On the Approximation of Toeplitz Operators for Nonparametric H∞\mathcal{H}_\infty-norm Estimation

Given a stable SISO LTI system $G$, we investigate the problem of estimating the $\mathcal{H}_\infty$-norm of $G$, denoted $||G||_\infty$, when $G$ is only accessible via noisy observations. Wahlberg et al. recently proposed a nonparametric algorithm based on the power method for estimating the top eigenvalue of a matrix. In particular, by applying a clever time-reversal trick, Wahlberg et al. implement the power method on the top left $n \times n$ corner $T_n$ of the Toeplitz (convolution) operator associated with $G$. In this paper, we prove sharp non-asymptotic bounds on the necessary length $n$ needed so that $||T_n||$ is an $\varepsilon$-additive approximation of $||G||_\infty$. Furthermore, in the process of demonstrating the sharpness of our bounds, we construct a simple family of finite impulse response (FIR) filters where the number of timesteps needed for the power method is arbitrarily worse than the number of timesteps needed for parametric FIR identification via least-squares to achieve the same $\varepsilon$-additive approximation

The quality of sustainability and the nature of open source software

The aim is to categorise Open Source Software as a commons based production process and resource. The definition of the commons is always accompanied by the doubt about its sustainability, the so-called "tragedy of the commons." Therefore it is worth to have a closer look on Open Source and why a "tragedy" does not appear

Embedded Markov chain approximations in Skorokhod topologies

In order to approximate a continuous time stochastic process by discrete time Markov chains one has several options to embed the Markov chains into continuous time processes. On the one hand there is the Markov embedding, which uses exponential waiting times. On the other hand each Skorokhod topology naturally suggests a certain embedding. These are the step function embedding for $J_1$, the linear interpolation embedding for $M_1$, the multi step embedding for $J_2$ and a more general embedding for $M_2$. We show that the convergence of the step function embedding in $J_1$ implies the convergence of the other embeddings in the corresponding topologies, respectively. For the converse statement a $J_1$-tightness condition for embedded Markov chains is given. The result relies on various representations of the Skorokhod topologies. Additionally it is shown that $J_1$ convergence is equivalent to the joint convergence in $M_1$ and $J_2$.Comment: To appear in Probability and Mathematical Statistic

Probing the Transition Between the Synchrotron and Inverse-compton Spectral Components of 1ES 1959+650

1ES 1959+650 is one of the most remarkable high-peaked BL Lacertae objects (HBL). In 2002, it exhibited a TeV gamma-ray flare without a similar brightening of the synchrotron component at lower energies. We present the results of a multifrequency campaign, triggered by the INTEGRAL IBIS detection of 1ES 1959+650. Our data range from the optical to hard X-ray energies, thus covering the synchrotron and inverse-Compton components simultaneously. We observed the source with INTEGRAL, the Swift X-Ray Telescope, and the UV-Optical Telescope, and nearly simultaneously with a ground-based optical telescope. The steep spectral component at X-ray energies is most likely due to synchrotron emission, while at soft gamma-ray energies the hard spectral index may be interpreted as the onset of the high-energy component of the blazar spectral energy distribution (SED). This is the first clear measurement of a concave X-ray-soft gamma-ray spectrum for an HBL. The SED can be well modeled with a leptonic synchrotron self-Compton model. When the SED is fitted this model requires a very hard electron spectral index of q ~ 1.85, possibly indicating the relevance of second-order Fermi acceleration.Comment: 5 pages, 2 postscript figure

Ramsey Properties of Permutations

The age of each countable homogeneous permutation forms a Ramsey class. Thus, there are five countably infinite Ramsey classes of permutations.Comment: 10 pages, 3 figures; v2: updated info on related work + some other minor enhancements (Dec 21, 2012