPGL(3) which appears as the unstable counterpart of the stable symmetric-square lifting fromSL(2) toPGL(3) (see [IV] for an introduction to this project, and [VI] for the final results). Thus spherical functions onPGL(2) andPGL(3) which correspond to each other by satisfying an elementary representation theoretic relation are shown to have matching orbital integrals. The proof of this local statement is based on an application of the global trace formula.Partially supported by NSF grants     

Otpically illuminated dielectric interfaces as high-speed far-infrared modulators

A new device for the gigahertz modulation of far-infrared radiation is analytically and numerically analyzed. It consists of a thin layer of a high-mobility, direct-bandgap semiconductor, such as GaAs, in which a high-density electron-hole plasma is rapidly created and destroyed, thereby rapidly changing the free-carrier reflectivity of the active layer. Illumination by a high-power, near-infrared laser diode array generates the plasma through intrinsic photoconduction. It is shown that this device acis primarily as an amplitude modulator, and that its efficiency increases sharply with increasing far-IR frequency, in contrast to a Schottky diode, which acts primarily as a phase modulator, and whose efficiency falls off sharply with far-IR frequency. The breakeven frequency lies at about 1.5 THz, depending slightly on the assumed device parameters. The relative advantage of the new device increases rapidly with increasing far-infrared frequency. At an operating frequency of 2.5 THz (119 m), for example, a 1 GHz modulation bandwidth may be achieved with a single-sideband conversion loss of only-21 db, versus a Schottky's loss of-39 db, assuming a laser diode power of 1 W, which is readily available from recently developed laser diode arrays.