Extremal Lipschitz functions in the deviation inequalities from the mean

We obtain an optimal deviation from the mean upper bound \begin{equation} D(x)\=\sup_{f\in \F}\mu\{f-\E_{\mu} f\geq x\},\qquad\ \text{for}\ x\in\R\label{abstr} \end{equation} where \F is the class of the integrable, Lipschitz functions on probability metric (product) spaces. As corollaries we get exact solutions of \eqref{abstr} for Euclidean unit sphere $S^{n-1}$ with a geodesic distance and a normalized Haar measure, for $\R^n$ equipped with a Gaussian measure and for the multidimensional cube, rectangle, torus or Diamond graph equipped with uniform measure and Hamming distance. We also prove that in general probability metric spaces the $\sup$ in \eqref{abstr} is achieved on a family of distance functions.Comment: 7 page

A Digital Signature Scheme for Long-Term Security

In this paper we propose a signature scheme based on two intractable problems, namely the integer factorization problem and the discrete logarithm problem for elliptic curves. It is suitable for applications requiring long-term security and provides a more efficient solution than the existing ones

Analysis of Schr\"odinger operators with inverse square potentials I: regularity results in 3D

Let $V$ be a potential on \RR^3 that is smooth everywhere except at a discrete set \maS of points, where it has singularities of the form $Z/\rho^2$, with $\rho(x) = |x - p|$ for $x$ close to $p$ and $Z$ continuous on \RR^3 with $Z(p) > -1/4$ for p \in \maS. Also assume that $\rho$ and $Z$ are smooth outside \maS and $Z$ is smooth in polar coordinates around each singular point. We either assume that $V$ is periodic or that the set \maS is finite and $V$ extends to a smooth function on the radial compactification of \RR^3 that is bounded outside a compact set containing \maS. In the periodic case, we let $\Lambda$ be the periodicity lattice and define \TT := \RR^3/ \Lambda. We obtain regularity results in weighted Sobolev space for the eigenfunctions of the Schr\"odinger-type operator $H = -\Delta + V$ acting on L^2(\TT), as well as for the induced \vt k--Hamiltonians \Hk obtained by restricting the action of $H$ to Bloch waves. Under some additional assumptions, we extend these regularity and solvability results to the non-periodic case. We sketch some applications to approximation of eigenfunctions and eigenvalues that will be studied in more detail in a second paper.Comment: 15 pages, to appear in Bull. Math. Soc. Sci. Math. Roumanie, vol. 55 (103), no. 2/201

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008