A stochastic Lagrangian proof of global existence of the Navier–Stokes equations for flows with small Reynolds number

We consider the incompressible Navier–Stokes equations with spatially periodic boundary conditions. If the Reynolds number is small enough we provide an elementary short proof of the existence of global in time Hölder continuous solutions. Our proof uses a stochastic representation formula to obtain a decay estimate for heat flows in Hölder spaces, and a stochastic Lagrangian formulation of the Navier–Stokes equations.     

Designing Packet Buffers for Router Linecards

Internet routers and Ethernet switches contain packet buffers to hold packets during times of congestion. Packet buffers are at the heart of every packet switch and router, which have a combined annual market of tens of billions of dollars, and equipment vendors spend hundreds of millions of dollars on memory each year. Designing packet buffers used to be easy: DRAM was cheap, low power and widely used. But something happened at 10 Gb/s when packets started to arrive and depart faster than the access time of a DRAM. Alternative memories were needed, but SRAM is too expensive and power-hungry. A caching solution is appealing, with a hierarchy of SRAM and DRAM, as used by the computer industry. However, in switches and routers it is not acceptable to have a ldquomiss-raterdquo as it reduces throughput and breaks pipelines. In this paper we describe how to build caches with 100% hit-rate under all conditions, by exploiting the fact that switches and routers always store data in FIFO queues. We describe a number of different ways to do it, with and without pipelining, with static or dynamic allocation of memory. In each case, we prove a lower bound on how big the cache needs to be, and propose an algorithm that meets, or comes close, to the lower bound. These techniques are practical and have been implemented in fast silicon; as a result, we expect the techniques to fundamentally change the way switches and routers use external memory.     

Critical age-dependent branching Markov processes and their scaling limits

This paper studies: (i) the long-time behaviour of the empirical distribution of age and normalized position of an age-dependent critical branching Markov process conditioned on non-extinction; and (ii) the super-process limit of a sequence of agedependent critical branching Brownian motions.     

Complexities and algorithms for synchronized scheduling of parallel machine assembly and air transportation in consumer electronics supply chain

In this paper, we study the problem of synchronized scheduling of assembly and air transportation to achieve accurate delivery with minimized cost in consumer electronics supply chain. This problem was motivated by a major PC manufacturer in consumer electronics industry. The overall problem is decomposed into two sub-problems, which consist of an air transportation allocation problem and an assembly scheduling problem. The air transportation allocation problem is formulated as an integer linear programming problem with the objective of minimizing transportation cost and delivery earliness tardiness penalties. The assembly scheduling problem seeks to determine a schedule ensuring that the orders are completed on time and catch the flights such that the waiting penalties between assembly and transportation is minimized. The problem is formulated as a parallel machine scheduling problem with earliness penalties. The computational complexities of the two sub-problems are investigated. The air transportation allocation problem with split delivery is shown to be solvable. The parallel machine assembly scheduling problem is shown to be NP-complete. Simulated annealing based heuristic algorithms are presented to solve the parallel machine problem.     

Embedding theorems on hyperelliptic varieties

In this paper, we investigate linear systems on hyperelliptic varieties. We prove analogues of well-known theorems on abelian varieties, like Lefschetz’s embedding theorem and higher k-jet embedding theorems. Syzygy or \(N_p\) -properties are also deduced for appropriate powers of ample line bundles. This is a first result on linear series, on hyperelliptic varieties.     

Application of two-tracer isotope dilution mass spectrometry in the determination of uranium in geological samples

The determination of the uranium concentration in mineral samples using two tracer (²³³U and ²³⁵U) mass spectrometric isotope dilution techniques is described. The precision and accuracy are discussed and results are compared with those obtained by x-ray fluorescence and instrumental neutron activation methods. Based on the two independent values obtained for the same dilution, parameters such as the chemical procedures adopted, effect of mass fractionation and uranium distribution in minerals are evaluated. The ability of the method to distinguish between the analytical errors and heterogeneous distribution of uranium is discussed.