Construction Techniques for Highly Accurate Quasi-Interpolation Operators

Under mild additional assumptions this paper constructs quasi-interpolants in the form

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with approximation order ℓ−1, where_h(x) is a linear combination of translatesψ(x−jh) of a functionψinC^ℓ( ). Thus the order of convergence of such operators can be pushed up to a limit that only depends on the smoothness of the functionψ. This approach can be generalized to the multivariate setting by using discrete convolutions with tensor products of odd-degreeB-splines.     

Path integral formulation of general diffusion processes

A method is given for the derivation of a path integral representation of the Green's function solutionP of equationsP/t=L P,L being some Liouville operator. The method is applied to general diffusion processes.Feynman's path integral representation of the Schrödinger equation and Stratonovich's path integral representation of multivariate Markovian processes are obtained as special cases if the metric of the general diffusion process is flat. For curved phase spaces our result is a nontrivial generalization and new. New applications e.g. to quantized motion in general relativity, to transport processes in inhomogeneous systems, or to nonlinear non-equilibrium thermodynamics are made possible. We expect applications to be fruitfull in all cases where (continuous) macroscopic transport processes in Riemann geometries have to be considered.     

Interferometric ultrafast SOA-based optical switches: From devices to applications

All-optical switches are fundamental building blocks for future, high-speed optical networks that utilize optical time division multiplexing (OTDM) techniques to achieve single channel data rates exceeding 100 Gb/s. Interferometric optical switches using semiconductor optical amplifier (SOA) non-linearities perform efficient optical switching with < 500 fJ of control energy and are approaching optical sampling bandwidths of nearly 1 THz. In this paper, we review work underway at Princeton University to characterize and demonstrate these optical switches as processing elements in practical networks and systems. Three interferometric optical switch geometries are presented and characterized. We discuss limitations on the minimum temporal width of the switching window and prospects for integrating the devices. Using these optical switches as demultiplexers, we demonstrate two 100-Gb/s testbeds for photonic packet switching. In addition to the optical networking applications, we have explored simultaneous wavelength conversion and pulse width management. We have also designed high bandwidth sampling systems using SOA-based optical switches as analog optical sampling gates capable of analyzing optical waveforms with bandwidths exceeding 100 GHz. We believe these devices represent a versatile approach to all-optical processing as a variety of applications can be performed without significantly changing the device architecture.     

Base drag effects on maximum lift-to-drag ratio airfoils at moderate supersonic speeds

Maximum lift-to-drag ratio airfoils at moderate supersonic speeds are determined using Ackeret's linear theory for the forebody pressure coefficient and Chapman's experimental results for the base pressure coefficient. Two cases are investigated for both laminar flow and turbulent flow: (i) given length and thickness and (ii) given length and enclosed area.For case (i), it is shown that the maximum lift-to-drag ratio airfoil is identical with the minimum drag airfoil. Furthermore, for turbulent flow and given Mach number and Reynolds number, two critical values of the thickness ratio exist, ₁ and ₂. For ₁, the optimum airfoil is a diamond shape with a closed trailing edge and exhibits maximum thickness at midchord; for ₁₂, the optimum airfoil is a diamond shape with a blunt trailing edge and exhibits maximum thickness between midchord and the trailing edge; finally, for ₂, the optimum airfoil is a wedge and exhibits maximum thickness at the trailing edge.For case (ii), it is shown that the maximum lift-to-drag ratio is identical with the minimum drag airfoil. Furthermore, for turbulent flow and given Mach number and Reynolds number, a critical value (A/l ²)₁ of the enclosed area ratio exists. ForA/l ²(A/l ²)₁, the optimum airfoil is biconvex with a closed trailing edge and exhibits maximum thickness at midchord; forA/l ²(A/l ²)₁, the optimum airfoil is biconvex with a blunt trailing edge and exhibits maximum thickness between midchord and the trailing edge.This research, supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-828-67, is a condensed version of the investigations described in Refs. 1–2. The author would like to thank Dr. Angelo Miele for suggesting the problem and helpful discussions.     

