Inverting the signature of a path

The aim of this article is to develop an explicit procedure that enables one to reconstruct any $C^1$ path (at natural parametrization) from its signature. We also explicitly quantify the distance between the reconstructed path and the original path in terms of the number of terms in the signature that are used for the construction and the modulus of continuity of the derivative of the path. A key ingredient in the construction is the use of a procedure of symmetrization that separates the behavior of the path at small and large scales.Comment: 31 pages; minor change

A simple proof of distance bounds for Gaussian rough paths

We derive explicit distance bounds for Stratonovich iterated integrals along two Gaussian processes (also known as signatures of Gaussian rough paths) based on the regularity assumption of their covariance functions. Similar estimates have been obtained recently in [Friz-Riedel, AIHP, to appear]. One advantage of our argument is that we obtain the bound for the third level iterated integrals merely based on the first two levels, and this reflects the intrinsic nature of rough paths. Our estimates are sharp when both covariance functions have finite 1-variation, which includes a large class of Gaussian processes. Two applications of our estimates are discussed. The first one gives the a.s. convergence rates for approximated solutions to rough differential equations driven by Gaussian processes. In the second example, we show how to recover the optimal time regularity for solutions of some rough SPDEs.Comment: 20 pages, updated abstract and introductio

Sprinklers: A Randomized Variable-Size Striping Approach to Reordering-Free Load-Balanced Switching

Internet traffic continues to grow exponentially, calling for switches that can scale well in both size and speed. While load-balanced switches can achieve such scalability, they suffer from a fundamental packet reordering problem. Existing proposals either suffer from poor worst-case packet delays or require sophisticated matching mechanisms. In this paper, we propose a new family of stable load-balanced switches called "Sprinklers" that has comparable implementation cost and performance as the baseline load-balanced switch, but yet can guarantee packet ordering. The main idea is to force all packets within the same virtual output queue (VOQ) to traverse the same "fat path" through the switch, so that packet reordering cannot occur. At the core of Sprinklers are two key innovations: a randomized way to determine the "fat path" for each VOQ, and a way to determine its "fatness" roughly in proportion to the rate of the VOQ. These innovations enable Sprinklers to achieve near-perfect load-balancing under arbitrary admissible traffic. Proving this property rigorously using novel worst-case large deviation techniques is another key contribution of this work

Structures and photoelectron spectroscopy of Cu-n(BO2)(m) - (n, m=1, 2) clusters: Observation of hyperhalogen behavior

The electronic structures of CuBO2 −, Cu(BO2)2 −, Cu2(BO2)−, and Cu2(BO2)2 − clusters were investigated using photoelectron spectroscopy. The measured vertical and adiabatic detachment energies of these clusters revealed unusual properties of Cu(BO2)2 cluster. With an electron affinity of 5.07 eV which is larger than that of its BO2 superhalogen (4.46 eV) building-block, Cu(BO2)2 can be classified as a hyperhalogen. Density functional theory based calculations were carried out to identify the ground stategeometries and study the electronic structures of these clusters. Cu(BO2) and Cu(BO2)2 clusters were found to form chainlike structures in both neutral and anionic forms. Cu2(BO2) and Cu2(BO2)2 clusters, on the other hand, preferred a chainlike structure in the anionic form but a closed ringlike structure in the neutral form. Equally important, substantial differences between adiabatic detachment energies and electron affinities were found, demonstrating that correct interpretation of the experimental photoelectron spectroscopy data requires theoretical support not only in determining the ground stategeometry of neutral and anionic clusters, but also in identifying their low lying isomers

Portable broadband cavity-enhanced spectrometer utilizing Kalman filtering: application to real-time, in situ monitoring of glyoxal and nitrogen dioxide

This article describes the development and field application of a portable broadband cavity enhanced spectrometer (BBCES) operating in the spectral range of 440-480 nm for sensitive, real-time, in situ measurement of ambient glyoxal (CHOCHO) and nitrogen dioxide (NO2). The instrument utilized a custom cage system in which the same SMA collimators were used in the transmitter and receiver units for coupling the LED light into the cavity and collecting the light transmitted through the cavity. This configuration realised a compact and stable optical system that could be easily aligned. The dimensions and mass of the optical layer were 676 × 74 × 86 mm3 and 4.5 kg, respectively. The cavity base length was about 42 cm. The mirror reflectivity at λ = 460 nm was determined to be 0.9998, giving an effective absorption pathlength of 2.26 km. The demonstrated measurement precisions (1σ) over 60 s were 28 and 50 pptv for CHOCHO and NO2 and the respective accuracies were 5% and 4%. By applying a Kalman adaptive filter to the retrieved concentrations, the measurement precisions of CHOCHO and NO2 were improved to 8 pptv and 40 pptv in 21 s

Subcritical approximations to stochastic defocusing mass-critical nonlinear Schrödinger equation on R

We show robustness of various truncated and subcritical approximations to the stochastic defocusing mass-critical nonlinear Schrödinger equation (NLS) in dimension d=1, whose solution was constructed in [8] with one particular such approximation. The key ingredient in the proof is a uniform bound of the solutions to the family of deterministic mass-subcritical defocusing NLS