Discrete alloy-type models: Regularity of distributions and recent results

We consider discrete random Schr\"odinger operators on $\ell^2 (\mathbb{Z}^d)$ with a potential of discrete alloy-type structure. That is, the potential at lattice site $x \in \mathbb{Z}^d$ is given by a linear combination of independent identically distributed random variables, possibly with sign-changing coefficients. In a first part we show that the discrete alloy-type model is not uniformly $\tau$-H\"older continuous, a frequently used condition in the literature of Anderson-type models with general random potentials. In a second part we review recent results on regularity properties of spectral data and localization properties for the discrete alloy-type model.Comment: 20 pages, 0 figure

Effects of Macroeconomic Shocks to the Quality of the Aggregate Loan Portfolio

The paper concerns macro-prudential analysis. It uses an unrestricted VAR model to empirically investigate transmission involving a set of macroeconomic variables describing the development of the Czech economy and the functioning of its credit channel in the past eleven years. Its novelty lies in the fact that it provides the first systematic assessment of the links between loan quality and macroeconomic shocks in the Czech context. The VAR methodology is applied to monthly data transformed into percentage change. The out-of-sample forecast indicates that the most likely outlook for the quality of the banking sector's loan portfolio is that up to the end of 2006 the share of non-performing loans in it will follow a slightly downward trend below double-digit rates. The impulse response is augmented by stress testing exercises that enable us to determine a macroeconomic early warning signal of any worsening in the quality of banks' loans. The paper suggests that the Czech banking sector has attained a considerable ability to withstand a credit risk shock and that the banking sector's stability is compatible both with price stability and with economic growth. Despite being devoted to empirical investigation, the paper pays great attention to methodological issues. At the same time it tries to present both the VAR model and its results transparently and to openly discuss their weak points, which to a large degree can be attributed to data constraints or to the evolutionary nature of an economy in transition.Czech Republic, Macro-prudential analysis, Non-performing loans, VAR model.

Equidistribution estimates for eigenfunctions and eigenvalue bounds for random operators

We discuss properties of $L^2$-eigenfunctions of Schr\"odinger operators and elliptic partial differential operators. The focus is set on unique continuation principles and equidistribution properties. We review recent results and announce new ones.Comment: Keywords: scale-free unique continuation property, equidistribution property, observability estimate, uncertainty relation, Carleman estimate, Schr\"odinger operator, elliptic differential equatio

Raking the Cocktail Party

We present the concept of an acoustic rake receiver---a microphone beamformer that uses echoes to improve the noise and interference suppression. The rake idea is well-known in wireless communications; it involves constructively combining different multipath components that arrive at the receiver antennas. Unlike spread-spectrum signals used in wireless communications, speech signals are not orthogonal to their shifts. Therefore, we focus on the spatial structure, rather than temporal. Instead of explicitly estimating the channel, we create correspondences between early echoes in time and image sources in space. These multiple sources of the desired and the interfering signal offer additional spatial diversity that we can exploit in the beamformer design. We present several "intuitive" and optimal formulations of acoustic rake receivers, and show theoretically and numerically that the rake formulation of the maximum signal-to-interference-and-noise beamformer offers significant performance boosts in terms of noise and interference suppression. Beyond signal-to-noise ratio, we observe gains in terms of the \emph{perceptual evaluation of speech quality} (PESQ) metric for the speech quality. We accompany the paper by the complete simulation and processing chain written in Python. The code and the sound samples are available online at \url{http://lcav.github.io/AcousticRakeReceiver/}.Comment: 12 pages, 11 figures, Accepted for publication in IEEE Journal on Selected Topics in Signal Processing (Special Issue on Spatial Audio