On an integral method for backward boundary layers

Zusammenfassung Eine Integralmethode für die Lösung von zweidimensionalen Grenzschichtgleichungen wird entwickelt für Fälle, in denen die Lösungen sich anfänglich wie diejenigen von End-Grenzschichten benehmen. Diese Lösungen sind von Goldstein behandelt worden. Die bedeutendste Anwendung betrifft den Fall, wo ein zweidimensionaler Strom gegen eine Öffnung fliesst, durch welche die Flüssigkeit entleert wird. In erster Annäherung wird die Grenzschichtströmung für grosse Entfernungen von der Öffnung durch die Ähnlichkeitsgesetze der konvergenten Strömung in einem keilförmigen Kanal beschrieben. Um Strömungen dieser Art zu studieren, die über lange Entfernungen laufen, wird die Geschwindigkeit am Rande der Grenzschicht an Stelle der Entfernung längs der Grenzschicht als unabhängige Veränderliche eingeführt. Die beschriebene Methode wird auf den Fall der Strömung in Richtung eines Schlitzes angewendet, ausgehend von einem unendlichen Halbraum, ferner auf den Fall einer Strömung um eine endliche Platte, die senkrecht zur Strömungsrichtung steht. In beiden Beispielen ist die Potentialströmung in der Form des Kirchhoff-Rayleighschen Gesetzes angenommen, wobei die freien Stromlinien an die scharfen Kanten gebunden sind. Die Methode wird verallgemeinert, so dass sie für allgemeinere End-Grenzschichten benutzt werden kann. Die Resultate stimmen ausgezeichnet mit denen der Methode von Thwaites sowie mit den Ähnlichkeitslösungen von Falkner-Skan überein.     

On the cohomology of ergodic group actions

We consider three problems concerning cocycles of ergodic group actions: behavior of cohomology under the restriction of an ergodic semi-simple Lie group action to a lattice subgroup; “compactness” of bounded cocyles; and the relation of relative almost periodicity to relative discrete spectrum for extensions of ergodic actions.     

A Fleming–Viot Particle Representation¶of the Dirichlet Laplacian

We consider a model with a large number N of particles which move according to independent Brownian motions. A particle which leaves a domain D is killed; at the same time, a different particle splits into two particles. For large N, the particle distribution density converges to the normalized heat equation solution in D with Dirichlet boundary conditions. The stationary distributions converge as N→∞ to the first eigenfunction of the Laplacian in D with the same boundary conditions. Received: 11 November 1999 / Accepted: 19 May 2000     

Harmonic maps with fixed singular sets

Suppose Ω is a smooth domain in R^m,N is a compact smooth Riemannian manifold, andZ is a fixed compact subset of Ω having finite (m − 3)-dimensional Minkowski content (e.g.,Z ism − 3 rectifiable). We consider various spaces of harmonic mapsu: Ω →N that have a singular setZ and controlled behavior nearZ. We study the structure of such spacesH and questions of existence, uniqueness, stability, and minimality under perturbation. In caseZ = 0,H is a Banach manifold locally diffeomorphic to a submanifold of the product of the boundary data space with a finite-dimensional space of Jacobi fields with controlled singular behavior. In this smooth case, the projection ofu εH tou |ϖΩ is Fredholm of index 0. R. H.’s research partially supported by the National Science Foundation.     

Determining the Profile of a Nonmonotonic Hill

In a well-known generalization of the tautochrone problem, the shape of a monotone hill is determined by observing the travel time of a particle on the surface of the hill. In this paper we present a technique for uniquely determining the shape of any smooth hill, whether monotone or not, by observing collisions of particles on the hill. It is shown that this inverse problem is well posed. Error bounds for the solution are given for the case in which there are incomplete data. The efficiency of these bounds is shown by means of an example.     

Finite dimensional approximation in infinite dimensional mathematical programming

We consider the problem of approximating an optimal solution to a separable, doubly infinite mathematical program (P) with lower staircase structure by solutions to the programs (P(N)) obtained by truncating after the firstN variables andN constraints of (P). Viewing the surplus vector variable associated with theNth constraint as a state, and assuming that all feasible states are eventually reachable from any feasible state, we show that the efficient set of all solutions optimal to all possible feasible surplus states for (P(N)) converges to the set of optimal solutions to (P). A tie-breaking algorithm which selects a nearest-point efficient solution for (P(N)) is shown (for convex programs) to converge to an optimal solution to (P). A stopping rule is provided for discovering a value ofN sufficiently large to guarantee any prespecified level of accuracy. The theory is illustrated by an application to production planning.The work of Robert L. Smith was partially supported by the National Science Foundation under Grant ECS-8700836.