Consider the operator H = −∆ + pω − V, where pω is the potential of the Andersson type and V ≥ 0 is a function that decays slowly at the infinity. We study the rate of accumulation of eigenvalues of H to the bottom of the essential spectrum. 1. STATEMENT OF THE MAIN RESULT The question we study is rooted in two different areas of mathematics: the spectral theory of differential operators and th...

We prove a general implicit function theorem for multifunctions with a metric estimate on the implicit multifunction and a characterization of its coderivative. Traditional open covering theorems, stability results, and sufficient conditions for a multifunction to be metrically regular or pseudo-Lipschitzian can be deduced from this implicit function theorem. We prove this implicit multifunctio...

We obtain large deviation estimates for a large class of nonuniformly hyperbolic systems: namely those modelled by Young towers with summable decay of correlations. In the case of exponential decay of correlations, we obtain exponential large deviation estimates given by a rate function. In the case of polynomial decay of correlations, we obtain polynomial large deviation estimates, and exhibit...

The aim of this note is twofold. First, we prove an analogue of the wellknownRobinson–Ursescu Theorem on the relative openness with a linear rate (restrictive metric regularity) of a multivalued mapping. Second, we prove a generalization of Graves Open Mapping Theorem for a class of mappings which can be approximated at a reference point by a bunch of linear mappings. The approximated non-linea...

When I was asked to comment on the benefits and costs of bridging cognitive psychology, my own field of research, with social psychology, I thought this was easy—the more so as I have a social-psychological background myself (Schultz-Gambard, Feierabend, & Hommel, 1988; Schultz-Gambard & Hommel, 1987). However, addressing this issue requires one to define what a cognitive and a social psycholog...

We review some recent results on eigenvalues of fractional Laplacians and fractional Schrödinger operators. We discuss, in particular, Lieb–Thirring inequalities and their generalizations, as well as semi-classical asymptotics.

The anomaly detection and localization problem can be treated as a multiple hypotheses testing (MHT) problem in the Bayesian framework. The Bayesian test with the 0-1 loss function is a standard solution for this problem, but the alternative hypotheses have quite different importance in practice. The 0-1 loss function does not reflect this fact while the quadratic loss function is more appropri...

We consider the problem of recommending relevant labels (items) for a given data point (user). In particular, we are interested in the practically important setting where the evaluation is with respect to non-decomposable (over labels) performance metrics like the F1 measure, and the training data has missing labels. To this end, we propose a generic framework that given a performance metric Ψ,...

Binary tomography deals with the problem of reconstructing a binary image from its projections. In particular, there is a focus on highly underdetermined reconstruction problems for which many solutions may exist. In such cases, it is important to have a quality measure for the reconstruction with respect to the unknown original image. In this article, we derive a series of upper bounds that ca...